  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.
