Solving Inverse Problems for Steady-State Equations using A Multiple Criteria Model with Collage Distance, Entropy, and Sparsity

TL;DR

This paper extends the collage theorem-based method for inverse steady-state problems by incorporating entropy and sparsity criteria, improving approximation quality and solution simplicity through a multi-criteria scalarization approach.

Contribution

It introduces a multi-criteria optimization framework combining collage error, entropy, and sparsity for inverse problems, with a scalarization technique for solution.

Findings

Entropy weighting improves approximation accuracy.

Sparsity reduces solution complexity.

Method yields good, though sub-optimal, results.

Abstract

In this paper, we extend the previous method for solving inverse problems for steady-state equations using the Generalized Collage Theorem by searching for an approximation that not only minimizes the collage error but also maximizes the entropy and minimize the sparsity. In this extended formulation, the parameter estimation minimization problem can be understood as a multiple criteria problem, with three different and conflicting criteria: The generalized collage error, the entropy associated with the unknown parameters, and the sparsity of the set of unknown parameters. We implement a scalarization technique to reduce the multiple criteria program to a single criterion one, by combining all objective functions with different trade-off weights. Numerical examples confirm that the collage method produces good, but sub-optimal, results. A relatively low-weighted entropy term allows for better approximations while the sparsity term decreases the complexity of the solution in terms of the number of elements in the basis.