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

Abstract

A goal of many inverse problems is to find unknown parameter values, λ ∈ Λ, so that the given observed data utrue agrees well with the solution data produced using these parameters uλ. Unfortunately finding uλ in terms of the parameters of the problem may be a difficult or even impossible task. Further, the objective function may be a complicated function of the parameters λ ∈ Λ and may require complex minimization techniques. In recent literature, the collage coding approach to solving inverse problems has emerged. This approach avoids the aforementioned difficulties by bounding the approximation error above by a more readily minimizable distance, thus making the approximation error small. The first of these methods was applied to first-order ordinary differential equations and gets its name from the “collage theorem” used in this setting to achieve an upperbound on the approximation error. A number of related ODE problems have been solved using this method and extensions thereof. More recently, collage-based methods for solving linear and nonlinear elliptic partial differential equations have been developed. In this paper we establish a collage-based method for solving inverse problems for nonlinear hyperbolic PDEs. We develop the necessary background material, discuss the complications introduced by the presence of time-dependence, establish sufficient conditions for using the collage-based approach in this setting and present examples of the theory in practice.