Inverse source problem and inverse diffusion coefficient problem for parabolic equations with applications in geology

Abstract

This dissertation deals with inverse problems for parabolic partial differential equations with an integral constraint and their applications in geology. We employ analytical and numerical methods to analyze and approximate solutions of these inverse problems. We consider an inverse diffusion coefficient problem for a parabolic equation that arises in science and engineering, specifically, in geochronology, a branch of geology in which radiometric ages are determined for rock formations and geological events. We investigate the corresponding direct problem, where we use a finite element approximation for a 2D model to describe the distribution of argon in mica crystals. Using a fixed point method, we show that the inverse coefficient problem has a unique classical solution that depends continuously on the data. We also consider an inverse source problem for a parabolic equation with Dirichlet and Neumann boundary conditions. We prove the existence of both weak and classical solutions to the inverse source problem subject to the Dirichlet boundary condition by means of the Rothe method and the semigroup method, respectively. An energy method is used to show uniqueness. In addition, we establish the existence and uniqueness of solutions to the inverse source problem with the Neumann boundary condition. We derive and implement numerical schemes that are used to compute approximate solutions of the inverse diffusion coefficient problem and solutions of the inverse source problem with the Dirichlet boundary condition. Our numerical algorithms employ a finite element discretization in space and the implicit Euler method in time. To assess the accuracy of the approximation, we compute the errors and estimate the rates of convergence. We conducted numerical experiments using the mathematical programming software MATLAB. We finally outline a discussion of possible future work.