Elenco seminari del ciclo di seminari
“SCUBE”

A nonlinear matrix is a matrix whose entries are nonlinear functions of a parameter. Such problems arise from physics (Schroedinger equation) and mechanical engineering (porous materials, boundary element method, e.g.). The last 20 years, rational approximation methods and linearization were proposed to approximate such matrices by linear pencils of much higher dimensions. Applications are the nonlinear eigenvalue problem, parametric linear systems, frequency sweeping, model order reduction and the solution of time dependent problems with nonlinear frequency dependencies. We give an overview of approximation methods with focus on AAA and Krylov methods that exploit the structure of the linear pencil.

Stochastic Galerkin (SG) methods provide a surrogate modelling technique for facilitating forward UQ in PDEs with uncertain inputs. Unlike conventional sampling methods, such as Monte Carlo, SG schemes yield approximations that are functions of the random input variables so that all realisations of the PDE solution are essentially approximated simultaneously. Since they use simple tensor product approximation spaces, standard SG schemes give rise to huge linear systems with coefficient matrices with a characteristic Kronecker product structure. Solving these systems is often a bottleneck when working on standard desktop computers. In this presentation, we will review two potential remedies. The first is to learn a lower-dimensional Galerkin approximation space using a bespoke a posteriori error estimator, leading to smaller linear systems that can be solved with modest computational resources. The second is to retain large tensor product spaces and recast the associated Kronecker system as a matrix equation. One can then apply low rank methods that iteratively construct a bespoke reduced basis and apply projection techniques to solve a problem of reduced size to determine a solution in factored form.