A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems

摘要: Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods indispensable. In this paper, we will introduce a framework for solution such problems. This comprises use adaptive finite element schemes, solvers large linear systems arising from discretization, and to treat additional information form inequality constraints on parameter be recovered. The developed based an all-at-once approach, which inverse problem is solved through Lagrangian formulation. main feature paper continuous (function space) setting formulate algorithms, order allow discretizations that adaptively refined as nonlinear iterations proceed. entails steps description Newton step or line search first formulated functions only then evaluated discrete functions. On other hand, approach avoids dependence dimensional norms mesh size, making individual algorithm comparable even if they used differently meshes. Numerical examples demonstrate applicability efficiency method with several million unknowns more than 10,000 parameters.