A collage-based approach to solving inverse problems for second-order nonlinear parabolic PDEs

Abstract

The essence of collage-based methods for solving inverse problems is to bound the approximation error above by a more readily minimizable distance. The original collage method applies to ordinary differential equations (ODEs) and makes use of Banach’s fixed point theorem, and the collage theorem (named for its usefulness in fractal imaging) to build an upper bound on the approximation error. A similar technique was established for solving inverse problems for second-order linear elliptic partial differential equations (PDEs). In this setting the Lax–Milgram representation theorem is the driving force, and a generalized collage theorem has been established and can be used to control the approximation error. These ideas have been further extended to include nonlinear elliptic PDEs using the nonlinear Lax–Milgram representation theorem, a more general version of its linear counterpart. In this paper we develop a collage-based method for solving inverse problems for a general class of second-order nonlinear parabolic PDEs. We develop necessary background theory, discuss the complications introduced by the presence of time-dependence, establish sufficient conditions for using this method, and present examples of the method in practice.