An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs

Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, unknown is often assumed to be piecewise constant a given pixel partition (corresponding desired resolution), only finitely many can made. This leads problem inverting finite-dimensional non-linear forward operator $\mathcal F:\ \mathcal D(\mathcal F)\subseteq \mathbb R^n\to R^m$, where evaluating F$ requires one or several PDE solutions. Numerical inversion methods require implementation this Jacobian. We show how efficiently implement both using standard FEM package prove convergence approximations against their true-solution counterparts. present simple example codes for Comsol with Matlab Livelink package, numerically demonstrate challenges that arise non-uniqueness, non-linearity instability issues. also discuss monotonicity convexity properties symmetric measurement settings.