Related papers: Convexification for an Inverse Parabolic Problem
We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense…
This review surveys previous and recent results on null controllability and inverse problems for parabolic systems with dynamic boundary conditions. We aim to demonstrate how classical methods such as Carleman estimates can be extended to…
The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit…
We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point,…
The inverse radiative transfer problem finds broad applications in medical imaging, atmospheric science, astronomy, and many other areas. This problem intends to recover the optical properties, denoted as absorption and scattering…
In this paper we consider the solution of monotone inverse problems using the particular example of a parameter identification problem for a semilinear parabolic PDE. For the regularized solution of this problem, we introduce a total…
In the development of controllability and inverse problem results for semi-discrete systems, by using Carleman estimates, it is required to estimate of the discrete operators applied to Carleman weight functions. This work aims to establish…
We investigate uniqueness in the inverse problem of reconstructing simultaneously a spacewise conductivity function and a heat source in the parabolic heat equation from the usual conditions of the direct problem and additional information…
There are two main approaches to solve inverse coefficient determination problems for wave equations: the Boundary Control method and an approach based on geometric optics. These notes focus on the Boundary Control method, but we will have…
An inverse problem of the determination of an initial condition in a hyperbolic equation from the lateral Cauchy data is considered. This problem has applications to the thermoacoustic tomography, as well as to linearized coefficient…
The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE…
A 3-D inverse medium problem in the frequency domain is considered. Another name for this problem is Coefficient Inverse Problem. The goal is to reconstruct spatially distributed dielectric constants from scattering data. Potential…
In the recent developments of regularization theory for inverse and ill-posed problems, a variational quasi-reversibility (QR) method has been designed to solve a class of time-reversed quasi-linear parabolic problems. Known as a PDE-based…
We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the non-homogeneous diffusion coefficient…
In this paper, we propose a novel, computationally efficient reduced order method to solve linear parabolic inverse source problems. Our approach provides accurate numerical solutions without relying on specific training data. The forward…
A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly…
This paper presents a concurrent global-local numerical method for solving multiscale parabolic equations in divergence form. The proposed method employs hybrid coefficient to provide accurate macroscopic information while preserving…
The main purpose of this work is to study an inverse coefficient problem for the telegrapher's equations on a tree-shaped network. To analyze the stability for this inverse problem, Carleman estimate is established first. Based upon this…
In this paper, a reconstruction method for the spatially distributed dielectric constant of a medium from the back scattering wave field in the frequency domain is considered. Our approach is to propose a globally convergent algorithm,…
This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.