Inverse scattering problems of the reconstructions of physical properties of a medium from boundary measurements are substantially challenging ones. This work aims to verify the performance on experimental data of a newly developed convexification method for a 3D coefficient inverse problem for the case of objects buried in a sandbox a fixed frequency and the point source moving along an interval of a straight line. Using a special Fourier basis, the method of this work strongly relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic equations. This problem, in turn, is solved via the minimization of a Tikhonov-like functional weighted by a Carleman Weight Function. The global convergence of the numerical procedure is established analytically. The numerical verification is performed using experimental data, which are raw backscatter data of the electric field. These data were collected using a microwave scattering facility at The University of North Carolina at Charlotte.
An $\left( n+1\right) -$D coefficient inverse problem for the radiative stationary transport equation is considered for the first time. A globally convergent so-called convexification numerical \ method is developed and its convergence analysis is provided. The analysis is based on a Carleman estimate. In particular, convergence analysis implies a certain uniqueness theorem. Extensive numerical studies in the 2-D case are presented.
A convexification-based numerical method for a Coefficient Inverse Problem for a parabolic PDE is presented. The key element of this method is the presence of the so-called Carleman Weight Function in the numerical scheme. Convergence analysis ensures the global convergence of this method, as opposed to the local convergence of the conventional least squares minimization techniques. Numerical results demonstrate a good performance.
For the first time, we develop in this paper the globally convergent convexification numerical method for a Coefficient Inverse Problem for the 3D Helmholtz equation for the case when the backscattering data are generated by a point source running along an interval of a straight line and the wavenumber is fixed. Thus, by varying the wavenumber, one can reconstruct the dielectric constant depending not only on spatial variables but the wavenumber (i.e. frequency) as well. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasilinear elliptic partial dif…
A Coefficient Inverse Problem for the radiative transport equation is considered. The globally convergent numerical method, the so-called convexification, is developed. For the first time, the viscosity solution is considered for a boundary value problem for the resulting system of two coupled partial differential equations. A Lipschitz stability estimate is proved for this boundary value problem using a Carleman estimate for the Laplace operator. Next, the global convergence analysis is provided via that Carleman estimate. Results of numerical experiments demonstrate a high computational effi…
A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonov-like functional. This functional is strictly convex on a certain bounded set in a Hilbert space, and the diameter of this set is an arbitrary positive number. The global convergence of the gradient projection method is established. Computational results dem…
The forward problem here is the Cauchy problem for a 1D hyperbolic PDE with a variable coefficient in the principal part of the operator. That coefficient is the spatially distributed dielectric constant. The inverse problem consists of the recovery of that dielectric constant from backscattering boundary measurements. The data depend on one variable, which is time. To address this problem, a new version of the convexification method is analytically developed. The theory guarantees the global convergence of this method. Numerical testing is conducted for both computationally simulated and expe…
To compute the spatially distributed dielectric constant from the backscattering data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve the inverse problem, we establish a new version of Carleman estimate and then employ this estimate to construct a cost functional which is strictly convex on a convex bounded set with an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and is considered as the central analytical result of this paper. Minimizing this convex functional by the …
This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation …
This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman Weight Function in it. In the previous works the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave fi…
This work extends the applicability of our recent convexification-based algorithm for constructing images of the dielectric constant of buried or occluded target. We are orientated towards the detection of explosive-like targets such as antipersonnel land mines and improvised explosive devices in the non-invasive inspections of buildings. In our previous work, the method is posed in the perspective that we use multiple source locations running along a line of source to get a 2D image of the dielectric function. Mathematically, we solve a 1D coefficient inverse problem for a hyperbolic equation…
A version of the convexification globally convergent numerical method is constructed for a coefficient inverse problem for a wave-like partial differential equation. The presence of the Carleman Weight Function in the corresponding Tikhonov-like cost functional ensures the global strict convexity of this functional. Numerical results are presented to illustrate the effectiveness and efficiency of the proposed method.
This paper is concerned with the study of a version of the globally convergent convexification method with direct application to synthetic aperture radar (SAR) imaging. Results of numerical testing are presented for experimentally collected data for a fake landmine. The SAR imaging technique is a common tool used to create maps of parts of the surface of the Earth or other planets. Recently, it has been applied in the context of non-invasive inspections of buildings in military and civilian services. Nowadays, any SAR imaging software is based on the Born approximation, which is a linearizatio…
This report extends our recent progress in tackling a challenging 3D inverse scattering problem governed by the Helmholtz equation. Our target application is to reconstruct dielectric constants, electric conductivities and shapes of front surfaces of objects buried very closely under the ground. These objects mimic explosives, like, e.g., antipersonnel land mines and improvised explosive devices. We solve a coefficient inverse problem with the backscattering data generated by a moving source at a fixed frequency. This scenario has been studied so far by our newly developed convexification meth…
Error analysis of the space-time interface-fitted finite element method for an inverse source problem for an advection-diffusion equation with moving subdomains
A frequency-domain method to inverse moving source problem with unknown radiating moment
School of Mathematical Sciences, Sichuan Normal University（四川师范大学数学科学学院）；School of Mathematical Sciences and Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学科学学院与数学交叉科学研究所）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院与理论物理研究中心）
An inverse scattering problem for SAR data in application to through-the-wall imaging is addressed. In contrast with the conventional algorithms of SAR imaging, that work with the linearized mathematical model based on the Born approximation, the fully nonlinear case is considered here. To avoid the local minima problem, the so-called "convexification" globally convergent inversion scheme is applied to approximate the distribution of the slant range (SR) dielectric constant in the 3-D domain. The benchmark scene of this paper comprises a homogeneous dielectric wall and different dielectric tar…
This article is concerned with the derivation of numerical reconstruction schemes for the inverse moving source problem on determining source profiles in (time-fractional) evolution equations. As a continuation of the theoretical result on the uniqueness, we adopt a minimization procedure with regularization to construct iterative thresholding schemes for the reduced backward problems on recovering one or two unknown initial value(s). Moreover, an elliptic approach is proposed to solve a convection equation in the case of two profiles.
We propose an eigensolver and the corresponding package, GCGE, for solving large scale eigenvalue problems. This method is the combination of damping idea, subspace projection method and inverse power method with dynamic shifts. To reduce the dimensions of projection subspaces, a moving mechanism is developed when the number of desired eigenpairs is large. The numerical methods, implementing techniques and the structure of the package are presented. Plenty of numerical results are provided to demonstrate the efficiency, stability and scalability of the concerned eigensolver and the package GCG…
This paper is concerned with the numerical simulation of three dimensional time-dependent inverse source problems of acoustic waves. The reconstructions of both multiple stationary point sources and a moving point source are considered. The modified method of fundamental solutions (MMFS), which expands the solution utilizing the time convolution of the Green's function and the signal function, is proposed to solve the problem. For the reconstruction of a moving point source, moreover, the MMFS is simplified as a simple sampling method at each time step. Numerical experiments are provided to sh…
Two-level space-time domain decomposition methods for unsteady inverse problems
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；University of Colorado Boulder（科罗拉多大学博尔德分校）；The Chinese University of Hong Kong（香港中文大学）
A Trace--Logarithmic Variational Functional for Equidistribution and Alignment in Moving Mesh Adaptation
Xiangtan University（湘潭大学）；Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Xiangtan University（湘潭大学基础学科基本算法理论与新型计算方法湖南省研究中心）；Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University（湘潭大学计算科学湖南省国际科技合作基地、教育部智能计算与信息处理重点实验室）；National Center of Applied Mathematics in Hunan, Hunan Key Laboratory for Computation and Simulation in Science and Engineering（湖南应用数学中心、湖南省计算与仿真科学重点实验室）
Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we fo…
