Solving inverse problems for variational equations using “generalized collage methods,” with applications to boundary value problems

A number of inverse problems involving deterministic and random differential equations may be viewed in terms of the problem of approximating a target element xx of a complete metric space (X,d)(X,d) by the fixed point x̄ of a contraction mapping T:X→XT:X→X. Most practical methods rely on a reformulation of this problem due to the “Collage Theorem,” a simple consequence of Banach’s Fixed Point Theorem: They search for a contraction mapping that minimizes the “collage distance” d(x,Tx)d(x,Tx). One may consider the collage method as a kind of regularization procedure for the inverse problem. In this paper, after recalling some applications of the Collage Theorem to the solution of inverse problems for fixed point equations and applications of it to initial value problems, with the help of the Lax–Milgram representation, we develop some generalizations of the collage method in order to solve inverse problems for variational equations. We consider both deterministic and stochastic problems. We then show some applications to inverse boundary value problems.