Related papers: An Inverse Source Problem for Semilinear Stochasti…
We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a…
We consider the first and half order time fractional equation with the zero initial condition. We investigate an inverse source problem of determining the time-independent source factor by the data at an arbitrarily fixed time and we…
This paper concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a…
The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose…
In this article, we propose a non-parametric Bayesian level-set method for simultaneous reconstruction of two different piecewise constant coefficients in an elliptic partial differential equation. We show that the Bayesian formulation of…
We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasi-reversibility method and a suitable Carleman weight…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
An approximately globally convergent numerical method for a 3d Coefficient Inverse Problem for a hyperbolic equation with backscattering data is presented. A new approximate mathematical model is presented. An approximation is used only on…
This paper concerns the inverse source scattering problems of recovering random sources for acoustic and elastic waves. The underlying sources are assumed to be random functions driven by an additive white noise. The inversion process aims…
We study the inverse source problem for the semilinear wave equation \[ (\Box_g + q_1)u + q_2 u^2 = F, \] on a globally hyperbolic Lorentzian manifold. We demonstrate that the coefficients $q_1$ and $q_2$, as well as the source term $F$,…
We establish Lipschitz stability for both the potential and the initial conditions from a single boundary measurement in the context of a hyperbolic boundary initial value problem. In our setting, the initial conditions are allowed to…
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and…
Carleman linearization is a technique that embeds systems of ordinary differential equations with polynomial nonlinearities into infinite dimensional linear systems in a procedural way. In this paper we generalize the method for systems of…
The inverse problem of estimating dielectric constants of explosives using boundary measurements of one component of the scattered electric field is addressed. It is formulated as a coefficient inverse problem for a hyperbolic differential…
We propose a data-assisted two-stage method for solving an inverse random source problem of the Helmholtz equation. In the first stage, the regularized Kaczmarz method is employed to generate initial approximations of the mean and variance…
This paper addresses null controllability for both forward and backward linear stochastic parabolic equations by introducing convection terms on the drift parts with bounded coefficients. Moreover, the forward stochastic parabolic equation…
The main objective of this paper is to establish the null controllability for the fourth order semilinear parabolic equations with the nonlinearities involving the state and its gradient up to second order. First of all, based on optimal…
Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems,…
We propose in this work a novel iterative direct sampling method for imaging moving inhomogeneities in parabolic problems using boundary measurements. It can efficiently identify the locations and shapes of moving inhomogeneities when very…
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…