Abstract : With the promotion by both mathematics itself and the practical requirements from engineering, the interest of researches on inverse problems and imaging, has been growing vigorously recent decades. The characteristic of the ill-posedness for inverse problems and imaging makes it hard to construct solutions. To overcome difficulties, various regularization techniques must be introduced which are closely related to many mathematical branches such as partial differential equations, differential geometry, numerical analysis, machine learning, image processing, functional analysis, optimizations and computer science. This minisymposium will bring together experts to discuss recent progresses in this area and related topics.
[03733] Increasing stability in the linearized inverse Schrodinger potential problems
Format : Talk at Waseda University
Author(s) :
Shuai Lu (Fudan University)
Abstract : Inverse Schrodinger potential problem concerns about the recovery of a potential function in the Schrodinger equation in a bounded domain throught the DtN map. In this talk, we introduce the linearized DtN map, and prove a stability estimate with explicit dependence on wavenumbers. This is an increasing stability result, in the sense that the logarithmic stable term decays when wavenumber increases. The talk is based on joint works with Victor Isakov (Wichita), Mikko Salo (Jyvaskyla), Boxi Xu (SUFE) and Sen Zou (Fudan).
[03422] High-order boundary integral equation solvers for layered-medium scattering problems
Format : Talk at Waseda University
Author(s) :
Tao Yin (Chinese Academy of Sciences)
Abstract : This talk will present our recent works on the fast and highly accurate boundary integral equation (BIE) methods, including the windowed Green function (WGF) method and perfectly-matched-layer (PML) BIE method, for solving the acoustic and elastic wave scattering problems in both two- and three-dimensional layered-medium. 1) The WGF method utilize the free-space fundamental solution to derive the BIEs on the whole unbounded surface which requires to be truncated in practical computing. Based on the solutions due to the scattering by flat surface, a correction strategy is introduced to ensure uniform accuracy for all incident angles. 2) For the half-space and two layered-medium case, the original scattering problem can be truncated onto a bounded domain by the PML. Assuming the vanishing of the scattered field on the PML boundary, BIEs on local defects are derived only in terms of using the PML-transformed free-space Green's function. For the considered two methods, a high-order Chebyshev-based rectangular-polar singular-integration solver is used in numerical implementation. Numerical experiments for both two- and three-dimensional problems are carried out to demonstrate the accuracy and efficiency of the proposed solvers. Potential applications to the inverse problems of reconstructing unbounded surfaces will also be discussed.
[04333] An inverse boundary value problem for a nonlinear elastic wave equation
Format : Talk at Waseda University
Author(s) :
Jian Zhai (Fudan University)
Abstract : We consider an inverse boundary value problem for a nonlinear model of elastic waves. We show that all the material parameters appearing in the equation can be uniquely determined from boundary measurements under certain geometric conditions. The proof is based on the construction of Gaussian beam solutions.
[03434] Inverse random scattering problems for stochastic wave equations
Format : Talk at Waseda University
Author(s) :
Jianliang Li (Hunan Normal University)
Peijun Li (Purdue University)
Xu Wang (Chinese Academy of Sciences)
Abstract : Inverse random scattering problems with a random source or potential will be introduced for time-harmonic wave equations. The unknown random source or potential is assumed to be a generalized isotropic Gaussian random field. With information of the data observed in a bounded domain, the strength of the random source or potential is shown to be uniquely determined by a single realization of the magnitude of the wave field averaged over the frequency band almost surely.
[04564] An Inverse Problem for Nonlinear Time-dependent Schrodinger Equations with Partial Data
Format : Talk at Waseda University
Author(s) :
Ting Zhou (Zhejiang University)
Ru-Yu Lai (University of Minnesota)
Xuezhu Lu (Northeastern University)
Abstract : In this talk, I will present some recent results on solving inverse boundary value problems for nonlinear PDEs, especially for a time-dependent Schrodinger equation with time-dependent potentials with partial boundary Dirichlet-to-Neumann map. After a higher order linearization step, the problem will be reduced to implementing special geometrical optics (GO) solutions to prove the uniqueness and stability of the reconstruction. This is a joint work with my PhD student Xuezhu Lu and Prof. Ru-Yu Lai.
[04099] Inverse scattering problems with incomplete data
Format : Talk at Waseda University
Author(s) :
Xiaodong Liu (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : Inverse scattering problems aim to determine unknown scatterers with wave fields measured around the scatterers. However, from the practical point of views, we have only limited information, e.g., limited aperture data phaseless data and sparse data. In this talk, we introduce some data retrieval techniques and the applications in the inverse scattering problems. The theoretical and numerical methods for inverse scattering problems with multi-frequency spase measurements will also be mentioned.
[03745] Imaging of penetrable locally rough surfaces from phaseless total-field data
Format : Talk at Waseda University
Author(s) :
Haiwen Zhang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Long Li (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Jiansheng Yang (Peking University)
Bo Zhang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : This talk is concerned with inverse scattering by a two-layered medium with a locally rough interface in 2D. We propose a direct imaging method to reconstruct the penetrable locally rough surface from phaseless total-field data. The theoretical analysis is mainly based on the results in our recent work (L. Li, J. Yang, B. Zhang and H. Zhang, arXiv:2208.00456) on the uniform far-field asymptotics of the scattered field for acoustic scattering in a two-layered medium.
[04075] A new approach to an inverse source problem for the wave equation
Format : Talk at Waseda University
Author(s) :
Haibing Wang (Southeast University)
Abstract : Consider an inverse problem of reconstructing a source term from boundary measurements for the wave equation. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann-Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have non-zero averaged eigenfunctions. By introducing a Riesz basis, we reconstruct the wave field, generated before injecting the particles, on the center of the particles. Finally, from these last fields, we reconstruct the source term. A significant advantage of our approach is that we only need the measurements for a single point away from the support of the source. This is a joint work with Prof. Mourad Sini from RICAM.
[04288] Recovering an infinite rough surface by acoustic measurements
Format : Talk at Waseda University
Author(s) :
Jiaqing Yang (Xi'an Jiaotong University)
Abstract : In this talk, I will report some recent advances on inverse scattering by rough surfaces, where new uniqueness results and inversion algorithms will be presented to recover the shape and location of the rough surface from the near-field measurements associated with incident point sources. Moreover, several numerical examples will be provided to illustrate the effectiveness of the algorithms.
[03405] Uniqueness on recovering coefficients from localized Dirichlet-to-Neumann map for piecewise homogeneous piezoelectricity
Format : Talk at Waseda University
Author(s) :
Xiang Xu (Zhejiang University)
Abstract : In this talk, we present an inverse problem on determining coefficients of piecewise homogeneous piezoelectric equations from a localized Dirichlet-to-Neumann map on partial boundaries. Assume the bounded domain can be divided into finite subdomains, in which the unknown coefficients including the anisotropic elastic tensor, the piezoelectric tensor, and the dielectric tensor are constants. Two different cases are considered: the subdomains are either known and Lipschitz or unknown and subanalytic. For both cases, the unknown coefficients can be uniquely determined from a given localized Dirichlet-to-Neumann map.
