Parallel overlapping domain decomposition methods for coupled inverse elliptic problems

Abstract

We study an overlapping domain decomposition method for solving the coupled nonlinear system of equations arising from the discretization of inverse elliptic problems. Most algorithms for solving inverse problems take advantage of the fact that the optimality system has a natural splitting into three components: the state equation for the constraints, the adjoint equation for the Lagrange multipliers, and the equation for the parameter to be identified. Such algorithms often involve interiterations between the three separate solvers, and the intercomponent iteration is sequential. Several fully coupled or so-called one-shot approaches exist, and the main challenges in these approaches are that the system has stronger nonlinearity, and the corresponding Jacobian system is more ill-conditioned, in addition to being three times larger. Here we investigate a class of overlapping Newton–Krylov–Schwarz algorithms for solving such coupled systems, obtained with a pointwise ordering of the variables, and show numerically that, with a reasonably large overlap, the algorithm is capable of finding the solution even with noise and discontinuous coefficients. More importantly, we show that this approach is fully parallel and scalable with respect to the size of the problems.