On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

Abstract

Parameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems (in divergence form) with Dirichlet, Neumann, Robin or mixed boundary conditions. In particular, we show that the corresponding parameter-to-state operators fulfil, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography.

A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations

“…Furthermore, u depends continuously on both a and F . In particular: ii) Given a ∈ B c (V, W ) and F ∈ W * , the existence and uniqueness of u a,F follow from a Banach space version of the Lions-Lax-Milgram theorem (see Lemma 3.1. in [34]). Furthermore, one also has the stability estimate ||u a,F || V ≤ (λ(a)) −1 ||F || * .…”

“…In general, this corresponds to exploring the so-called solution manifold [23,41], that is S := {u µ } µ∈Θ . The map µ → u µ is known under many equivalent names such as the parametric map [55], the parameter-to-state map [34] or the solution map [50]. Approximating the parametric map in a highly-efficient way is a challenging task that can be encountered in several contexts, from optimal control problems with parametric PDEs constraints [8] to multiscale fluid mechanics [37], or Bayesian inversion and uncertainty quantification [11].…”