How to lighten parametric inverse computations if the misfit is non-convex and the forward solver needs stabilization

Abstract

Solving parametric inverse problems (IPs) for partial differential equations (PDEs) plays an important role in medical diagnosis, resource investigation, nondestructive testing and many other human activities. Real-world IPs formulated as misfit functional minimization tasks are frequently ill-conditioned. The origins of this ill-posedness are the misfit multimodality and insensitivity as well as the instability of the utilized numerical PDE solver. We propose a complex multi-population memetic strategy HMS combined with the Petrov–Galerkin method stabilized by the Demkowicz operator to overcome these obstacles. The paper delivers a rigorous mathematical formulation of the strategy as well as a theoretical motivation for common inverse/forward error scaling, which significantly reduces the computational cost of the strategy. The presented theory is illustrated with two examples. The first one shows the analytical construction of the Demkowicz operator. The second one is an application of HMS in solving an IP utilizing the approximate stabilization with a forward solver that uses the isogeometric residual minimization (iGRM) method.