## 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

# Perfusion estimate for noisy DCE measurments.

# Estimate of probability for low perfusion.

#### Kernel-based stochastic collocation for an elliptic problem

# Mean solution field of elliptic model problem with random coefficient.

# 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.

# Convergence results for this application.

## Related Work

- M. Griebel, Ch. Rieger, P. Zaspel. Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations. Available as arXiv:1810.11270, October 2018.
- P. Zaspel. Ensemble Kalman filters for reliability estimation in perfusion inference. Accepted for publication in International Journal for Uncertainty Quantification, December 2018; also available as arXiv:1810.09290. (A previous revision is available as Preprint 2017-04, Fachbereicht Mathematik, Universität Basel).
- 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. 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)