Inverse Problems and Imaging最新文献

{"title":"Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds","authors":"Lili Yan","doi":"10.3934/ipi.2022034","DOIUrl":"https://doi.org/10.3934/ipi.2022034","url":null,"abstract":"<p style='text-indent:20px;'>We prove that a continuous potential <inline-formula><tex-math id=\"M1\">begin{document}$ q $end{document}</tex-math></inline-formula> can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator <inline-formula><tex-math id=\"M2\">begin{document}$ Delta_g^2+q $end{document}</tex-math></inline-formula> on a conformally transversally anisotropic Riemannian manifold of dimension <inline-formula><tex-math id=\"M3\">begin{document}$ ge 3 $end{document}</tex-math></inline-formula> with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [<xref ref-type=\"bibr\" rid=\"b56\">56</xref>]. In particular, our result is applicable and new in the case of smooth bounded domains in the <inline-formula><tex-math id=\"M4\">begin{document}$ 3 $end{document}</tex-math></inline-formula>–dimensional Euclidean space as well as in the case of <inline-formula><tex-math id=\"M5\">begin{document}$ 3 $end{document}</tex-math></inline-formula>–dimensional admissible manifolds.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46048137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Direct sampling methods for isotropic and anisotropic scatterers with point source measurements","authors":"I. Harris, Dinh-Liem Nguyen, Thi-Phong Nguyen","doi":"10.3934/ipi.2022015","DOIUrl":"https://doi.org/10.3934/ipi.2022015","url":null,"abstract":"In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48281174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination","authors":"Pardeep Kumar, M. T. Mohan","doi":"10.3934/ipi.2022024","DOIUrl":"https://doi.org/10.3934/ipi.2022024","url":null,"abstract":"<p style='text-indent:20px;'>In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{align*} boldsymbol{u}_t-mu Deltaboldsymbol{u}+(boldsymbol{u}cdotnabla)boldsymbol{u}+alphaboldsymbol{u}+beta|boldsymbol{u}|^{r-1}boldsymbol{u}+nabla p = boldsymbol{F}: = boldsymbol{f} g, nablacdotboldsymbol{u} = 0, end{align*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in bounded domains <inline-formula><tex-math id=\"M1\">begin{document}$ Omegasubsetmathbb{R}^d $end{document}</tex-math></inline-formula> (<inline-formula><tex-math id=\"M2\">begin{document}$ d = 2, 3 $end{document}</tex-math></inline-formula>) with smooth boundary, where <inline-formula><tex-math id=\"M3\">begin{document}$ alpha, beta, mu>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M4\">begin{document}$ rin[1, infty) $end{document}</tex-math></inline-formula>. The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function <inline-formula><tex-math id=\"M5\">begin{document}$ boldsymbol{u} $end{document}</tex-math></inline-formula>, the pressure gradient <inline-formula><tex-math id=\"M6\">begin{document}$ nabla p $end{document}</tex-math></inline-formula> and the vector-valued function <inline-formula><tex-math id=\"M7\">begin{document}$ boldsymbol{f} $end{document}</tex-math></inline-formula>. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for <inline-formula><tex-math id=\"M8\">begin{document}$ rgeq 1 $end{document}</tex-math></inline-formula> in two dimensions and for <inline-formula><tex-math id=\"M9\">begin{document}$ r geq 3 $end{document}</tex-math></inline-formula> in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45361120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas","authors":"Venky Krishnan, V. Sharafutdinov","doi":"10.3934/ipi.2021076","DOIUrl":"https://doi.org/10.3934/ipi.2021076","url":null,"abstract":"<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id=\"M1\">begin{document}$ rge0 $end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id=\"M2\">begin{document}$ r^{mathrm{th}} $end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id=\"M3\">begin{document}$ m $end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id=\"M4\">begin{document}$ {{mathbb R}}^n $end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id=\"M5\">begin{document}$ f $end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id=\"M6\">begin{document}$ r $end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id=\"M7\">begin{document}$ L^2 $end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id=\"M8\">begin{document}$ widehat f $end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id=\"M9\">begin{document}$ A^{(m,r,l)} (0le lle r) $end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id=\"M10\">begin{document}$ {{mathbb S}}^{n-1} $end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id=\"M11\">begin{document}$ r $end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id=\"M12\">begin{document}$ r $end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id=\"M13\">begin{document}$ r $end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id=\"M14\">begin{document}$ 0,1,2 $end{document}</tex-math></inline-formula> are presented in an explicit form.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"51 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86152836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Direct regularized reconstruction for the three-dimensional Calderón problem","authors":"K. Knudsen, A. K. Rasmussen","doi":"10.3934/ipi.2022002","DOIUrl":"https://doi.org/10.3934/ipi.2022002","url":null,"abstract":"Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderón problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"29 3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83387648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An inverse problem for a fractional diffusion equation with fractional power type nonlinearities","authors":"Li Li","doi":"10.3934/ipi.2021064","DOIUrl":"https://doi.org/10.3934/ipi.2021064","url":null,"abstract":"We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88082760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On new surface-localized transmission eigenmodes","authors":"Youjun Deng, Yan Jiang, Hongyu Liu, Kai Zhang","doi":"10.3934/ipi.2021063","DOIUrl":"https://doi.org/10.3934/ipi.2021063","url":null,"abstract":"<p style='text-indent:20px;'>Consider the transmission eigenvalue problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ (Delta+k^2mathbf{n}^2) w = 0, (Delta+k^2)v = 0 mbox{in} Omega;quad w = v, partial_nu w = partial_nu v mbox{on} partialOmega. $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown in [<xref ref-type=\"bibr\" rid=\"b16\">16</xref>] that there exists a sequence of eigenfunctions <inline-formula><tex-math id=\"M1\">begin{document}$ (w_m, v_m)_{minmathbb{N}} $end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id=\"M2\">begin{document}$ k_mrightarrow infty $end{document}</tex-math></inline-formula> such that either <inline-formula><tex-math id=\"M3\">begin{document}$ {w_m}_{minmathbb{N}} $end{document}</tex-math></inline-formula> or <inline-formula><tex-math id=\"M4\">begin{document}$ {v_m}_{minmathbb{N}} $end{document}</tex-math></inline-formula> are surface-localized, depending on <inline-formula><tex-math id=\"M5\">begin{document}$ mathbf{n}>1 $end{document}</tex-math></inline-formula> or <inline-formula><tex-math id=\"M6\">begin{document}$ 0<mathbf{n}<1 $end{document}</tex-math></inline-formula>. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions <inline-formula><tex-math id=\"M7\">begin{document}$ (w_m, v_m)_{minmathbb{N}} $end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id=\"M8\">begin{document}$ k_mrightarrow infty $end{document}</tex-math></inline-formula> such that both <inline-formula><tex-math id=\"M9\">begin{document}$ {w_m}_{minmathbb{N}} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M10\">begin{document}$ {v_m}_{minmathbb{N}} $end{document}</tex-math></inline-formula> are surface-localized, no matter <inline-formula><tex-math id=\"M11\">begin{document}$ mathbf{n}>1 $end{document}</tex-math></inline-formula> or <inline-formula><tex-math id=\"M12\">begin{document}$ 0<mathbf{n}<1 $end{document}</tex-math></inline-formula>. Though our study is confined within the radial geometry, the construction is subtle and technical.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89807226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation","authors":"X. Jing, Masahiro Yamamoto","doi":"10.3934/ipi.2022019","DOIUrl":"https://doi.org/10.3934/ipi.2022019","url":null,"abstract":"<p style='text-indent:20px;'>We consider two kinds of inverse problems on determining multiple parameters simultaneously for one-dimensional time-fractional diffusion-wave equations with derivative order <inline-formula><tex-math id=\"M1\">begin{document}$ alpha in (0, 2) $end{document}</tex-math></inline-formula>. Based on the analysis of the poles of Laplace transformed data and a transformation formula, we first prove the uniqueness in identifying multiple parameters, including the order of the derivative in time, a spatially varying potential, initial values, and Robin coefficients simultaneously from boundary measurement data, provided that no eigenmodes are zero. Our main results show that the uniqueness of four kinds of parameters holds simultaneously by such observation for the time-fractional diffusion-wave model where unknown orders <inline-formula><tex-math id=\"M2\">begin{document}$ alpha $end{document}</tex-math></inline-formula> vary order (0, 2) including 1, restricted to neither <inline-formula><tex-math id=\"M3\">begin{document}$ alpha in (0, 1] $end{document}</tex-math></inline-formula> nor <inline-formula><tex-math id=\"M4\">begin{document}$ alpha in (1, 2) $end{document}</tex-math></inline-formula>. Furthermore, for another formulation of the fractional diffusion-wave equation with input source term in place of the initial value, we can also prove the simultaneous uniqueness of multiple parameters, including a spatially varying potential and Robin coefficients by means of the uniqueness result in the case of non-zero initial value and Duhamel's principle.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49002422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random tree Besov priors – Towards fractal imaging","authors":"Hanne Kekkonen, M. Lassas, E. Saksman, S. Siltanen","doi":"10.3934/ipi.2022059","DOIUrl":"https://doi.org/10.3934/ipi.2022059","url":null,"abstract":"We propose alternatives to Bayesian a priori distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well defined infinite-dimensional random variables, and can be approximated by finite-dimensional random variables. We introduce a new wavelet-based model, where the non zero coefficient are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in some Besov spaces and have singularities only on a small set τ that has a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in denoising problem.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42250381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counterexamples to inverse problems for the wave equation","authors":"Tony Liimatainen, L. Oksanen","doi":"10.3934/ipi.2021058","DOIUrl":"https://doi.org/10.3934/ipi.2021058","url":null,"abstract":"<p style='text-indent:20px;'>We construct counterexamples to inverse problems for the wave operator on domains in <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{R}^{n+1} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">begin{document}$ n ge 2 $end{document}</tex-math></inline-formula>, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On <inline-formula><tex-math id=\"M3\">begin{document}$ mathbb{R}^{n+1} $end{document}</tex-math></inline-formula> the metrics are conformal to the Minkowski metric.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":"69 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90580032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}