Using the Generalized Collage Theorem for Estimating Unknown Parameters in Perturbed Mixed Variational Equations

TL;DR

This paper extends the Collage Theorem to perturbed mixed variational equations, providing existence, uniqueness, stability, and a Galerkin method, with applications to inverse problems and numerical validation.

Contribution

It introduces a generalized Collage Theorem for perturbed mixed variational problems, including stability analysis, a Galerkin approximation, and solutions to inverse problems using Schauder bases.

Findings

Existence and uniqueness of solutions under perturbations

Convergence of the Galerkin method

Effective solution of inverse problems with numerical examples

Abstract

In this paper, we study a mixed variational problem subject to perturbations, where the noise term is modelled by means of a bilinear form that has to be understood to be "small" in some sense. Indeed, we consider a family of such problems and provide a result that guarantees existence and uniqueness of the solution. Moreover, a stability condition for the solutions yields a Generalized Collage Theorem, which extends previous results by the same authors. We introduce the corresponding Galerkin method and study its convergence. We also analyze the associated inverse problem and we show how to solve it by means of the mentioned Generalized Collage Theorem and the use of adequate Schauder bases. Numerical examples show how the method works in a practical context.