Solving inverse problems for PDEs in terms of Lax-Milgram functional and a generalized collage method

In this paper, we develop a general collage coding framework for inverse problems in partial differential equations (PDEs) with boundary conditions. Although a general PDEs inverse problem can be very complicated, via the Generalized Collage Theorem in this paper, many such problems can be reduced to an optimization problem which can be solved at least approximately. We study a general theory for variational formulation of PDEs and then we show an application to a one-dimensional steady-state diffusion equation. We give many numerical examples and we analyze stability results under perturbation of data.