Identification of Sparsely Representable Diffusion Parameters in Elliptic Problems

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 61-90, March 2024.
Abstract. We consider the task of estimating the unknown diffusion parameter in an elliptic PDE as a model problem to develop and test the effectiveness and robustness to noise of reconstruction schemes with sparsity regularization. To this end, the model problem is recast as a nonlinear infinite dimensional optimization problem, where the logarithm of the unknown diffusion parameter is modeled using a linear combination of the elements of a dictionary, i.e., a known bounded sequence of [math] functions, with unknown coefficients that form a sequence in [math]. We show that the regularization of this nonlinear optimization problem using a weighted [math]-norm has minimizers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimizer of this weighted [math]-norm regularized nonlinear optimization problem that accounts for the fact that in general the smooth part of the functional being optimized is a functional only defined over [math]. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimizer, which locally uses Gauss–Newton type surrogate models that additionally are stabilized by means of a Levenberg–Marquardt type approach. Our numerical examples show that the regularization with the weighted [math]-norm indeed does make the estimation more robust with respect to noise. Moreover, the numerical examples also demonstrate that the ASISTA and FASISTA methods are quite efficient, outperforming both ISTA and FISTA.