Fully Coupled Domain Decomposition Methods for Inverse Elliptic Problems

Xiao-Chuan Cai, Si Liu, and Jun Zou

Email: cai@cs.colorado.edu
Postal address: Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309-0430

In this talk we discuss a class of parallel full space Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithms for inverse elliptic problems. In LNKSz, a Lagrangian functional is first formed according to the inverse elliptic problem with a proper regularization, and then differentiated to obtain an optimality system of nonlinear equations. Inexact Newton's method with line search is then applied directly to the fully coupled nonlinear optimality system and at each Newton's iteration the Jacobian system is solved with a Krylov subspace method preconditioned with an overlapping additive Schwarz method. We apply LNKSz to some parameter identification problems described as minimization problems constrained by elliptic partial differential equations. We report some promising results of a PETSc based parallel implementation of LNKSz for several different types of inverse problems.