Applied Mathematics Seminar | Davide La Torre, Inverse Problems for Differential Equations via the "Collage Theorem”: Recent Results and Future Directions

Many inverse problems may be recast as the approximation of a target element x in a complete metric space (X,d) by the fixed point x* of a contraction mapping T:X->X. Thanks to a simple consequence of Banach's Fixed Point Theorem known as the Collage Theorem, most practical methods of solving the inverse problem for fixed point equations seek an operator T for which the collage distance d(x,Tx) is as small as possible. This is the essence of the method of collage coding which has been the basis of most, if not all, fractal image coding and compression methods [11].

This method may be applied in other situations where contractive mappings are encountered: These ideas have been extended to inverse problems for Initial Value Problems [6], [8], [10], [14],[15].

In a manner analogous to the Collage Theorem, the Generalized Collage Theorem has been formulated for solving Boundary Value Problems [7], [12],[13], replacing the minimization of the true error by the minimization of something akin to the collage distance. In place of Banach's fixed point theorem for contraction maps on a complete metric space, we have appealed to the Lax-Milgram representation theorem.

These results have been recently extended to a wider class of partial differential equations [3],[9] and boundary valued problems by considering not only Hilbert but also reflexive Banach spaces, and even replacing the primal variational formulation of such a problem with a more general constrained variational one [1],[5]. Other recent extensions of the Generalized Collage Theorem deal with inverse problems for vector-valued Lax-Milgram systems [2] and inverse problems on perforated domains [4].

Main references:

[1]  M.I. Berenguer, H. Kunze, D. La Torre, M. Ruiz Galan (2016)  Galerkin method for constrained variational equations and a collage-based approach to related inverse problems, Journal of Computational and Applied Mathematics, 292, 67–75.

[2]  H. Kunze, D. La Torre, K. Levere, M. Ruiz Galan (2015) Inverse problems via the ”Generalized Collage Theorem“ for vector-valued Lax Milgram-based variational problems (2015), Mathematical Problems in Engineering, Volume 2015, Article ID 764643, 8 pages.

[3]  K.Levere, H.Kunze, D. La Torre (2015), A collage-based approach to solving inverse problems for second-order nonlinear hyperbolic PDEs, Communications in Nonlinear Science and Numerical Simulations, 29 (1–3), 283–299.

[4] H.Kunze, D. La Torre (2015), Collage-type approach to inverse problems for elliptic PDEs on perforated domains (2015), Electronic Journal of Differential Equations, 48, 1-11.

[5] M.I. Berenguer, H. Kunze, D. La Torre, M. Ruiz Galan (2015), Galerkin schemes and inverse boundary value problems in reflexive Banach spaces (2015), Journal of Computational and Applied Mathematics, 275, 100-112.

[6]  H.Kunze, D. La Torre, F.Mendivil, M.Ruiz Galan, R.Zaki (2014), Fractal-based methods and inverse problems for differential equations: current state-of-the-art (2014),  Mathematical Problems in Engineering, Volume 2014, Article ID 737694, 11 pages.

[7] V.Capasso, H.Kunze, D. La Torre, E.R.Vrscay (2014), Solving inverse problems for differential equations by a ``generalized collage'' method and application to a mean field stochastic model (2014), Nonlinear Analysis: Real World Applications. 15, 276-289.

[8] V.Capasso, H.Kunze, D. La Torre, E.R.Vrscay (2013), Solving inverse problems for biological models using the collage method for differential equations, Journal of Mathematical Biology, 67, 1, 25-38.

[9] K.Levere, H.Kunze, D. La Torre, A collage-based approach to solving inverse problems for second-order nonlinear parabolic PDEs (2013),  Journal of Mathematical Analysis and Applications, 406, 1, 120-133.

[10] H.Kunze, D. La Torre, E.R.Vrscay (2012), Solving inverse problems for DEs using the collage theorem and entropy maximization, Applied Mathematics Letters, 25, 2306-2311.

[11] H.Kunze, D. La Torre, F.Mendivil, E.R.Vrscay (2012), Fractal-based methods in analysis, Springer, ISBN 1461418909.

[12] H.Kunze, D. La Torre, E.R. Vrscay (2010), Solving inverse problems for variational equations using "generalized collage methods," with applications to boundary value problems,  (2010), Nonlinear Analysis Real World Applications, 11 (5), 3734-3743.

[13]  H.Kunze, D. La Torre, E.R. Vrscay (2009), A generalized collage method based upon the Lax-Milgram functional for solving boundary value inverse problems, Nonlinear Analysis: Theory, Methods \& Applications, 71 (12), e1337-e1343.

[14] H.Kunze, D. La Torre, E.R. Vrscay (2009), Inverse problems for random differential equations using the collage method for random contraction mappings, Journal of Computational and Applied Mathematics, 223 (2), 853-861.

[15] H.Kunze, D. La Torre, E.R. Vrscay (2007), Random fixed point equations and inverse problems using "the collage method" for contraction mappings,  Journal of Mathematical Analysis and Applications, 334 (2), 1116--1129.