Numerical solutions of boundary inverse problems for some elliptic partial differential equations

In this dissertation, we study boundary inverse problems for some elliptic partial differential equations. These are problems arising from quantitative analysis of various non-destructive testing techniques in applications. In such a problem, we are interested in using boundary measurements of the solution to recover either an unknown coefficient function in the boundary condition, or a portion of the boundary, or an unknown interior interface. We first introduce formulations of the boundary value problems into integral equations, then design numerical algorithms for solving each of these inverse problems. Numerical implementation and examples are presented to illustrate the feasibility and effectiveness of the numerical methods.;Keywords. Robin inverse problem, inverse linear source problem, boundary integral equation, Tikhonov regularization, Nystrom method

Dimension-Independent MCMC Sampling for Inverse Problems with Non-Gaussian Priors

The computational complexity of MCMC methods for the exploration of complex probability measures is a challenging and important problem. A challenge of particular importance arises in Bayesian inverse problems where the target distribution may be supported on an infinite dimensional space. In practice this involves the approximation of measures defined on sequences of spaces of increasing dimension. Motivated by an elliptic inverse problem with non-Gaussian prior, we study the design of proposal chains for the Metropolis-Hastings algorithm with dimension independent performance. Dimension-independent bounds on the Monte-Carlo error of MCMC sampling for Gaussian prior measures have already been established. In this paper we provide a simple recipe to obtain these bounds for non-Gaussian prior measures. To illustrate the theory we consider an elliptic inverse problem arising in groundwater flow. We explicitly construct an efficient Metropolis-Hastings proposal based on local proposals, and we provide numerical evidence which supports the theory.Comment: 26 pages, 7 figure

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems

Robust preconditioners for PDE-constrained optimization with limited observations

Regularization robust preconditioners for PDE-constrained optimization problems have been successfully developed. These methods, however, typically assume that observation data is available throughout the entire domain of the state equation. For many inverse problems, this is an unrealistic assumption. In this paper we propose and analyze preconditioners for PDE-constrained optimization problems with limited observation data, e.g. observations are only available at the boundary of the solution domain. Our methods are robust with respect to both the regularization parameter and the mesh size. That is, the condition number of the preconditioned optimality system is uniformly bounded, independently of the size of these two parameters. We first consider a prototypical elliptic control problem and thereafter more general PDE-constrained optimization problems. Our theoretical findings are illuminated by several numerical results

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

In this article, we consider a Bayesian inverse problem associated to elliptic partial differential equations in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic sequential Monte Carlo (SMC) method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the SMC methods for inverse problems which were introduced in Kantas et al. (SIAM/ASA J Uncertain Quantif 2:464–489, 2014); the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology and its desirable theoretical properties, are demonstrated for numerical examples in both two and three dimensions