European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   Uncertainty Quantification and Stochastic Models (Part 2)
Chair: Fred Wubs
25 mins
An Improved Multilevel Particle Filter
Marco Ballesio, Ajay Jasra, Erik Von Schwerin, Raul Tempone
Abstract: Hidden Markov models have a wide range of applications in engineering, finance and weather forecasting. We consider filtering problems connected to partially observed diffusions that are regularly observed at discrete times. In most cases of practical interest, a particle filter (PF) is used to evaluate the filtering distribution. A multilevel particle filter (MLPF) uses a hierarchy of discretizations to reduce the cost of the PF. It is essential for MLPFs that the resamplings, which are performed in PFs when the effective sample size becomes too small, do not destroy the coupling between the particles on coarse-fine discretization pairs; see e.g. [1]. We suggest an alternative resampling method, based on optimal Wasserstein coupling, that outperforms existing MLPFs in certain cases. In many applications of the MLPFs a change of measure technique, similar to [2], is used to overcome the instability of the underlying diffusion process. [1] A. Jasra, K. J. H. Law, K. Katamani, Y. Zhou, Multilevel Particle Filter, SIAM J, Numer. Anal., 55, 3068-3096 (2017). [2] W. Fang, M. B. Giles, Multilevel Monte Carlo Method for Ergodic SDEs without Contractivity, arXiv:1803.05932, 2018.
25 mins
A general DO-field method for PDEs with quadratic nonlinearity
Fred Wubs, Sourabh Kotnala
Abstract: We will present a method to perform uncertainty quantification of flows subject to noise.
25 mins
Numerical methods for stochastic conservation laws with multiplicative rough drivers in the flux
Haakon Hoel, Nils Henrik Risebro, Kenneth Karlsen, Erlend Briseid Storrosten
Abstract: Stochastic conservation laws (SCL) with quasilinear multiplicative ``rough'' path dependence in the flux arise in modeling of mean field games. An impressive collection of theoretical results has been developed for SCL in recent years by Gess, Lions, Perthame, and Souganidis. We present the first fully computable numerical methods for pathwise solutions of scalar SCL with, for instance, "rough" paths in the form of Wiener processes. Convergence rates are derived for the numerical methods and we show that for strictly convex flux functions, "rough" path oscillations lead to cancellations in the flow map solution; a property we take advantage of to develop more efficient numerical methods.
25 mins
A stochastic Galerkin reduced basis method for parametrized elliptic PDEs based on adaptive snapshots
Christopher Mueller, Jens Lang
Abstract: We consider model order reduction by proper orthogonal decomposition for elliptic boundary value problems with parametrized random and deterministic inputs. When a value for the deterministic parameter is fixed, the solution of the remaining problem can be approximated by a stochastic Galerkin finite element (SGFE) method at the cost of solving a large block-structured system of equations. In scenarios where the solution must be computed for numerous different values of the deterministic parameter, a stochastic Galerkin reduced basis (SGRB) method can be applied to reduce the overall computational costs. This also results from the fact that the reduced order model is constructed with respect to the weak formulation over the spatial and the stochastic space. As a consequence, the reduced order model needs to be solved only once for each new deterministic parameter. Setting up the SGRB model is based on snapshots, i.e. SGFE solutions of the boundary value problem with random data for different values of the deterministic parameter. In order to lower the associated computational burden, especially when the solution exhibits features that change with respect to the deterministic parameter, we use an adaptive SGFE method to compute the snapshots. Due to the adaptive procedure, every snapshot belongs to a different SGFE space in the general case. This fact entails different theoretical and numerical issues which we address. We illustrate the properties of the SGRB model based on a convection-diffusion-reaction test case where the convective velocity is the deterministic parameter and the parametrized reactivity field is the random input.