Statement

The extraction of stochastic information from PDEs with random data requires the solution of parametric PDE problems. Very often, the non-intrusive parametric solution using existing PDE solvers is preferred. Here, many simulations are executed for changing input parameters. For large-scale problems, time restrictions limit the number of snapshots and therefore the achievable accuracy. RBF kernel-based approximation might mitigate these limitations.

Beyond pure forward-problems, the inference under uncertainties is of great interest. Many real-world applications, such as medical imaging, are inverse problems and inherently subject to uncertainties.

Exemplary results

Ensemble Kalman filter - based estimation of perfusion in DCE imaging

Estimate of probability for low perfusion.

Perfusion estimate for noisy DCE measurments.

Estimate of probability for low perfusion.

Estimate of probability for low perfusion.

Kernel-based stochastic collocation for an elliptic problem

Mean solution field of elliptic model problem with random coefficient.

Mean solution field of elliptic model problem with random coefficient.

Convergence measurements for different kernels in the kernel-base stochastich collocation method.

Convergence measurements for different kernels in the kernel-base stochastich collocation method.

Kernel-based stochastic collocation for a two-phase flow problem

Streamline visualization of a 2D slice through the mean solution of a 3D velocity field in a two-phase flow simulation with rising bubble under random volume forcing.

Streamline visualization of a 2D slice through the mean solution of a 3D velocity field in a two-phase flow simulation with rising bubble under random volume forcing.

Convergence results for this application.

Convergence results for this application.

Related Work

  • H. Harbrecht, P. Zaspel. On the algebraic construction of sparse multilevel approximations of elliptic tensor product problems, accepted for publication in Journal of Scientific Computing, Springer, August 2018. Also available as Preprint 2018-01, Fachbereich Mathematik, Universität Basel, Switzerland, 2018 and as arXiv:1801.10532.
  • P. Zaspel. Ensemble Kalman filters for reliability estimation in perfusion inference, Preprint 2017-04, Fachbereicht Mathematik, Universität Basel, Switzerland, 2017.

  • P. Zaspel. Parallel RBF Kernel-Based Stochastic Collocation for Large-Scale Random PDEs, PhD Thesis, Institute for Numerical Simulation, University of Bonn, Germany, Apr. 2015

  • V. Heuveline, M. Schick, C. Webster, P. Zaspel. Uncertainty Quantification and High Performance Computing, Dagstuhl Reports, Vol. 6, Issue 9, pp. 59-73.

Software

  • Multi-GPU support and uncertainty quantification for two-phase Navier Stokes (NaSt3DGPF)