Abstract : Inverse problems for partial differential equations (PDEs) concern recovery of unknown coefficients or geometries/topologies within the equations by knowledge of certain observables. These problems sit at the intersection of mathematical analysis, PDE theory, and scientific computing, with broader application to modern imaging science and technology.
This minisymposium aims to highlight recent advances in inverse problems for PDEs. It will bring together international scientific researchers to discuss recent developments and emerging challenges in this fast-evolving field. Major topics include (1) inverse problems in wave-based imaging; (2) inverse scattering theory; (3) data-driven inverse methods, and their applications to medical and geophysical imaging.
Organizer(s) : Huaian Diao, Hongyu Liu, Yang Yang, Minghui Song
[00556] Deterministic-Statistical Approach for Inverse Problems with Partial Data
Format : Online Talk on Zoom
Author(s) :
Jiguang Sun (Michigan Technological University)
Abstract : We propose a deterministic-statistical approach for inverse problems with partial data. Certain deterministic method is first used to obtain useful (qualitative) information for the unknowns. Then the inverse problem is recast as a statistical inference problem and the Bayesian inversion is employed to obtain more (quantitative) information of the unknowns. Several examples are presented for demonstration. Furthermore, we introduce new statistical estimators to characterize the non-unique solutions of several inverse problems.
[00553] Quantitative PAT with simplified PN approximation
Format : Online Talk on Zoom
Author(s) :
Yimin Zhong (Auburn University)
Hongkai Zhao (Duke University)
Abstract : The photoacoustic tomography (PAT) is a hybrid modality that combines the optics and acoustics to obtain high resolution and high contrast imaging of heterogeneous media. In this work, our objective is to study the inverse problem in the quantitative step of PAT which aims to reconstruct the optical coefficients of the governing radiative transport equation from the ultrasound measurements. In our analysis, we take the simplified P N approximation of the radiative transport equation as the physical model and then show the uniqueness and stability for this modified inverse problem. Numerical simulations based on synthetic data are presented to validate our analysis.
[02856] Adaptive Mesh-free Approach for Gravity Inversion
Format : Talk at Waseda University
Author(s) :
Yan Liu (Chinese Academy of Geological Sciences)
Abstract : We proposes a method of gravity inversion based on an adaptive mesh-free approach by using a modified radial basis function. As the subsurface space is generally discretized into regular grid cells, and this unstructured nodal discretization bring the expensive mesh generation and manipulation, we use a mesh-free discretization strategy to establish a mapping of subsurface grid cells to a cloud of discrete points. The nodes are adaptively refined during the inversion process to better recover abnormal bodies. In addition, the hybrid basis function and the modified radial basis function are used to improve the accuracy and stability of the solution.
[00358] A neural network method for inverse source problem with limited-aperture data
Format : Talk at Waseda University
Author(s) :
Weishi Yin (Changchun University of Science and Technology)
Ping Zhang (Changchun University of Science and Technology)
Pinchao Meng (Changchun University of Science and Technology)
Hongyu Liu ( City University of Hong Kong)
Abstract : This talk is concerned with an inverse moving source problem, that is, one identifies and predicts
the trajectory of a moving point source by measuring the corresponding wave field. First, for the practical consideration,the dynamical wave field data are collected in a limited aperture and full aperture respectively. Second, we design a parameter inversion model by neural network to reconstruct the trajectory of the moving point source. This model solves the problem of information loss caused by data acquisition in limited aperture and has certain robustness with respect to noise. Third, we consider the trajectory prediction of the moving point source for the inverse source problem associated with the novel input/instruction approach, and construct a trajectory prediction model by neural network to predict the trajectory of the moving point source. Numerical experiments show that the proposed device works effectively and efficiently in some practical scenarios.
[00236] On plasmon modes in multi-layer structures
Format : Talk at Waseda University
Author(s) :
Youjun Deng (Central South University)
Abstract : We consider the plasmon resonances in multi-layer structures. We show the plasmon modes are equivalent to the eigenvalue problem of a matrix, whose order is the same to the number of layers. For any number of layers, the exact characteristic polynomial is derived by a conjecture, which is verified by using induction. It is shown that all the solutions to the characteristic polynomial are real and exist in the span [-1, 2]. Numerical examples are presented for finding all the plasmon modes.
[00405] Uniqueness and non-uniqueness for inverse source problems of elliptic equations
Format : Talk at Waseda University
Author(s) :
Yi-Hsuan Lin (Department of Applied Mathematics, National Yang Ming Chiao Tung University)
Abstract : We study inverse source problems associated to second order elliptic equations on a bounded domain. We demonstrate both uniqueness and non-uniqueness for inverse source problems of different type elliptic equations.
[00361] Mathematical analysis of microscale hydrodynamic cloaking and shielding using electro-osmosis
Format : Talk at Waseda University
Author(s) :
Guang-Hui Zheng (Hunan University)
Abstract : Rendering objects invisible by cloaking them with metamaterials have made rapid progress in the past decade. However, the difficulties of metamaterials manufacturing have limited its development. In this talk, we discuss the mathematical analysis of hydrodynamic cloaking and shielding via electro-osmosis in a microfluidic chamber that does not rely on metamaterials. Based on layer potential technique, the conditions that can ensure the occurrence of the microscale hydrodynamic cloaking and shielding are established. Finally, several numerical examples are served to validate our theoretical analysis. (joint works with Hongyu Liu (CityU) and Zhiqiang Miao (HNU))
[00920] Regularizing Effect of Damping Mechanisms in Inverse Problems of Evolution Equations
Format : Talk at Waseda University
Author(s) :
Sakthivel Kumarasamy (Indian Institute of Space Science and Technology )
Anjuna Dileep (Indian Institute of Space Science and Technology )
Abstract : It is known that the undamped evolution models, such as the Euler-Bernoulli beam, and Kirchhoff-Love plate, don’t support the unique determination of unknown spatial load from the measured displacement at the final time. We solve this issue by introducing the viscous external damping and Kelvin-Voigt damping effects in the basic governing equations. The damping terms play a pivotal role in getting more regular solutions with less regular data on the direct problem, while in the context of inverse problems, it has a similar effect to a regularization term in the Tikhonov functional of the quasi-solution approach.
[00359] A NOVEL QUANTITATIVE INVERSE SCATTERING SCHEME USING INTERIOR RESONANT MODES
Format : Talk at Waseda University
Author(s) :
Xianchao Wang (Harbin Institute of Technology)
Abstract : In this talk, we introduce a novel quantitative imaging scheme to identify impenetrable obstacles in time-harmonic acoustic scattering from the associated far-field data. The proposed method consists of two phases. In the first phase, we determine the interior eigenvalues of the underlying unknown obstacle from the far-field data via the indicating behaviour of the linear sampling method. Then we further determine the associated interior eigenfunctions by solving a constrained optimization problem, again only involving the far-field data. In the second phase, we propose a novel iteration scheme of Newton’s type to identify the boundary surface of the obstacle. By using the interior eigenfunctions determined in the first phase, we can avoid computing any direct scattering problem at each Newton’s iteration. The proposed method is particularly valuable for recovering a sound-hard obstacle, where the Newton’s formula involves the geometric quantities of the unknown boundary surface in a natural way.
[00333] Simultaneous recovery of a scattering cavity and its internal sources
Format : Talk at Waseda University
Author(s) :
Deyue Zhang (Jilin University)
Yukun Guo (Harbin Institute of Technology)
Yinglin Wang (Jilin University)
Yan Chang (Harbin Institute of Technology)
Abstract : We consider the simultaneous reconstruction of a sound-soft cavity and its excitation sources from the total-field data. Using the single-layer potential representations on two measurement curves, this problem can be decoupled into an inverse cavity scattering problem and an inverse source problem. Then the uncoupled subproblems are respectively solved by the modified optimization and sampling method. Numerical examples will be presented to demonstrate the effectiveness of the method.
[00259] The anisotropic Calderón problem at large fixed frequency on manifolds with invertible ray transform
Format : Talk at Waseda University
Author(s) :
Shiqi Ma (Jilin University)
Abstract : We consider the inverse problem of recovering a potential from the Dirichlet to Neumann map at a large fixed frequency on certain Riemannian manifolds. We extend the earlier result of [ G. Uhlmann and Y. Wang, arXiv:2104.03477] to the case of simple manifolds, and more generally to manifolds where the geodesic ray transform is stably invertible.
[00245] Minnaert resonances for bubbles in soft elastic materials
Format : Online Talk on Zoom
Author(s) :
Hongjie LI (The Chinese University of Hong Kong)
Abstract : In this talk, the low-frequency resonance for acoustic bubbles embedded in soft elastic materials is discussed. This is a hybrid physical process that couples the acoustic and elastic wave propagations. By delicately and subtly balancing the acoustic and elastic parameters as well as the geometry of the bubble, we show that Minnaert resonance can occur for rather general constructions. This study poses a great potential for the effective realisation of negative elastic materials by using bubbly elastic media.
