Multi-Index Stochastic Collocation for PDEs with Imperfect Solvers
Please login to view abstract download link
This talk considers the construction of surrogate models (response surfaces) for parametric PDEs using multi-fidelity collocation methods, namely Multi-Index Stochastic Collocation (MISC). In some scenarios, in addition to “standard” discretization errors, the PDE approximations used to build a MISC response surface are affected by “noise” (e.g. due to iterative method tolerances, pre-asymptotic meshes, time-stepping). This noise is particularly problematic in low fidelity models; it might be parameter-dependent and hard to estimate and control a priori. Noise is interpolated by MISC and corrupts the approximated response surface (loss of monotonicity, spurious high-frequency oscillations), spoiling any subsequent UQ analysis. We propose an improved version of MISC that can detect such phenomena. Within our updated adaptive algorithm, at each iteration, we inspect the spectral content of the response surface and consequently stop exploring fidelities once the decay of their spectral coefficients stagnates due to such noise. Numerical experiments show the effectiveness of our approach in preventing the MISC approximation from becoming corrupted.