Research-Young Scholar Workshop on Inverse Problems, Imaging and PDEs

Abstracts

Abstracts

1. CAI Yongyong, Beijing Computational Science Research Center

Title: Non-relativistic limit of the nonlinear Dirac equation and its numerical methods

Abstract: We consider the nonlinear Dirac equation in the non-relativistic limit regime, involving a small parameter inversely proportional to the speed of light. The nonlinear Dirac equation converges to the nonlinear Schroedinger equation in the non-relativistic limit. By a careful analysis, we obtain a semi-relativistic limit of the nonlinear Dirac equation, which enables a design of uniformly accurate multi-scale numerical method. The major difficulty of the problem is that the solution has a rapid oscillation in time depending on the small parameter.

2. CHEN Junqing, Tsinghua University

Title: A direct imaging method for inverse electromagnetic scattering problem in waveguide

Abstract: I will talk about a direct imaging method based on reverse time migration (RTM) algorithm for imaging extended targets using electromagnetic waves at a fixed frequency in the rectangular waveguide. The imaging functional is defined as the imaginary part of the cross-correlation of the Green function for Helmholtz equation and the back-propagated electromagnetic field. The resolution of our RTM method for penetrable extended targets is studied by virtue of Helmholtz-Kirchhoff identity in the rectangular domain, which implies that the imaging functional always peaks in the target. Numerical examples are provided to demonstrate the powerful imaging quality and confirm our theoretical results.

3. DENG Youjun, Central South University

Title: Simultaneous recovery of an internal current and the surrounding medium in Maxwell's system

Abstract: We consider the inverse problem of recovering both an unknown electric current and the surrounding electromagnetic parameters of a medium from boundary measurements. This inverse problem arises in brain imaging. We show that we can recover both the source and the electromagnetic parameters if these are piecewise constant and the current source is invariant in a fixed direction or a harmonic function.

4. GONG Rongfang, Nanjing University of Aeronautics and Astronautics

Title: A modified coupled complex boundary method for an inverse chromatography problem

Abstract: Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes.

In this talk, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem, [Cheng, et al. A novel coupled complex boundary method for inverse source problems, Inverse Problems, 2014, 30, 055002]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method. This is a joint work with Xiaoliang Cheng and Guangliang Lin of Zhejiang University, China, Marten. Gulliksson and Ye Zhang of Orebro University, Sweden.

5. GUO Ling, Shanghai Normal University

Title: Stochastic collocation methods via nonconvex minimizations and its applications in UQ

Abstract: Stochastic computation has received intensive attention in recent years, due to the pressing need to conduct uncertainty quantification (UQ) in practical computing. One of the most widely used techniques in UQ is generalized polynomial chaos. In this talk, we will discuss collocation method via nonconvex minimizations for recovering sparse polynomial chaos expansions (PCE). We present several theoretical estimates regarding the recoverability for both sparse and nonsparse signals via the transformed L1 minimization. We then apply the method to sparse orthogonal polynomial approximations for stochastic collocation. Various numerical examples, including sparse polynomial functions and non-sparse analytical functions recovery, demonstrate the recoverability and efficiency of the method.

6. GUO Yukun, Harbin Institute of Technology

Title: Fourier method for the inverse source problem with multi-frequency far-field data.

Abstract: This talk is concerned with an inverse source problem of determining a source term in the Helmholtz equation from multi-frequency far-field measurements. Based on the Fourier series expansion, we develop a novel non-iterative reconstruction method for solving the problem. A promising feature of this method is that it utilizes the data from only a few observation directions for each frequency. Theoretical uniqueness and stability analysis will be provided. Numerical experiments will be presented to illustrate the effectiveness and efficiency of the proposed method in both two and three dimensions. This is a joint work with Xianchao Wang, Deyue Zhang and Hongyu Liu.

7. JI Xia, Institute of Computational Math, CAS

Title: A multi-correction method for the source identification problem

Abstract: We design a multi-correction method for the source identification problems in which the state variable is the solution of the 2D variable-coefficient Poisson problem with full and partial observations. In this method, we replace solving the PDE constrained optimization problem on the finest finite element space by solving a series of linear boundary value problems with multigrid scheme in the corresponding series of finite element spaces and a series of PDE constrained optimization problems in the coarsest finite element space. Based on the analysis, the proposed method can obtain optimal errors with an almost optimal computational complexity.

8. JIANG Daijun, Central China Normal University

Title: Quadratic convergence of L-M method for parabolic inverse Robin problems and DDMs for convex minimizations

Abstract: Consider the interior transmission problem arising in inverse boundary value problems for the heat equation with discontinuous heat conductivity. The unique solvability of this problem is justified in the following way. First, we construct the local parametrix for the interior transmission problem in the Laplace domain, by using the theory of pseudodifferential operators with a large parameter. Second, the local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all local parametrices and the fundamental solution of the heat operator to have a parametrix for the parabolic interior transmission problem, and then compensate it to get the Green function. Thus, the unique solvability of our interior transmission problem can be proved in a standard way. Based on this result, we develop a sampling method for an inverse boundary value problem for the heat equation.

9. XIAO Jingni, Hong Kong Baptist University

Title: On Electromagnetic Scattering from a Penetrable Corner

Abstract: We show that an inhomogeneous electromagnetic(EM) medium with a right corner will always scatter the incident waves, by assuming certain regularity conditions. This result implies that one cannot make such a medium with corner invisible. This is a joint work with Hongyu Liu.

10. XIE Feng, Shanghai Jiao Tong University

Title: MHD Boundary layers in Sobolev spaces without monotonicity.

Abstract: We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl type equations that are derived from the incompressible MHD system with nonslip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-in-time existence, uniqueness of solution for the nonlinear MHD boundary layer equations. Compared with the well-posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. This talk is based on the joint work with Dr. Chengjie Liu and Professor Tong Yang from City U of Hong Kong.

11. XU Zhenli, Shanghai Jiao Tong University

Title: Ion distribution near layered dielectric media

Abstract: We develop a fast algorithm for the ion interaction in media with layered dielectric structure, in particular, a core-shell structured sphere. The algorithm is derived by analytically solving the Green’s function, which represents the solution as an image line charge and thus it can be approximated by point charges using numerical quadrature. Based on this algorithm, we performed Monte Carlo simulations for ion structure near a core-shell dielectric nanoparticle, and observed that the ion distribution significantly depends on the dielectric ratios and the shell thickness. Monte Carlo simulations show the influence of the layered structure on the capacitance of the electric double layer.

12. LAI Junjiang, Minjiang University

Title: Some numerical experiments for anomalous localized resonance of the elastostatic system

Abstract: In this talk, we give some numerical experiments for anomalous localized resonance of the elastostatic system. These numerical results confirm some related anomalous localized resonance theory, such as resonance with no core and resonance for a core. We also obtain some new conclusions through these experiments. This is joint work with Hongyu Liu.

13. LI Hongjie, Hong Kong Baptist University

Title: On anomalous localized resonance for the elastostatic system

Abstract: In this talk, I will talk about our recent results on anomalous localized resonances (ALR) due to a plasmonic structure for the elastostatic system in two and three dimensions. I shall first talk about our study from a variational perspective. Then, I will discuss the ALR from a spectral perspective. Since the Neumann-Poincare(N-P) operator for the Lame system is not compact even if the domain has a smooth boundary, the main difficulty is to calculate the spectrum of that operator. Using the spectrum of N-P operator, the occurrence and non-occurrence of ALR will be discussed in three dimensions.

14. LI Jinglai, Shanghai Jiao Tong University

Title: Bayesian inference and uncertainty quantification for infinite dimensional inverse problems

Abstract: Bayesian inference has become increasingly popular as a tool to solve inverse problems, largely due to its ability to quantify the uncertainty in the solutions obtained. In many practical problems such as image reconstructions, the unknowns are often of infinite dimension, i.e., functions of space and/or time. Theories and methods developed for finite dimensional problems may become problematic in the infinite dimensional setting and thus new theories and methods must be developed for such problems. In this talk we shall discuss several critical issues associated with the infinite dimensional problems and some efforts made to address them. First we discuss the Maximum a Posterior (MAP) estimation in this setting and its numerical implementations. Next we introduce a non-Gaussian prior for modeling functions that are subject to sharp jumps. We then present an efficient adaptive MCMC algorithm that is specifically designed for function space inference. Finally, we apply the Bayesian inference methods to a medical image reconstruction problem.

15. LIU Keji, Shanghai University of Finance and Economics

Title: Direct recovery of wave-penetrable scatterers in a stratified ocean waveguide

Abstract: We shall extend the direct sampling method for more general recovery of wave-penetrable inhomogeneous scatterers in a three-dimensional stratified ocean waveguide. The proposed method is simple, straightforward and computationally efficient since no matrix inversions, optimizations or solutions of ill-posed linear systems are involved. Moreover, it is highly tolerant to noise and is applicable with a few scattered fields corresponding to only two or four incidences. In addition, a priori knowledge of either the number of disconnected components or the physical features of the unknown obstacles is not required by the direct sampling method. A mathematical derivation is provided for its validation and several extensive numerical experiments are shown to evaluate the effectiveness and feasibility of the newly proposed method.

16. LIU Xiaodong, Institute of Applied Math, CAS

Title: Recovery of an embedded obstacle and its surrounding medium by formally-determined scattering data

Abstract: We consider an inverse acoustic scattering problem in simultaneously recovering an embedded obstacle and its surrounding inhomogeneous medium by formally determined far field data. It is shown that the knowledge of the scattering amplitude with a fixed incident direction and all observation angles along with frequencies from an open interval can be used to uniquely identify the embedded obstacle, sound-soft or sound-hard disregarding the surrounding medium. Furthermore, if the surrounding inhomogeneous medium is from an admissible class (still general), then the medium can be recovered as well. Our argument is based on deriving certain integral identities involving the unknowns and then inverting them by certain harmonic analysis techniques. Finally, based on our theoretical study, a fast and robust sampling method is proposed to reconstruct the shape and location of the buried targets and the support of the surrounding inhomogeneities. This is a joint work with Prof. Hongyu Liu from HongKong Baptist University.

17. WANG Haibing, Southeast University

Title: Solvability of interior transmission problem for the heat equation with applications

18. WANG Yuliang, Hong Kong Baptist University

Title: Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking

Abstract: This talk is concerned with the invisibility cloaking in electromagnetic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. Our study is based on an interior transmission eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that there exists an infinite set of incident waves such that the cloaking device is nearly-invisible under the corresponding wave interrogation. The set of waves is generated from the Maxwell-Herglotz approximation of the associated interior transmission eigenfunctions. We provide the mathematical design of the cloaking device and sharply quantify the cloaking performance.

19. XIE Chunjing, Shanghai Jiao Tong University

Title: Steady subsonic Euler flows in nozzles or past a body

Abstract: We first discuss the well-posedness theory for subsonic Euler flows in nozzles or past a symmetric body. One of the key observations to achieve these results is that the streamlines in these flows have simple topological structure. This helps us to introduce the stream function

formulation so that the compressible Euler system can be reduced into a single second order equation. Finally, we will also discuss some related open problems.

20. YANG Jiaqing, Xi’an Jiaotong University　

Title: Inverse obstacle scattering in a two-layered medium with locally rough surfaces

Abstract: In this talk, I will discuss an inverse scattering problem by an impenetrable obstacle in a two-layered medium with locally rough surfaces. The global uniqueness result on the surface, buried obstacles and the wavenumber is obtained from the knowledge of near-field data measured only on the upper-half space. Moreover, a sampling method is developed to numerically solve the inverse problem of recovering the local rough surface using such data. This is a join work with Dr. Jianliang Li and Prof. Bo Zhang.

21. YING Wenjun, Shanghai Jiao Tong University

Title: A simple method for computing singular or nearly singular integrals on closed surfaces

Abstract: In this talk, I will present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is first computed with a regularized kernel and then discretized with a new quadrature using surface points which project onto grid points in coordinate planes. Leading order terms of the regularization and discretization errors are computed from asymptotic analysis near the singular point and added to the computed value so that the result has high order accuracy. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The method is accelerated by the treecode algorithm of Duan and Krasny for Ewald summation. I will also present numerical examples with different surfaces.

22. ZHANG Haiwen, Institute of Applied Math, CAS

Title: A direct imaging method for inverse scattering problem by unbounded rough surfaces: the case of incident point sources

Abstract: In this talk, we consider the inverse acoustic scattering problems by an unbounded rough surface. A direct imaging method is proposed to reconstruct the rough surfaces from Cauchy data of the scattered field for incident point sources. The performance analysis is also studied by using integral equation method. The reconstruction method is very robust to noises of measured data and works for penetrable surfaces and impenetrable surfaces with Dirichlet or impedance boundary conditions. Finally, numerical examples are carried out to illustrate that our method is fast, accurate and stable.