Identification of sparsely representable diffusion parameters in elliptic problems

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 regularisation. To this end, the model problem is recasted as a nonlinear optimal control problem, where the unknown diffusion parameter is modelled using a linear combination of the elements of a known bounded sequence of functions with unknown coefficients. We show that the regularisation of this nonlinear optimal control problem using a weighted $\ell^1$-norm has minimisers that are finitely supported. We then propose modifications of well-known algorithms (ISTA and FISTA) to find a minimiser of this weighted $\ell^1$-norm regularised nonlinear optimal control problem that account for the fact that in general the coefficients need to be $\ell^1$ and not only $\ell^2$ summable. We also introduce semismooth methods (ASISTA and FASISTA) for finding a minimiser, which locally use Gauss-Newton type surrogate models that additionally are stabilised by means of a Levenberg-Marquardt type approach. Our numerical examples show that the regularisation with the weighted $\ell^1$-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.