Uncertainty Quantification


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 from Ensemble Kalman filter - based estimation of perfusion in DCE imaging

Perfusion estimate for noisy DCE measurments.
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Estimate of probability for low perfusion.

Exemplary results from Kernel-based stochastic collocation for an elliptic problem

Mean solution field of elliptic model problem with random coefficient.
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Convergence measurements for different kernels in the kernel-based stochastic collocation method.

Exemplary results from 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.
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Convergence results for this application.

Related work