Solving inverse problems for differential equations by a “generalized collage” method and application to a mean field stochastic model

In the first part of this paper, after recalling how to solve inverse problems for deterministic and random differential equations using the collage method, we switch to the analysis of stochastic differential equations. Here inverse problems can be solved by minimizing the collage distance in an appropriate metric space. In the second part, we develop a general collage coding framework for inverse problems for boundary value problems. Although a general inverse problem can be very complicated, via the Generalized Collage Theorem presented in this paper, many such problems can be reduced to an optimization problem which can be solved at least approximately. We recall some previous results by some of the authors on the same topic, but we provide more numerical examples to analyze the stability of the generalized collage method under perturbation of data. We then extend these results to the case of diffusion equations. Finally, we show an application of this methodology to a system of coupled stochastic differential equations which describes the interaction between particles in a physical system.