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href="https://www.academia.edu/94214664/Sampling_large_hyperplane_truncated_multivariate_normal_distributions"><img alt="Research paper thumbnail of Sampling large hyperplane-truncated multivariate normal distributions" class="work-thumbnail" src="https://attachments.academia-assets.com/96734476/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214664/Sampling_large_hyperplane_truncated_multivariate_normal_distributions">Sampling large hyperplane-truncated multivariate normal distributions</a></div><div class="wp-workCard_item"><span>Le Centre pour la Communication Scientifique Directe - HAL - ENSM-SE</span><span>, Aug 2, 2022</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a 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In this paper, simulating large multivariate normal distributions truncated on the intersection of a set of hyperplanes is investigated. The proposed methodology focuses on Gaussian vectors extracted from a Gaussian process (GP) in one dimension. It is based on combining both Karhunen-Loève expansions (KLE) and Matheron's update rules (MUR). The KLE requires the computation of the decomposition of the covariance matrix of the random variables. This step becomes expensive when the random vector is too large. To deal with this issue, the input domain is split in smallest subdomains where the eigendecomposition can be computed. By this strategy, the computational complexity is drastically reduced. The mean-square truncation and block errors have been calculated. 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Le Riche et al. GPs in mechanics March 2022 3 / 45 R. Le Riche et al. GPs in mechanics March 2022 4 / 45 Context (cont.): small data and explanability Kriging is a machine learning technique that applies to small data, is expressive and somewhat explainable as we will argue here.","publication_date":{"day":10,"month":3,"year":2022,"errors":{}},"grobid_abstract_attachment_id":96734540},"translated_abstract":null,"internal_url":"https://www.academia.edu/94214663/A_discussion_of_the_current_and_potential_uses_of_Gaussian_Processes_in_mechanics","translated_internal_url":"","created_at":"2023-01-02T23:53:01.923-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734540,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734540/thumbnails/1.jpg","file_name":"talk_SF2M_LeRiche_march2022.pdf","download_url":"https://www.academia.edu/attachments/96734540/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_discussion_of_the_current_and_potentia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734540/talk_SF2M_LeRiche_march2022-libre.pdf?1672739696=\u0026response-content-disposition=attachment%3B+filename%3DA_discussion_of_the_current_and_potentia.pdf\u0026Expires=1732751545\u0026Signature=hKgM9i-GDlVXTqzw7bd-rI-0MIDw2qURSCPV4TuXLy2O5zdRVfpaXW4YHFoAWTSnixsT4hj5oVL2G6Cf6h2IROC127hiMFVuvrFCpYUlE7-HZDVDGVU5ZYcLkJ0sc4RTFQ61YDOeG6rDsJ4i45Q~eyzCT5dHO9westdA7klCdUDJtiQVaphsho~bCuIWHEgNr6PrrrSCa5vd36QhEEYQHe8AjK6ygzW3KJIGgP8-tQsx5kNTwuhKsu7W97Ej3eLlgJtnljyO8B0FYKduQqSp~KDHj27CfhnXLtpqfCBlOhtQIV52yYlXCFu9Xym8wc~U1vW~EEQkqmiG813zxv8d9A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_discussion_of_the_current_and_potential_uses_of_Gaussian_Processes_in_mechanics","translated_slug":"","page_count":45,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734540,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734540/thumbnails/1.jpg","file_name":"talk_SF2M_LeRiche_march2022.pdf","download_url":"https://www.academia.edu/attachments/96734540/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_discussion_of_the_current_and_potentia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734540/talk_SF2M_LeRiche_march2022-libre.pdf?1672739696=\u0026response-content-disposition=attachment%3B+filename%3DA_discussion_of_the_current_and_potentia.pdf\u0026Expires=1732751545\u0026Signature=hKgM9i-GDlVXTqzw7bd-rI-0MIDw2qURSCPV4TuXLy2O5zdRVfpaXW4YHFoAWTSnixsT4hj5oVL2G6Cf6h2IROC127hiMFVuvrFCpYUlE7-HZDVDGVU5ZYcLkJ0sc4RTFQ61YDOeG6rDsJ4i45Q~eyzCT5dHO9westdA7klCdUDJtiQVaphsho~bCuIWHEgNr6PrrrSCa5vd36QhEEYQHe8AjK6ygzW3KJIGgP8-tQsx5kNTwuhKsu7W97Ej3eLlgJtnljyO8B0FYKduQqSp~KDHj27CfhnXLtpqfCBlOhtQIV52yYlXCFu9Xym8wc~U1vW~EEQkqmiG813zxv8d9A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":342314,"name":"Gaussian","url":"https://www.academia.edu/Documents/in/Gaussian"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"}],"urls":[{"id":27685974,"url":"https://hal.archives-ouvertes.fr/hal-03605952"}]}, dispatcherData: dispatcherData }); 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We then propose a methodology to construct interpolators that take into account an innite number of informations. As an application, we illustrate how boundary constraints can be enforced in Gaussian process models.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2c84d4745fce227fbdae2557a47181fb" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734472,"asset_id":94214662,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734472/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214662"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214662"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214662; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214662]").text(description); $(".js-view-count[data-work-id=94214662]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214662; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214662']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214662, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2c84d4745fce227fbdae2557a47181fb" } } $('.js-work-strip[data-work-id=94214662]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214662,"title":"Spectral Decomposition of Integral Operators for Optimal Interpolation in Hilbert Subspaces","translated_title":"","metadata":{"abstract":"The orthogonal projection associated to optimal interpolation in a Hilbert subspace is characterized by the spectral decomposition of problem adapted integral operators. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214661"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214661/Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines"><img alt="Research paper thumbnail of Simulation of Gaussian processes with interpolation and inequality constraints - A correspondence with optimal smoothing splines" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214661/Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines">Simulation of Gaussian processes with interpolation and inequality constraints - A correspondence with optimal smoothing splines</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Complex physical phenomena are observed in many fields (sciences and engineering) and are often s...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the con...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214661"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214661"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214661; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214661]").text(description); $(".js-view-count[data-work-id=94214661]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214661; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214661']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214661, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=94214661]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214661,"title":"Simulation of Gaussian processes with interpolation and inequality constraints - A correspondence with optimal smoothing splines","translated_title":"","metadata":{"abstract":"Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the con...","publication_date":{"day":null,"month":null,"year":2014,"errors":{}}},"translated_abstract":"Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the con...","internal_url":"https://www.academia.edu/94214661/Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines","translated_internal_url":"","created_at":"2023-01-02T23:53:01.713-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":225304,"name":"Smoothing","url":"https://www.academia.edu/Documents/in/Smoothing"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214660"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214660/An_analytic_comparison_of_regularization_methods_for_Gaussian_Processes"><img alt="Research paper thumbnail of An analytic comparison of regularization methods for Gaussian Processes" class="work-thumbnail" src="https://attachments.academia-assets.com/96734470/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214660/An_analytic_comparison_of_regularization_methods_for_Gaussian_Processes">An analytic comparison of regularization methods for Gaussian Processes</a></div><div class="wp-workCard_item"><span>arXiv: Optimization and Control</span><span>, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experime...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the inversion of a covariance matrix that, in practice, is often ill-conditioned. Regularization methodologies are then employed with consequences on the GPs that need to be better understood.The two principal methods to deal with ill-conditioned covariance matrices are i) pseudoinverse and ii) adding a positive constant to the diagonal (the so-called nugget regularization).The first part of this paper provides an algebraic comparison of PI and nugget regularizations. Redundant points, responsible for covariance matrix singularity, are defined. It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the vari...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b999d93bc72df98c2a49a4da8ed8a594" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734470,"asset_id":94214660,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734470/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214660"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214660"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214660; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214660]").text(description); $(".js-view-count[data-work-id=94214660]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214660; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214660']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214660, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "b999d93bc72df98c2a49a4da8ed8a594" } } $('.js-work-strip[data-work-id=94214660]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214660,"title":"An analytic comparison of regularization methods for Gaussian Processes","translated_title":"","metadata":{"abstract":"Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214659"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214659/Optimal_Interpolation_in_RKHS_Spectral_Decomposition_of_Integral_Operators_and_Application"><img alt="Research paper thumbnail of Optimal Interpolation in RKHS, Spectral Decomposition of Integral Operators and Application" class="work-thumbnail" src="https://attachments.academia-assets.com/96734471/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214659/Optimal_Interpolation_in_RKHS_Spectral_Decomposition_of_Integral_Operators_and_Application">Optimal Interpolation in RKHS, Spectral Decomposition of Integral Operators and Application</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The orthogonal projection associated to optimal interpolation in a reproducing kernel Hilbert spa...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The orthogonal projection associated to optimal interpolation in a reproducing kernel Hilbert space is characterized by the spectral decomposition of an integral operator. This operator is built from the reproducing kernel and a measure on the interpolation set. As an application, we illustrate how boundary value constraints can be enforced in Gaussian process models.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2858495f41c798b4d54fc6c2e3d387c9" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734471,"asset_id":94214659,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734471/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214659"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214659"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214659; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214659]").text(description); $(".js-view-count[data-work-id=94214659]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214659; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214659']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214659, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2858495f41c798b4d54fc6c2e3d387c9" } } $('.js-work-strip[data-work-id=94214659]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214659,"title":"Optimal Interpolation in RKHS, Spectral Decomposition of Integral Operators and Application","translated_title":"","metadata":{"abstract":"The orthogonal projection associated to optimal interpolation in a reproducing kernel Hilbert space is characterized by the spectral decomposition of an integral operator. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214658"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214658/Constrained_Optimal_Smoothing_and_Bayesian_Estimation"><img alt="Research paper thumbnail of Constrained Optimal Smoothing and Bayesian Estimation" class="work-thumbnail" src="https://attachments.academia-assets.com/96734469/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214658/Constrained_Optimal_Smoothing_and_Bayesian_Estimation">Constrained Optimal Smoothing and Bayesian Estimation</a></div><div class="wp-workCard_item"><span>ArXiv</span><span>, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [7] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c0261a88dff5e764387a4e15a856a633" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734469,"asset_id":94214658,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734469/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214658"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214658"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214658; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214658]").text(description); $(".js-view-count[data-work-id=94214658]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214658; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214658']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214658, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c0261a88dff5e764387a4e15a856a633" } } $('.js-work-strip[data-work-id=94214658]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214658,"title":"Constrained Optimal Smoothing and Bayesian Estimation","translated_title":"","metadata":{"abstract":"In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214656"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214656/Karhunen_Lo%C3%A8ve_decomposition_of_Gaussian_measures_on_Banach_spaces"><img alt="Research paper thumbnail of Karhunen–Loève decomposition of Gaussian measures on Banach spaces" class="work-thumbnail" src="https://attachments.academia-assets.com/96734466/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214656/Karhunen_Lo%C3%A8ve_decomposition_of_Gaussian_measures_on_Banach_spaces">Karhunen–Loève decomposition of Gaussian measures on Banach spaces</a></div><div class="wp-workCard_item"><span>Probability and Mathematical Statistics</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The study of Gaussian measures on Banach spaces is of active interest both in pure and applied ma...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="06232e2cf85787db2680740074564ba4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734466,"asset_id":94214656,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734466/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214656"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214656"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214656; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214656]").text(description); $(".js-view-count[data-work-id=94214656]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214656; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214656']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214656, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "06232e2cf85787db2680740074564ba4" } } $('.js-work-strip[data-work-id=94214656]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214656,"title":"Karhunen–Loève decomposition of Gaussian measures on Banach spaces","translated_title":"","metadata":{"abstract":"The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. 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In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214655"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214655/An_analysis_of_covariance_parameters_in_Gaussian_process_based_optimization"><img alt="Research paper thumbnail of An analysis of covariance parameters in Gaussian process-based optimization" class="work-thumbnail" src="https://attachments.academia-assets.com/96734546/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214655/An_analysis_of_covariance_parameters_in_Gaussian_process_based_optimization">An analysis of covariance parameters in Gaussian process-based optimization</a></div><div class="wp-workCard_item"><span>Croatian Operational Research Review</span><span>, 2018</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="21e81b44bbc590b7284b95d9c7eae56c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734546,"asset_id":94214655,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734546/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214655"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214655"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214655; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214655]").text(description); $(".js-view-count[data-work-id=94214655]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214655; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214655']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214655, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "21e81b44bbc590b7284b95d9c7eae56c" } } $('.js-work-strip[data-work-id=94214655]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214655,"title":"An analysis of covariance parameters in Gaussian process-based optimization","translated_title":"","metadata":{"publisher":"Croatian Operational Research Society","grobid_abstract":"The need for globally optimizing expensive-to-evaluate functions frequently occurs in many real-world applications. Among the methods developed for solving such problems, the Efficient Global Optimization (EGO) is regarded as one of the state-of-the-art unconstrained continuous optimization algorithms. The surrogate model used in EGO is a Gaussian process (GP) conditional on data points. The most important control on the efficiency of the EGO algorithm is the GP covariance function (or kernel), which is taken as a parameterized function. In this paper, we theoretically and empirically analyze the effect of the covariance parameters, the so-called \"characteristic length scale\" and \"nugget\", on EGO performance. More precisely, we analyze the EGO algorithm with fixed covariance parameters and compare them to the standard setting where they are statistically estimated. The limit behavior of EGO with very small or very large characteristic length scales is identified. Experiments show that a \"small\" nugget should be preferred to its maximum likelihood estimate. Overall, this study contributes to a better theoretical and practical understanding of a key optimization algorithm.","publication_date":{"day":null,"month":null,"year":2018,"errors":{}},"publication_name":"Croatian Operational Research Review","grobid_abstract_attachment_id":96734546},"translated_abstract":null,"internal_url":"https://www.academia.edu/94214655/An_analysis_of_covariance_parameters_in_Gaussian_process_based_optimization","translated_internal_url":"","created_at":"2023-01-02T23:53:00.926-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734546,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734546/thumbnails/1.jpg","file_name":"703e2be797615b3f9ee6fe2c26578b4b155c.pdf","download_url":"https://www.academia.edu/attachments/96734546/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_analysis_of_covariance_parameters_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734546/703e2be797615b3f9ee6fe2c26578b4b155c-libre.pdf?1672739686=\u0026response-content-disposition=attachment%3B+filename%3DAn_analysis_of_covariance_parameters_in.pdf\u0026Expires=1732751545\u0026Signature=Qgd~bDIw728A~25QfJHVi~OgzhUpuwl0GfuTo~X5elVb8Q-gamHs3XBgdeMuXMDwMrQhMQ1XPsVnjS9tlxR~TxWltc~1rWxIeLheVcbtKDJjddVMDlgjRFZgjg6C3kiTiUlUQJmB6g0TWahnUZsEffa8TnRRYy5kYoqCjVH9BXsI3Kxxd-ayywq8NRHXnu8U5Vuk7vrgfmwWAz98UlmZbMBEQriEgphaEVjQKKegX-J6i12QTRHd-1y13DaUgvOhm1BtuvbKeNyS7lKmZw63cK5N2a~xF0zUe4~2ajfdINfFHS7Vz9Zy5utxUJay8yAohvVf1jkN7ELmk5bzUNqKzg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"An_analysis_of_covariance_parameters_in_Gaussian_process_based_optimization","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734546,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734546/thumbnails/1.jpg","file_name":"703e2be797615b3f9ee6fe2c26578b4b155c.pdf","download_url":"https://www.academia.edu/attachments/96734546/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_analysis_of_covariance_parameters_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734546/703e2be797615b3f9ee6fe2c26578b4b155c-libre.pdf?1672739686=\u0026response-content-disposition=attachment%3B+filename%3DAn_analysis_of_covariance_parameters_in.pdf\u0026Expires=1732751545\u0026Signature=Qgd~bDIw728A~25QfJHVi~OgzhUpuwl0GfuTo~X5elVb8Q-gamHs3XBgdeMuXMDwMrQhMQ1XPsVnjS9tlxR~TxWltc~1rWxIeLheVcbtKDJjddVMDlgjRFZgjg6C3kiTiUlUQJmB6g0TWahnUZsEffa8TnRRYy5kYoqCjVH9BXsI3Kxxd-ayywq8NRHXnu8U5Vuk7vrgfmwWAz98UlmZbMBEQriEgphaEVjQKKegX-J6i12QTRHd-1y13DaUgvOhm1BtuvbKeNyS7lKmZw63cK5N2a~xF0zUe4~2ajfdINfFHS7Vz9Zy5utxUJay8yAohvVf1jkN7ELmk5bzUNqKzg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":107131,"name":"Global Optimization","url":"https://www.academia.edu/Documents/in/Global_Optimization"},{"id":272592,"name":"Mathematical Optimization","url":"https://www.academia.edu/Documents/in/Mathematical_Optimization"},{"id":282522,"name":"Ego","url":"https://www.academia.edu/Documents/in/Ego"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"},{"id":1473341,"name":"Covariance","url":"https://www.academia.edu/Documents/in/Covariance"},{"id":3017674,"name":"Covariance function","url":"https://www.academia.edu/Documents/in/Covariance_function"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214654"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214654/A_new_method_for_interpolating_in_a_convex_subset_of_a_Hilbert_space"><img alt="Research paper thumbnail of A new method for interpolating in a convex subset of a Hilbert space" class="work-thumbnail" src="https://attachments.academia-assets.com/96734541/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214654/A_new_method_for_interpolating_in_a_convex_subset_of_a_Hilbert_space">A new method for interpolating in a convex subset of a Hilbert space</a></div><div class="wp-workCard_item"><span>Computational Optimization and Applications</span><span>, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="efdcd504f5c81671c666e788a99a0547" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734541,"asset_id":94214654,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734541/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214654"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214654"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214654; 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We propose a new approximation method based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. An algorithm is derived and numerical examples with boundedness and monotonicity constraints in one and two dimensions are given. Keywords Optimization • RKHS • Interpolation • Inequality constraints 1 Introduction Let X be a nonempty set of R d (d ≥ 1) and E = C 0 (X) the linear (topological) space of real valued continuous functions on X. Given n distinct points x (1) ,. .. , x (n) ∈ X and y 1 ,. .. , y n ∈ R, we define the set I of interpolating functions by I := f ∈ E, f x (i) = y i , i = 1,. .. , n .","publication_date":{"day":null,"month":null,"year":2017,"errors":{}},"publication_name":"Computational Optimization and Applications","grobid_abstract_attachment_id":96734541},"translated_abstract":null,"internal_url":"https://www.academia.edu/94214654/A_new_method_for_interpolating_in_a_convex_subset_of_a_Hilbert_space","translated_internal_url":"","created_at":"2023-01-02T23:53:00.752-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734541/thumbnails/1.jpg","file_name":"Maatouk_H.pdf","download_url":"https://www.academia.edu/attachments/96734541/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_new_method_for_interpolating_in_a_conv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734541/Maatouk_H-libre.pdf?1672739693=\u0026response-content-disposition=attachment%3B+filename%3DA_new_method_for_interpolating_in_a_conv.pdf\u0026Expires=1732751545\u0026Signature=VRLz~TD88oVueia-vnrC8pEl19ahVi7FeUV~16wSQ25SgVKfy7PI89-PU-nncLTxHyRLTzPhZ1kGmxwN75fIzbgnM6M~3JldEYNXSo~yYdJolAQsj1B-ZWRk5ken5kntf5O1GdNPGMjnGNb-HLbSptqEzSSrI9tkgP5wmDHvMtEIn-68nAUzyVegEv4~D1xN3hBnsUpNFn2rMwuITZjx9RnApKw0pdIV1lL-1EAbQvJeArSVJWR7G8epH4NJ1WSmXcpQ4UflHolixV-w7oTTTsAmBT0zYXrkQDNY~xAp4831O2gwFKG8GiY-pEHl60cGnCKTiA7kJnTrrrqWI5qrJQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_new_method_for_interpolating_in_a_convex_subset_of_a_Hilbert_space","translated_slug":"","page_count":34,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734541/thumbnails/1.jpg","file_name":"Maatouk_H.pdf","download_url":"https://www.academia.edu/attachments/96734541/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_new_method_for_interpolating_in_a_conv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734541/Maatouk_H-libre.pdf?1672739693=\u0026response-content-disposition=attachment%3B+filename%3DA_new_method_for_interpolating_in_a_conv.pdf\u0026Expires=1732751545\u0026Signature=VRLz~TD88oVueia-vnrC8pEl19ahVi7FeUV~16wSQ25SgVKfy7PI89-PU-nncLTxHyRLTzPhZ1kGmxwN75fIzbgnM6M~3JldEYNXSo~yYdJolAQsj1B-ZWRk5ken5kntf5O1GdNPGMjnGNb-HLbSptqEzSSrI9tkgP5wmDHvMtEIn-68nAUzyVegEv4~D1xN3hBnsUpNFn2rMwuITZjx9RnApKw0pdIV1lL-1EAbQvJeArSVJWR7G8epH4NJ1WSmXcpQ4UflHolixV-w7oTTTsAmBT0zYXrkQDNY~xAp4831O2gwFKG8GiY-pEHl60cGnCKTiA7kJnTrrrqWI5qrJQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":43981,"name":"Optimization","url":"https://www.academia.edu/Documents/in/Optimization"},{"id":254570,"name":"Interpolation","url":"https://www.academia.edu/Documents/in/Interpolation"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"},{"id":988387,"name":"Hilbert Space","url":"https://www.academia.edu/Documents/in/Hilbert_Space"},{"id":1970623,"name":"RKHS","url":"https://www.academia.edu/Documents/in/RKHS"},{"id":3933497,"name":"Inequality Constraints","url":"https://www.academia.edu/Documents/in/Inequality_Constraints"}],"urls":[{"id":27685967,"url":"http://link.springer.com/article/10.1007/s10589-017-9906-9/fulltext.html"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214653"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214653/Gaussian_Process_Emulators_for_Computer_Experiments_with_Inequality_Constraints"><img alt="Research paper thumbnail of Gaussian Process Emulators for Computer Experiments with Inequality Constraints" class="work-thumbnail" src="https://attachments.academia-assets.com/96734543/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214653/Gaussian_Process_Emulators_for_Computer_Experiments_with_Inequality_Constraints">Gaussian Process Emulators for Computer Experiments with Inequality Constraints</a></div><div class="wp-workCard_item"><span>Mathematical Geosciences</span><span>, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="902f9a5843e1dd444b2e7234b4ef2a20" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734543,"asset_id":94214653,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734543/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214653"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214653"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214653; 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These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. Our aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, we propose a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain. The inequality mean and mode (i.e. mean and maximum a posteriori) of the conditional Gaussian process are calculated and prediction intervals are quantified. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214652"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214652/Short_Term_Load_Forecasting_in_the_Industry_for_Establishing_Consumption_Baselines_A_French_Case"><img alt="Research paper thumbnail of Short Term Load Forecasting in the Industry for Establishing Consumption Baselines: A French Case" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214652/Short_Term_Load_Forecasting_in_the_Industry_for_Establishing_Consumption_Baselines_A_French_Case">Short Term Load Forecasting in the Industry for Establishing Consumption Baselines: A French Case</a></div><div class="wp-workCard_item"><span>Lecture Notes in Statistics</span><span>, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The estimation of baseline electricity consumptions for energy efficiency and load management mea...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. Perspectives for exploring different forecasting methods for these applications are considered for future works, as well as their application to different load curves from diverse industrial sectors and equipments.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214652"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214652"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214652; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214652]").text(description); $(".js-view-count[data-work-id=94214652]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214652; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214652']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214652, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=94214652]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214652,"title":"Short Term Load Forecasting in the Industry for Establishing Consumption Baselines: A French Case","translated_title":"","metadata":{"abstract":"The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. Perspectives for exploring different forecasting methods for these applications are considered for future works, as well as their application to different load curves from diverse industrial sectors and equipments.","publication_date":{"day":null,"month":null,"year":2015,"errors":{}},"publication_name":"Lecture Notes in Statistics"},"translated_abstract":"The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. Perspectives for exploring different forecasting methods for these applications are considered for future works, as well as their application to different load curves from diverse industrial sectors and equipments.","internal_url":"https://www.academia.edu/94214652/Short_Term_Load_Forecasting_in_the_Industry_for_Establishing_Consumption_Baselines_A_French_Case","translated_internal_url":"","created_at":"2023-01-02T23:53:00.510-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Short_Term_Load_Forecasting_in_the_Industry_for_Establishing_Consumption_Baselines_A_French_Case","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[],"research_interests":[{"id":48,"name":"Engineering","url":"https://www.academia.edu/Documents/in/Engineering"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":11397,"name":"Energy Consumption","url":"https://www.academia.edu/Documents/in/Energy_Consumption"},{"id":43986,"name":"Electricity","url":"https://www.academia.edu/Documents/in/Electricity"},{"id":131237,"name":"Cluster Analysis","url":"https://www.academia.edu/Documents/in/Cluster_Analysis"}],"urls":[]}, dispatcherData: dispatcherData }); 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Les lignes qui suivent montrent l'évaluation des commandes utilisées en R (voir la remarque sur R page 2). 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As a step by step application, we revisit the problem of prediction for the Brownian sheet knowing its values on a separation line.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="1039ed00d8c56958919a8003dededf18" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734406,"asset_id":94214577,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734406/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214577"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214577"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214577; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214577]").text(description); $(".js-view-count[data-work-id=94214577]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214577; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214577']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214577, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "1039ed00d8c56958919a8003dededf18" } } $('.js-work-strip[data-work-id=94214577]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214577,"title":"Extended Formula for Kriging Interpolation","translated_title":"","metadata":{"abstract":"In many elds, Kriging interpolation techniques are used within a nite discrete set of known values of a real function (no evaluation or measurement error). 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="6180136" id="papers"><div class="js-work-strip profile--work_container" data-work-id="94214664"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214664/Sampling_large_hyperplane_truncated_multivariate_normal_distributions"><img alt="Research paper thumbnail of Sampling large hyperplane-truncated multivariate normal distributions" class="work-thumbnail" src="https://attachments.academia-assets.com/96734476/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214664/Sampling_large_hyperplane_truncated_multivariate_normal_distributions">Sampling large hyperplane-truncated multivariate normal distributions</a></div><div class="wp-workCard_item"><span>Le Centre pour la Communication Scientifique Directe - HAL - ENSM-SE</span><span>, Aug 2, 2022</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="05366825f8abdd4713ec634883cd7239" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734476,"asset_id":94214664,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734476/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214664"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214664"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214664; 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In this paper, simulating large multivariate normal distributions truncated on the intersection of a set of hyperplanes is investigated. The proposed methodology focuses on Gaussian vectors extracted from a Gaussian process (GP) in one dimension. It is based on combining both Karhunen-Loève expansions (KLE) and Matheron's update rules (MUR). The KLE requires the computation of the decomposition of the covariance matrix of the random variables. This step becomes expensive when the random vector is too large. To deal with this issue, the input domain is split in smallest subdomains where the eigendecomposition can be computed. By this strategy, the computational complexity is drastically reduced. The mean-square truncation and block errors have been calculated. 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Le Riche et al. GPs in mechanics March 2022 3 / 45 R. Le Riche et al. GPs in mechanics March 2022 4 / 45 Context (cont.): small data and explanability Kriging is a machine learning technique that applies to small data, is expressive and somewhat explainable as we will argue here.","publication_date":{"day":10,"month":3,"year":2022,"errors":{}},"grobid_abstract_attachment_id":96734540},"translated_abstract":null,"internal_url":"https://www.academia.edu/94214663/A_discussion_of_the_current_and_potential_uses_of_Gaussian_Processes_in_mechanics","translated_internal_url":"","created_at":"2023-01-02T23:53:01.923-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734540,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734540/thumbnails/1.jpg","file_name":"talk_SF2M_LeRiche_march2022.pdf","download_url":"https://www.academia.edu/attachments/96734540/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_discussion_of_the_current_and_potentia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734540/talk_SF2M_LeRiche_march2022-libre.pdf?1672739696=\u0026response-content-disposition=attachment%3B+filename%3DA_discussion_of_the_current_and_potentia.pdf\u0026Expires=1732751545\u0026Signature=hKgM9i-GDlVXTqzw7bd-rI-0MIDw2qURSCPV4TuXLy2O5zdRVfpaXW4YHFoAWTSnixsT4hj5oVL2G6Cf6h2IROC127hiMFVuvrFCpYUlE7-HZDVDGVU5ZYcLkJ0sc4RTFQ61YDOeG6rDsJ4i45Q~eyzCT5dHO9westdA7klCdUDJtiQVaphsho~bCuIWHEgNr6PrrrSCa5vd36QhEEYQHe8AjK6ygzW3KJIGgP8-tQsx5kNTwuhKsu7W97Ej3eLlgJtnljyO8B0FYKduQqSp~KDHj27CfhnXLtpqfCBlOhtQIV52yYlXCFu9Xym8wc~U1vW~EEQkqmiG813zxv8d9A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_discussion_of_the_current_and_potential_uses_of_Gaussian_Processes_in_mechanics","translated_slug":"","page_count":45,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734540,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734540/thumbnails/1.jpg","file_name":"talk_SF2M_LeRiche_march2022.pdf","download_url":"https://www.academia.edu/attachments/96734540/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_discussion_of_the_current_and_potentia.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734540/talk_SF2M_LeRiche_march2022-libre.pdf?1672739696=\u0026response-content-disposition=attachment%3B+filename%3DA_discussion_of_the_current_and_potentia.pdf\u0026Expires=1732751545\u0026Signature=hKgM9i-GDlVXTqzw7bd-rI-0MIDw2qURSCPV4TuXLy2O5zdRVfpaXW4YHFoAWTSnixsT4hj5oVL2G6Cf6h2IROC127hiMFVuvrFCpYUlE7-HZDVDGVU5ZYcLkJ0sc4RTFQ61YDOeG6rDsJ4i45Q~eyzCT5dHO9westdA7klCdUDJtiQVaphsho~bCuIWHEgNr6PrrrSCa5vd36QhEEYQHe8AjK6ygzW3KJIGgP8-tQsx5kNTwuhKsu7W97Ej3eLlgJtnljyO8B0FYKduQqSp~KDHj27CfhnXLtpqfCBlOhtQIV52yYlXCFu9Xym8wc~U1vW~EEQkqmiG813zxv8d9A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":16460,"name":"Statistical Physics","url":"https://www.academia.edu/Documents/in/Statistical_Physics"},{"id":342314,"name":"Gaussian","url":"https://www.academia.edu/Documents/in/Gaussian"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"}],"urls":[{"id":27685974,"url":"https://hal.archives-ouvertes.fr/hal-03605952"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214662"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214662/Spectral_Decomposition_of_Integral_Operators_for_Optimal_Interpolation_in_Hilbert_Subspaces"><img alt="Research paper thumbnail of Spectral Decomposition of Integral Operators for Optimal Interpolation in Hilbert Subspaces" class="work-thumbnail" src="https://attachments.academia-assets.com/96734472/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214662/Spectral_Decomposition_of_Integral_Operators_for_Optimal_Interpolation_in_Hilbert_Subspaces">Spectral Decomposition of Integral Operators for Optimal Interpolation in Hilbert Subspaces</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The orthogonal projection associated to optimal interpolation in a Hilbert subspace is characteri...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The orthogonal projection associated to optimal interpolation in a Hilbert subspace is characterized by the spectral decomposition of problem adapted integral operators. We then propose a methodology to construct interpolators that take into account an innite number of informations. As an application, we illustrate how boundary constraints can be enforced in Gaussian process models.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2c84d4745fce227fbdae2557a47181fb" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734472,"asset_id":94214662,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734472/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214662"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214662"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214662; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214662]").text(description); $(".js-view-count[data-work-id=94214662]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214662; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214662']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214662, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2c84d4745fce227fbdae2557a47181fb" } } $('.js-work-strip[data-work-id=94214662]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214662,"title":"Spectral Decomposition of Integral Operators for Optimal Interpolation in Hilbert Subspaces","translated_title":"","metadata":{"abstract":"The orthogonal projection associated to optimal interpolation in a Hilbert subspace is characterized by the spectral decomposition of problem adapted integral operators. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214661"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214661/Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines"><img alt="Research paper thumbnail of Simulation of Gaussian processes with interpolation and inequality constraints - A correspondence with optimal smoothing splines" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214661/Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines">Simulation of Gaussian processes with interpolation and inequality constraints - A correspondence with optimal smoothing splines</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Complex physical phenomena are observed in many fields (sciences and engineering) and are often s...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the con...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214661"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214661"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214661; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214661]").text(description); $(".js-view-count[data-work-id=94214661]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214661; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214661']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214661, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=94214661]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214661,"title":"Simulation of Gaussian processes with interpolation and inequality constraints - A correspondence with optimal smoothing splines","translated_title":"","metadata":{"abstract":"Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the con...","publication_date":{"day":null,"month":null,"year":2014,"errors":{}}},"translated_abstract":"Complex physical phenomena are observed in many fields (sciences and engineering) and are often studied by time - consuming computer codes. These codes are analyzed with faster statistical models, often called emulators. The Gaussian process (GP) emulator is one of the most popular choice (Sacks et al., 1989) . In many situations, the physical system (computer model output) may be known to satisfy some inequality constraints with respect to some or all input variables. Incorporating inequality constraints into a GP emulator, the problem becomes more challenging since the resulting conditional process is not a GP. To this end, we suggest to approximate the original GP by a finite dimensional Gaussian process Y N such that all conditional simulations satisfy the inequality constraints in the whole domain. In the second part of the talk, we investigate the convergence of the proposed approach and the relationship with thin plate splines (Duchon, 1976) . We show that the mode of the con...","internal_url":"https://www.academia.edu/94214661/Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines","translated_internal_url":"","created_at":"2023-01-02T23:53:01.713-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Simulation_of_Gaussian_processes_with_interpolation_and_inequality_constraints_A_correspondence_with_optimal_smoothing_splines","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":225304,"name":"Smoothing","url":"https://www.academia.edu/Documents/in/Smoothing"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214660"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214660/An_analytic_comparison_of_regularization_methods_for_Gaussian_Processes"><img alt="Research paper thumbnail of An analytic comparison of regularization methods for Gaussian Processes" class="work-thumbnail" src="https://attachments.academia-assets.com/96734470/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214660/An_analytic_comparison_of_regularization_methods_for_Gaussian_Processes">An analytic comparison of regularization methods for Gaussian Processes</a></div><div class="wp-workCard_item"><span>arXiv: Optimization and Control</span><span>, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experime...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the inversion of a covariance matrix that, in practice, is often ill-conditioned. Regularization methodologies are then employed with consequences on the GPs that need to be better understood.The two principal methods to deal with ill-conditioned covariance matrices are i) pseudoinverse and ii) adding a positive constant to the diagonal (the so-called nugget regularization).The first part of this paper provides an algebraic comparison of PI and nugget regularizations. Redundant points, responsible for covariance matrix singularity, are defined. It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the vari...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b999d93bc72df98c2a49a4da8ed8a594" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734470,"asset_id":94214660,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734470/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214660"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214660"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214660; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214660]").text(description); $(".js-view-count[data-work-id=94214660]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214660; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214660']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214660, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "b999d93bc72df98c2a49a4da8ed8a594" } } $('.js-work-strip[data-work-id=94214660]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214660,"title":"An analytic comparison of regularization methods for Gaussian Processes","translated_title":"","metadata":{"abstract":"Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. 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It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the vari...","publication_date":{"day":null,"month":null,"year":2016,"errors":{}},"publication_name":"arXiv: Optimization and Control"},"translated_abstract":"Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the inversion of a covariance matrix that, in practice, is often ill-conditioned. Regularization methodologies are then employed with consequences on the GPs that need to be better understood.The two principal methods to deal with ill-conditioned covariance matrices are i) pseudoinverse and ii) adding a positive constant to the diagonal (the so-called nugget regularization).The first part of this paper provides an algebraic comparison of PI and nugget regularizations. Redundant points, responsible for covariance matrix singularity, are defined. It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the vari...","internal_url":"https://www.academia.edu/94214660/An_analytic_comparison_of_regularization_methods_for_Gaussian_Processes","translated_internal_url":"","created_at":"2023-01-02T23:53:01.587-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734470,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734470/thumbnails/1.jpg","file_name":"document.pdf","download_url":"https://www.academia.edu/attachments/96734470/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_analytic_comparison_of_regularization.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734470/document-libre.pdf?1672743210=\u0026response-content-disposition=attachment%3B+filename%3DAn_analytic_comparison_of_regularization.pdf\u0026Expires=1732751545\u0026Signature=TaXw1jQH6tMN9kUy2AgBPGygOqV5uiz9xhp~p8tgFo5cqBtGJGo6aLoA2wu6wIDFvmFEJSNfnjfOvWR5vHaswl7RR6oZjfNdF9--CZrC1tsiHHJmjZjjsB5ZJTORVJieGcorzOIUUllyK7gkKecBHXNOpgJ7iy8Io-gTLd3v1vNQbjoYbZEb59teXbpw8rVzwMOrY31hcRgNvCSA2AYGF31mPZ5evjw7f2yI36QVCJoJHs8J~xxn0QdyJAbBSpkYZgv6~mAxblGakyt0O7PpO5dXdSU2RLbZRutjn5BGsmf~khDJA-3WH65mAFQRlUlGlkvK1Hfe0WiHdhYPBPRY3g__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"An_analytic_comparison_of_regularization_methods_for_Gaussian_Processes","translated_slug":"","page_count":31,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734470,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734470/thumbnails/1.jpg","file_name":"document.pdf","download_url":"https://www.academia.edu/attachments/96734470/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_analytic_comparison_of_regularization.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734470/document-libre.pdf?1672743210=\u0026response-content-disposition=attachment%3B+filename%3DAn_analytic_comparison_of_regularization.pdf\u0026Expires=1732751545\u0026Signature=TaXw1jQH6tMN9kUy2AgBPGygOqV5uiz9xhp~p8tgFo5cqBtGJGo6aLoA2wu6wIDFvmFEJSNfnjfOvWR5vHaswl7RR6oZjfNdF9--CZrC1tsiHHJmjZjjsB5ZJTORVJieGcorzOIUUllyK7gkKecBHXNOpgJ7iy8Io-gTLd3v1vNQbjoYbZEb59teXbpw8rVzwMOrY31hcRgNvCSA2AYGF31mPZ5evjw7f2yI36QVCJoJHs8J~xxn0QdyJAbBSpkYZgv6~mAxblGakyt0O7PpO5dXdSU2RLbZRutjn5BGsmf~khDJA-3WH65mAFQRlUlGlkvK1Hfe0WiHdhYPBPRY3g__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm"},{"id":85880,"name":"Singular value decomposition","url":"https://www.academia.edu/Documents/in/Singular_value_decomposition"},{"id":146286,"name":"Kriging","url":"https://www.academia.edu/Documents/in/Kriging"},{"id":568878,"name":"Covariance Matrix","url":"https://www.academia.edu/Documents/in/Covariance_Matrix"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"},{"id":711810,"name":"Regularization","url":"https://www.academia.edu/Documents/in/Regularization"},{"id":958135,"name":"Gaussian Process Regression","url":"https://www.academia.edu/Documents/in/Gaussian_Process_Regression"},{"id":1473341,"name":"Covariance","url":"https://www.academia.edu/Documents/in/Covariance"},{"id":2961164,"name":"nugget","url":"https://www.academia.edu/Documents/in/nugget"}],"urls":[{"id":27685972,"url":"https://hal.archives-ouvertes.fr/hal-01264192v2/document"}]}, dispatcherData: dispatcherData }); 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This operator is built from the reproducing kernel and a measure on the interpolation set. As an application, we illustrate how boundary value constraints can be enforced in Gaussian process models.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2858495f41c798b4d54fc6c2e3d387c9" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734471,"asset_id":94214659,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734471/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214659"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214659"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214659; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214659]").text(description); $(".js-view-count[data-work-id=94214659]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214659; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214659']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214659, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2858495f41c798b4d54fc6c2e3d387c9" } } $('.js-work-strip[data-work-id=94214659]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214659,"title":"Optimal Interpolation in RKHS, Spectral Decomposition of Integral Operators and Application","translated_title":"","metadata":{"abstract":"The orthogonal projection associated to optimal interpolation in a reproducing kernel Hilbert space is characterized by the spectral decomposition of an integral operator. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214658"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214658/Constrained_Optimal_Smoothing_and_Bayesian_Estimation"><img alt="Research paper thumbnail of Constrained Optimal Smoothing and Bayesian Estimation" class="work-thumbnail" src="https://attachments.academia-assets.com/96734469/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214658/Constrained_Optimal_Smoothing_and_Bayesian_Estimation">Constrained Optimal Smoothing and Bayesian Estimation</a></div><div class="wp-workCard_item"><span>ArXiv</span><span>, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [7] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c0261a88dff5e764387a4e15a856a633" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734469,"asset_id":94214658,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734469/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214658"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214658"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214658; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214658]").text(description); $(".js-view-count[data-work-id=94214658]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214658; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214658']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214658, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "c0261a88dff5e764387a4e15a856a633" } } $('.js-work-strip[data-work-id=94214658]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214658,"title":"Constrained Optimal Smoothing and Bayesian Estimation","translated_title":"","metadata":{"abstract":"In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [7] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution.","publisher":"ArXiv","publication_date":{"day":null,"month":null,"year":2021,"errors":{}},"publication_name":"ArXiv"},"translated_abstract":"In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [7] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution.","internal_url":"https://www.academia.edu/94214658/Constrained_Optimal_Smoothing_and_Bayesian_Estimation","translated_internal_url":"","created_at":"2023-01-02T23:53:01.260-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734469,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734469/thumbnails/1.jpg","file_name":"2107.05275v1.pdf","download_url":"https://www.academia.edu/attachments/96734469/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Constrained_Optimal_Smoothing_and_Bayesi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734469/2107.05275v1-libre.pdf?1672743195=\u0026response-content-disposition=attachment%3B+filename%3DConstrained_Optimal_Smoothing_and_Bayesi.pdf\u0026Expires=1732751545\u0026Signature=MU8DTlDg~l28kNCKufj8UnZleHZmQR-aWSVSTiBchYbatuccRd1IpqjZTvnxN-gTrbP8j~7eBcQoIwud-vVt2lCjsHoUn3r0ETDewJEOid~enNBYg6sd4vzfHnlzhJ3rcA1xwueoM~mYNEBZbwbTZA0bbHTqav0OEJDhlP8NBZp6CPM3cgFJy7I3pzqnb1dVeMNMfwzaYRi82Ka2w852tWMziAG9BxpLGK10hsnfGnLFD4~1RT6OWtJHBJTuXnVvJYpLumfSTBSHmXMUJr8RLoLsx4afKUu6wSrUSKm2X1xZkbcT2zSAhMJqnKeqW90XnaFBjEdtxNVGMkKyCe3CsA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Constrained_Optimal_Smoothing_and_Bayesian_Estimation","translated_slug":"","page_count":15,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734469,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734469/thumbnails/1.jpg","file_name":"2107.05275v1.pdf","download_url":"https://www.academia.edu/attachments/96734469/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Constrained_Optimal_Smoothing_and_Bayesi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734469/2107.05275v1-libre.pdf?1672743195=\u0026response-content-disposition=attachment%3B+filename%3DConstrained_Optimal_Smoothing_and_Bayesi.pdf\u0026Expires=1732751545\u0026Signature=MU8DTlDg~l28kNCKufj8UnZleHZmQR-aWSVSTiBchYbatuccRd1IpqjZTvnxN-gTrbP8j~7eBcQoIwud-vVt2lCjsHoUn3r0ETDewJEOid~enNBYg6sd4vzfHnlzhJ3rcA1xwueoM~mYNEBZbwbTZA0bbHTqav0OEJDhlP8NBZp6CPM3cgFJy7I3pzqnb1dVeMNMfwzaYRi82Ka2w852tWMziAG9BxpLGK10hsnfGnLFD4~1RT6OWtJHBJTuXnVvJYpLumfSTBSHmXMUJr8RLoLsx4afKUu6wSrUSKm2X1xZkbcT2zSAhMJqnKeqW90XnaFBjEdtxNVGMkKyCe3CsA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":34109,"name":"Bayesian estimation","url":"https://www.academia.edu/Documents/in/Bayesian_estimation"},{"id":132898,"name":"Estimation","url":"https://www.academia.edu/Documents/in/Estimation"},{"id":220768,"name":"Correspondence","url":"https://www.academia.edu/Documents/in/Correspondence"},{"id":225304,"name":"Smoothing","url":"https://www.academia.edu/Documents/in/Smoothing"},{"id":884835,"name":"Reproducing Kernel Hilbert Space","url":"https://www.academia.edu/Documents/in/Reproducing_Kernel_Hilbert_Space"},{"id":1032783,"name":"Bayesian Probability","url":"https://www.academia.edu/Documents/in/Bayesian_Probability"},{"id":1665120,"name":"Bayes estimator","url":"https://www.academia.edu/Documents/in/Bayes_estimator"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"},{"id":3933497,"name":"Inequality Constraints","url":"https://www.academia.edu/Documents/in/Inequality_Constraints"}],"urls":[{"id":27685970,"url":"https://arxiv.org/pdf/2107.05275v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214656"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214656/Karhunen_Lo%C3%A8ve_decomposition_of_Gaussian_measures_on_Banach_spaces"><img alt="Research paper thumbnail of Karhunen–Loève decomposition of Gaussian measures on Banach spaces" class="work-thumbnail" src="https://attachments.academia-assets.com/96734466/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214656/Karhunen_Lo%C3%A8ve_decomposition_of_Gaussian_measures_on_Banach_spaces">Karhunen–Loève decomposition of Gaussian measures on Banach spaces</a></div><div class="wp-workCard_item"><span>Probability and Mathematical Statistics</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The study of Gaussian measures on Banach spaces is of active interest both in pure and applied ma...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="06232e2cf85787db2680740074564ba4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734466,"asset_id":94214656,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734466/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214656"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214656"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214656; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214656]").text(description); $(".js-view-count[data-work-id=94214656]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214656; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214656']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214656, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "06232e2cf85787db2680740074564ba4" } } $('.js-work-strip[data-work-id=94214656]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214656,"title":"Karhunen–Loève decomposition of Gaussian measures on Banach spaces","translated_title":"","metadata":{"abstract":"The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.","publisher":"Wydawnictwo Uniwersytetu Wroclawskiego","publication_date":{"day":null,"month":null,"year":2019,"errors":{}},"publication_name":"Probability and Mathematical Statistics"},"translated_abstract":"The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.","internal_url":"https://www.academia.edu/94214656/Karhunen_Lo%C3%A8ve_decomposition_of_Gaussian_measures_on_Banach_spaces","translated_internal_url":"","created_at":"2023-01-02T23:53:01.088-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734466,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734466/thumbnails/1.jpg","file_name":"6681.pdf","download_url":"https://www.academia.edu/attachments/96734466/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Karhunen_Loeve_decomposition_of_Gaussian.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734466/6681-libre.pdf?1672743206=\u0026response-content-disposition=attachment%3B+filename%3DKarhunen_Loeve_decomposition_of_Gaussian.pdf\u0026Expires=1732751545\u0026Signature=B5GejbvvLGWNVwkbxgIIWdlL1sm2UBacFox~zUhlrA-dNSTghxvpoDw6aBHPK6kryWtX6QQf7UMyrf8GepHW~56WOchkV2oUrxDqXBUgZZKypNgkFb7p4sO-K3ImIFWm7KfbU-weYlyg94WoD-ZwcNb42nTKHEctQULIElA2flRFeTS6~07KS0Md3nFkkKuCg5nTO0AnwgOMnH56b1JGEfesFIrSHzTtcJk4RTk6TIQR5dAlmhrCd4lyh3sd2d8Vmp6ESEfXjQl0ALpydmtToCnx~Fm1IWnjHteoIJH2TulCvXdzd2Mu5GdRSqndv-~4PON6vWzzocY6xhQ4JntfYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Karhunen_Loève_decomposition_of_Gaussian_measures_on_Banach_spaces","translated_slug":"","page_count":19,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734466,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734466/thumbnails/1.jpg","file_name":"6681.pdf","download_url":"https://www.academia.edu/attachments/96734466/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Karhunen_Loeve_decomposition_of_Gaussian.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734466/6681-libre.pdf?1672743206=\u0026response-content-disposition=attachment%3B+filename%3DKarhunen_Loeve_decomposition_of_Gaussian.pdf\u0026Expires=1732751545\u0026Signature=B5GejbvvLGWNVwkbxgIIWdlL1sm2UBacFox~zUhlrA-dNSTghxvpoDw6aBHPK6kryWtX6QQf7UMyrf8GepHW~56WOchkV2oUrxDqXBUgZZKypNgkFb7p4sO-K3ImIFWm7KfbU-weYlyg94WoD-ZwcNb42nTKHEctQULIElA2flRFeTS6~07KS0Md3nFkkKuCg5nTO0AnwgOMnH56b1JGEfesFIrSHzTtcJk4RTk6TIQR5dAlmhrCd4lyh3sd2d8Vmp6ESEfXjQl0ALpydmtToCnx~Fm1IWnjHteoIJH2TulCvXdzd2Mu5GdRSqndv-~4PON6vWzzocY6xhQ4JntfYA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":96734467,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734467/thumbnails/1.jpg","file_name":"6681.pdf","download_url":"https://www.academia.edu/attachments/96734467/download_file","bulk_download_file_name":"Karhunen_Loeve_decomposition_of_Gaussian.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734467/6681-libre.pdf?1672743204=\u0026response-content-disposition=attachment%3B+filename%3DKarhunen_Loeve_decomposition_of_Gaussian.pdf\u0026Expires=1732751545\u0026Signature=hRRKF7iobiHoCs16uDMB-6a-vQzMPpIFmCQMi43GF6ItWFd0hmsXlEco0IpD1muL5cvv2SG9Tl9CUOnmr5uYvhNNFk6SkNy-sAXO4Z-gcRdFDSDXwPdQy7mk34Bn8vWH38Z3yClayO1mtFybk~3ZR1r~~7wmQtheTcrD185tPk2meWKVWNSh0PxJrQzxhRND0-B46kRE8oXI5ysSz69QH3v-fImOF8HOOpttv1J~GI8IDoq3tKv4FWfZUM1E7suvIp40VmmMoXGi-g7~B9Qkbs-Si4q7o2hulstBDJtYHkAtfo4c~Te2-~TA1yIS3mDAAlFo~Kwdr208~pOx2BmNHA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":31412,"name":"Probability and Mathematical Statistics","url":"https://www.academia.edu/Documents/in/Probability_and_Mathematical_Statistics"},{"id":342314,"name":"Gaussian","url":"https://www.academia.edu/Documents/in/Gaussian"},{"id":359655,"name":"Proper Orthogonal Decomposition","url":"https://www.academia.edu/Documents/in/Proper_Orthogonal_Decomposition"},{"id":988387,"name":"Hilbert Space","url":"https://www.academia.edu/Documents/in/Hilbert_Space"},{"id":1196982,"name":"Gaussian Measure","url":"https://www.academia.edu/Documents/in/Gaussian_Measure"}],"urls":[{"id":27685968,"url":"http://wuwr.pl/pms/article/download/7035/6681"}]}, dispatcherData: dispatcherData }); 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Among the methods developed for solving such problems, the Efficient Global Optimization (EGO) is regarded as one of the state-of-the-art unconstrained continuous optimization algorithms. The surrogate model used in EGO is a Gaussian process (GP) conditional on data points. The most important control on the efficiency of the EGO algorithm is the GP covariance function (or kernel), which is taken as a parameterized function. In this paper, we theoretically and empirically analyze the effect of the covariance parameters, the so-called \"characteristic length scale\" and \"nugget\", on EGO performance. More precisely, we analyze the EGO algorithm with fixed covariance parameters and compare them to the standard setting where they are statistically estimated. The limit behavior of EGO with very small or very large characteristic length scales is identified. Experiments show that a \"small\" nugget should be preferred to its maximum likelihood estimate. Overall, this study contributes to a better theoretical and practical understanding of a key optimization algorithm.","publication_date":{"day":null,"month":null,"year":2018,"errors":{}},"publication_name":"Croatian Operational Research Review","grobid_abstract_attachment_id":96734546},"translated_abstract":null,"internal_url":"https://www.academia.edu/94214655/An_analysis_of_covariance_parameters_in_Gaussian_process_based_optimization","translated_internal_url":"","created_at":"2023-01-02T23:53:00.926-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734546,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734546/thumbnails/1.jpg","file_name":"703e2be797615b3f9ee6fe2c26578b4b155c.pdf","download_url":"https://www.academia.edu/attachments/96734546/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_analysis_of_covariance_parameters_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734546/703e2be797615b3f9ee6fe2c26578b4b155c-libre.pdf?1672739686=\u0026response-content-disposition=attachment%3B+filename%3DAn_analysis_of_covariance_parameters_in.pdf\u0026Expires=1732751545\u0026Signature=Qgd~bDIw728A~25QfJHVi~OgzhUpuwl0GfuTo~X5elVb8Q-gamHs3XBgdeMuXMDwMrQhMQ1XPsVnjS9tlxR~TxWltc~1rWxIeLheVcbtKDJjddVMDlgjRFZgjg6C3kiTiUlUQJmB6g0TWahnUZsEffa8TnRRYy5kYoqCjVH9BXsI3Kxxd-ayywq8NRHXnu8U5Vuk7vrgfmwWAz98UlmZbMBEQriEgphaEVjQKKegX-J6i12QTRHd-1y13DaUgvOhm1BtuvbKeNyS7lKmZw63cK5N2a~xF0zUe4~2ajfdINfFHS7Vz9Zy5utxUJay8yAohvVf1jkN7ELmk5bzUNqKzg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"An_analysis_of_covariance_parameters_in_Gaussian_process_based_optimization","translated_slug":"","page_count":10,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734546,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734546/thumbnails/1.jpg","file_name":"703e2be797615b3f9ee6fe2c26578b4b155c.pdf","download_url":"https://www.academia.edu/attachments/96734546/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_analysis_of_covariance_parameters_in.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734546/703e2be797615b3f9ee6fe2c26578b4b155c-libre.pdf?1672739686=\u0026response-content-disposition=attachment%3B+filename%3DAn_analysis_of_covariance_parameters_in.pdf\u0026Expires=1732751545\u0026Signature=Qgd~bDIw728A~25QfJHVi~OgzhUpuwl0GfuTo~X5elVb8Q-gamHs3XBgdeMuXMDwMrQhMQ1XPsVnjS9tlxR~TxWltc~1rWxIeLheVcbtKDJjddVMDlgjRFZgjg6C3kiTiUlUQJmB6g0TWahnUZsEffa8TnRRYy5kYoqCjVH9BXsI3Kxxd-ayywq8NRHXnu8U5Vuk7vrgfmwWAz98UlmZbMBEQriEgphaEVjQKKegX-J6i12QTRHd-1y13DaUgvOhm1BtuvbKeNyS7lKmZw63cK5N2a~xF0zUe4~2ajfdINfFHS7Vz9Zy5utxUJay8yAohvVf1jkN7ELmk5bzUNqKzg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":107131,"name":"Global Optimization","url":"https://www.academia.edu/Documents/in/Global_Optimization"},{"id":272592,"name":"Mathematical Optimization","url":"https://www.academia.edu/Documents/in/Mathematical_Optimization"},{"id":282522,"name":"Ego","url":"https://www.academia.edu/Documents/in/Ego"},{"id":688446,"name":"Gaussian Process","url":"https://www.academia.edu/Documents/in/Gaussian_Process"},{"id":1473341,"name":"Covariance","url":"https://www.academia.edu/Documents/in/Covariance"},{"id":3017674,"name":"Covariance function","url":"https://www.academia.edu/Documents/in/Covariance_function"}],"urls":[]}, dispatcherData: dispatcherData }); 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We propose a new approximation method based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. An algorithm is derived and numerical examples with boundedness and monotonicity constraints in one and two dimensions are given. Keywords Optimization • RKHS • Interpolation • Inequality constraints 1 Introduction Let X be a nonempty set of R d (d ≥ 1) and E = C 0 (X) the linear (topological) space of real valued continuous functions on X. Given n distinct points x (1) ,. .. , x (n) ∈ X and y 1 ,. .. , y n ∈ R, we define the set I of interpolating functions by I := f ∈ E, f x (i) = y i , i = 1,. .. , n .","publication_date":{"day":null,"month":null,"year":2017,"errors":{}},"publication_name":"Computational Optimization and Applications","grobid_abstract_attachment_id":96734541},"translated_abstract":null,"internal_url":"https://www.academia.edu/94214654/A_new_method_for_interpolating_in_a_convex_subset_of_a_Hilbert_space","translated_internal_url":"","created_at":"2023-01-02T23:53:00.752-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":56976577,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":96734541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734541/thumbnails/1.jpg","file_name":"Maatouk_H.pdf","download_url":"https://www.academia.edu/attachments/96734541/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_new_method_for_interpolating_in_a_conv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734541/Maatouk_H-libre.pdf?1672739693=\u0026response-content-disposition=attachment%3B+filename%3DA_new_method_for_interpolating_in_a_conv.pdf\u0026Expires=1732751545\u0026Signature=VRLz~TD88oVueia-vnrC8pEl19ahVi7FeUV~16wSQ25SgVKfy7PI89-PU-nncLTxHyRLTzPhZ1kGmxwN75fIzbgnM6M~3JldEYNXSo~yYdJolAQsj1B-ZWRk5ken5kntf5O1GdNPGMjnGNb-HLbSptqEzSSrI9tkgP5wmDHvMtEIn-68nAUzyVegEv4~D1xN3hBnsUpNFn2rMwuITZjx9RnApKw0pdIV1lL-1EAbQvJeArSVJWR7G8epH4NJ1WSmXcpQ4UflHolixV-w7oTTTsAmBT0zYXrkQDNY~xAp4831O2gwFKG8GiY-pEHl60cGnCKTiA7kJnTrrrqWI5qrJQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_new_method_for_interpolating_in_a_convex_subset_of_a_Hilbert_space","translated_slug":"","page_count":34,"language":"en","content_type":"Work","owner":{"id":56976577,"first_name":"Xavier","middle_initials":null,"last_name":"Bay","page_name":"XavierBay","domain_name":"independent","created_at":"2016-11-20T22:44:54.254-08:00","display_name":"Xavier Bay","url":"https://independent.academia.edu/XavierBay"},"attachments":[{"id":96734541,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/96734541/thumbnails/1.jpg","file_name":"Maatouk_H.pdf","download_url":"https://www.academia.edu/attachments/96734541/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_new_method_for_interpolating_in_a_conv.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/96734541/Maatouk_H-libre.pdf?1672739693=\u0026response-content-disposition=attachment%3B+filename%3DA_new_method_for_interpolating_in_a_conv.pdf\u0026Expires=1732751545\u0026Signature=VRLz~TD88oVueia-vnrC8pEl19ahVi7FeUV~16wSQ25SgVKfy7PI89-PU-nncLTxHyRLTzPhZ1kGmxwN75fIzbgnM6M~3JldEYNXSo~yYdJolAQsj1B-ZWRk5ken5kntf5O1GdNPGMjnGNb-HLbSptqEzSSrI9tkgP5wmDHvMtEIn-68nAUzyVegEv4~D1xN3hBnsUpNFn2rMwuITZjx9RnApKw0pdIV1lL-1EAbQvJeArSVJWR7G8epH4NJ1WSmXcpQ4UflHolixV-w7oTTTsAmBT0zYXrkQDNY~xAp4831O2gwFKG8GiY-pEHl60cGnCKTiA7kJnTrrrqWI5qrJQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":43981,"name":"Optimization","url":"https://www.academia.edu/Documents/in/Optimization"},{"id":254570,"name":"Interpolation","url":"https://www.academia.edu/Documents/in/Interpolation"},{"id":556845,"name":"Numerical Analysis and Computational Mathematics","url":"https://www.academia.edu/Documents/in/Numerical_Analysis_and_Computational_Mathematics"},{"id":988387,"name":"Hilbert Space","url":"https://www.academia.edu/Documents/in/Hilbert_Space"},{"id":1970623,"name":"RKHS","url":"https://www.academia.edu/Documents/in/RKHS"},{"id":3933497,"name":"Inequality Constraints","url":"https://www.academia.edu/Documents/in/Inequality_Constraints"}],"urls":[{"id":27685967,"url":"http://link.springer.com/article/10.1007/s10589-017-9906-9/fulltext.html"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214653"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214653/Gaussian_Process_Emulators_for_Computer_Experiments_with_Inequality_Constraints"><img alt="Research paper thumbnail of Gaussian Process Emulators for Computer Experiments with Inequality Constraints" class="work-thumbnail" src="https://attachments.academia-assets.com/96734543/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214653/Gaussian_Process_Emulators_for_Computer_Experiments_with_Inequality_Constraints">Gaussian Process Emulators for Computer Experiments with Inequality Constraints</a></div><div class="wp-workCard_item"><span>Mathematical Geosciences</span><span>, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="902f9a5843e1dd444b2e7234b4ef2a20" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":96734543,"asset_id":94214653,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/96734543/download_file?st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&st=MTczMjc0Nzk0NSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214653"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214653"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214653; 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These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. Our aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, we propose a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain. The inequality mean and mode (i.e. mean and maximum a posteriori) of the conditional Gaussian process are calculated and prediction intervals are quantified. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214652"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214652/Short_Term_Load_Forecasting_in_the_Industry_for_Establishing_Consumption_Baselines_A_French_Case"><img alt="Research paper thumbnail of Short Term Load Forecasting in the Industry for Establishing Consumption Baselines: A French Case" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214652/Short_Term_Load_Forecasting_in_the_Industry_for_Establishing_Consumption_Baselines_A_French_Case">Short Term Load Forecasting in the Industry for Establishing Consumption Baselines: A French Case</a></div><div class="wp-workCard_item"><span>Lecture Notes in Statistics</span><span>, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The estimation of baseline electricity consumptions for energy efficiency and load management mea...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. Perspectives for exploring different forecasting methods for these applications are considered for future works, as well as their application to different load curves from diverse industrial sectors and equipments.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="94214652"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="94214652"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 94214652; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=94214652]").text(description); $(".js-view-count[data-work-id=94214652]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 94214652; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='94214652']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 94214652, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=94214652]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":94214652,"title":"Short Term Load Forecasting in the Industry for Establishing Consumption Baselines: A French Case","translated_title":"","metadata":{"abstract":"The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. Perspectives for exploring different forecasting methods for these applications are considered for future works, as well as their application to different load curves from diverse industrial sectors and equipments.","publication_date":{"day":null,"month":null,"year":2015,"errors":{}},"publication_name":"Lecture Notes in Statistics"},"translated_abstract":"The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. 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Les lignes qui suivent montrent l'évaluation des commandes utilisées en R (voir la remarque sur R page 2). 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="94214577"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/94214577/Extended_Formula_for_Kriging_Interpolation"><img alt="Research paper thumbnail of Extended Formula for Kriging Interpolation" class="work-thumbnail" src="https://attachments.academia-assets.com/96734406/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/94214577/Extended_Formula_for_Kriging_Interpolation">Extended Formula for Kriging Interpolation</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In many elds, Kriging interpolation techniques are used within a nite discrete set of known value...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In many elds, Kriging interpolation techniques are used within a nite discrete set of known values of a real function (no evaluation or measurement error). 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