Numerical Gradient Computation for Simultaneous Detection of Geometry and Spatial Random Fields in a Statistical Framework

Abstract

The target of this chapter is the evaluation of gradients in inverse problems where spatial field parameters and geometry parameters are treated separately. Such an approach can be beneficial especially when the geometry needs to be detected accurately using L2-norm-based regularization. Emphasis is laid upon the computation of the gradients directly from the governing equations. Working in a statistical framework, the Karhunen-Loève (K-L) expansion is used for discretization of the spatial random field and inversion is done using the gradient-based Hamiltonian Monte Carlo (HMC) algorithm. The HMC gradients involve sensitivities w.r.t the random spatial field and geometry parameters. Building on a method developed by the authors, a procedure is developed which considers the gradients of the associated integral eigenvalue problem (IEVP) as well as the interaction between the gradients w.r.t random spatial field parameters and the gradients w.r.t the geometry parameters. The same mesh and linear shape functions are used in the finite element method employed to solve the forward problem, the artificial elastic deformation problem and the IEVP. Analysis of the rate of convergence using seven different meshes of increasing density indicates a linear rate of convergence of the gradients of the log posterior.