Document Zbl 1419.35248

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Appl. Math.</a> <a href="/?q=in%3A426650" title="Articles in this Issue">79, No. 1, 1-27 (2019)</a>. </div> <div class="abstract">The authors consider a scattering problem of the Helmholtz equation in three space dimensions. The aim is to determine the space-dependent dielectric constant using only phaseless measurements on a two-dimensional plane. On the one hand, just the intensity of the scattered wave is measured, while the phase is unknown. On the other hand, the measurements represent multi-frequency data, since the intensities are known for a large sample size of wave numbers at each space point. Thus an inverse scattering problem is given.<br class="zbmathjax-paragraph">The authors prove an asymptotic formula for the solution of the Helmholtz equation in the case of high wave numbers. If the intensities are given on the measurement plane in an entire interval of wave numbers, then the functions are uniquely determined in the asymptotic formula.<br class="zbmathjax-paragraph">A numerical method is derived for the inverse problem consisting of two stages. In the first stage, the authors compute an approximation of the functions in the asymptotic formula using a finite sample set of wave numbers, where a linear least squares problem is arranged. Consequently, both the intensity and the phase of the solution are approximately known on the measurement plane now. An error bound or an error estimate is not discussed. In the second stage, the authors determine the dielectric coefficient by the data on the measurement plane. A globally convergent iterative algorithm is applied from <span class="zbmathjax-textit">M. V. Klibanov</span> et al. [Inverse Probl. Imaging 12, No. 2, 493–523 (2018; <a href="/1395.35216">Zbl 1395.35216</a>)]. Therein, boundary value problems of elliptic equations have to be solved. The boundary value problem of the Helmholtz equation is considered in form of the integral Lippmann-Schwinger equation.<br class="zbmathjax-paragraph">Finally, the authors perform numerical simulations, where three cases of scattering configurations are arranged in the micrometer scale. The measurement data is artificially generated by sums of predetermined functions and random numbers to model measurement errors of 5%. The above numerical method yields adequate approximations. In some known data of the exact solution, the relative error becomes less or equal 8%, which is rated good with respect to the magnitude of the measurement errors. Comparisons to other methods are not included.<br class="zbmathjax-paragraph">In an appendix, the authors present numerical simulations, which illustrate that the Helmholtz equation yields a suitable approximation of the electric field satisfying the Maxwell equations under specific assumptions.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11183">Roland Pulch (Greifswald)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7004611">Cited in <strong>14</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35R30" title="MSC2020">35R30</a> </td> <td class="space"> Inverse problems for PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65N21" title="MSC2020">65N21</a> </td> <td class="space"> Numerical methods for inverse problems for boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J05" title="MSC2020">35J05</a> </td> <td class="space"> Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A46" title="MSC2020">78A46</a> </td> <td class="space"> Inverse problems (including inverse scattering) in optics and electromagnetic theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHelmholtz+equation">Helmholtz equation</a>; <a href="/?q=ut%3ALippmann-Schwinger+equation">Lippmann-Schwinger equation</a>; <a href="/?q=ut%3AMaxwell+equation">Maxwell equation</a>; <a href="/?q=ut%3Adielectric+constant">dielectric constant</a>; <a href="/?q=ut%3Ainverse+scattering+problem">inverse scattering problem</a>; <a href="/?q=ut%3Aelliptic+equation">elliptic equation</a>; <a href="/?q=ut%3Amulti-frequency+data">multi-frequency data</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1395.35216">Zbl 1395.35216</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1419.35248" data-ciurl="/ci/07004611" data-biburl="/bibtex/07004611.bib" data-amsurl="/amsrefs/07004611.bib" data-xmlurl="/xml/07004611.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07004611.pdf" title="Zbl 1419.35248 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1137/18M1168303" aria-label="DOI for “A coefficient inverse problem with a single measurement of phaseless scattering data”" title="10.1137/18M1168303">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1710.04804"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">T. 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