Efficient Numerical Methods For Inverse Problems

Inverse problems are often ill-posed in the sense that the solution may not exist and be unique, and more importantly, it fails to depend continuously on the data such that a small perturbation in the data may case an enormous de-viation of the solution. However, in practical applications, the data are always noisy and uncertain due to corruption by inherent measurement errors. Mean-while, the forward model may be imperfect and imprecise due to the presence of unmodeled physics. Therefore, the numerical solution of inverse problems is very challenging. Regularization methods are the standard approach for inverse problems. Algorithmically speaking, existing techniques roughly divide into two categories:deterministic and stochastic. There exist numerous mathematically elegant theoretical results and computationally efficient numerical algorithms for deterministic inverse techniques. However, they yield only a point estimate of the solution, without quantifying the associated uncertainties or rigorously considering the stochastic nature of data noise. Stochastic approaches are necessary for inverse problems under model uncertainties and for probabilistic calibration. This thesis attempts to design efficient numerical methods for the inverse heat source problems and the inverse Robin problems associated with the parabolic problem. It consist of three parts:Part 1 considers deterministic methods for the inverse heat source problem; Part 2 discusses Bayesian inference approach for inverse problems and Part 3 studies the stochastic collocation method via l1 minimization for stochastic partial differential equations.Part 1 discussed the use of the methods of fundamental solution (MFS) for reconstructing the unknown heat source in parabolic problems. The main idea of MFS is to approximate the unknown solution by a linear combination of fun-damental solutions whose singularities are located outsider the solution domain. Since the matrix arising from the MFS discretization is severely ill-conditioned, a regularization solution is obtained by employing the Tikhonov regularization, while the regularization parameter is determined by the GCV criterion. For nu- merical verification, several examples for solving inverse heat source problems with smooth and non-smooth geometries in two-and three dimensional space are given.Part 2 studies the Bayesian inference approach for uncertainly quantification of inverse Robin problems associated with the parabolic equation. With assump-tion on the prior belief about the form of the parameter and an assignment of normal error in sensor measurements, we derive the solution to the statistical in-verse problem analytically. A hierarchical Bayesian model is adopted for selecting the regularization parameter. The posterior probability density depends implicitly on the parameter through the forward model, and is exploring using the Markov chain Monte Carlo for obtaining relevant statistics. Then an augmented Tikhonov regularization(a-TR) method that could determine the regularization parameter and noise level is using to reconstruct the unknown heat source. Numerical results for several benchmark test problems are indicate that the Bayesian inference ap-proach and a-TR method are accurate and flexible methods for inverse problems.Part 3 consider a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output. The new approach differs from the standard stochastic collocation methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. We provide theoretical anal-ysis on the validity of the approach. Numerical tests are provided to examine the performance of the method and validate the theoretical finding. This opens an avenue for constructing stochastic surrogate models to accelerate the Bayesian inference approach for parameter estimation problems associated with partial dif-ferential equations.