CFD under uncertainty: combining model order reduction with spatial adaptivity
Sebastian Ullmann | TU Darmstadt | Germany
We consider unsteady incompressible Navier-Stokes problems at low Reynolds numbers, driven by uncertain boundary conditions. We estimate the statistics of relevant output quantities by stochastic sampling methods, which require the solution of time-dependent flow problems for each realization of the parametrized random input.
We propose the combination of two approaches for reducing computational complexity in this context: Firstly, we adjust the spatial resolution to the local features of the solution by means of an adaptive finite element method. This means an individual finite element space for each time instance and each parameter realization. Secondly, we use POD-Galerkin reduced-order modeling to decrease the stochastic sampling error associated with the number of parameter realizations. This means a few adaptive finite element simulations are performed to create a reduced basis. The reduced-order model is then solved for a large number of parameter realizations in a stochastic sampling loop.
The combination of model order reduction with spatial adaptivity raises some challenges. In general, spatial adaptivity destroys the Galerkin orthogonality with respect to the snapshot space(s), which is a challenge in view of error estimation. In particular for the Navier-Stokes case, the loss of a discrete divergence-free property due to adaptivity requires additional stabilization of the reduced-order model. We illustrate these issues and possible remedies with a test case involving the flow in a laterally heated cavity.
Wave Diffraction by Random Surfaces: Non-Conforming Sparse Tensor Boundary Elements
Prof. Carlos Jerez-Hanckes | Pontificia Universidad Católica de Chile | Chile
We consider the numerical solution of time-harmonic scattering of acoustic and electromagnetic waves from impenetrable and penetrable obstacles with uncertain geometries. Using first-order shape derivatives, we derive deterministic boundary integral equations for the mean field and the two-point correlation function of the random solution for a soft-obstacle Dirichlet problem. Sparse tensor Galerkin discretizations of these equations are implemented with the so-called combination technique. We generalize the method to non-nested meshes using a nodal transfer operator. Similar discretization errors for the covariance is achieved with O(N log N) degrees of freedom instead of O(N^2). Performance comparison of our approach to classic Monte-Carlo Galerkin formulation is given for different shapes. Finally, we verify the robustness of the sparse tensor approximation and compare it to low-rank approximations techniques.
Multilevel Monte Carlo for transmission problems with geometric uncertainties
Dr. Laura Scarabosio | TU München | Germany
We consider a Helmholtz transmission problem describing the scattering of a plane wave from an object whose shape is subject to random perturbations. We model these shape variations through a high-dimensional parameter. If we consider the point evaluation of the solution at locations that are very close to the interface, then this quantity of interest does not depend smoothly on the high-dimensional parameter, and high-order methods for the estimation of moments fail to converge with full rate. In this talk, I will show that Multilevel Monte Carlo (MLMC) offers a robust treatment for such problem, and provide numerical experiments that confirm this. The methodology used clearly conveys that MLMC is a viable aproach also for other problems lacking smoothness with respect to the stochastic parameter.
MLQMC with product weights for elliptic PDEs with lognormal coefficients parametrized in multiresolution representations
Dr. Lukas Herrmann | ETH Zurich | Switzerland
Parametric diffusions are considered with lognormal coefficients that are given by multiresolution representations. Approximations by quasi-Monte Carlo (QMC) with randomly shifted lattice rules for first order are analyzed with dimension independent convergence rates. The local support structure in the multiresolution expansion are known to allow product weights for QMC rules, cp.~[Herrmann, Schwab, SAM-report 2016-39]. Product weights allow for linear scaling in the dimension of integration in the cost to create QMC rules by the CBC construction, cp. [Nuyens, Cools, Math. Comp. 2006]. Multilevel QMC quadratures are considered to reduce the work of the QMC approximation in general polyhedral spatial domains, cp.~[Herrmann, Schwab, SAM-report 'MLQMC' 2017-(in preparation) and SAM-report 2017-04]. Analogous results hold for affine-parametric operator equations, cp.~[Gantner, Herrmann, Schwab, SAM-reports 2016-32 and 2016-54]. This research is supported in part by the Swiss National Science Foundation (SNSF) under grant SNF 159940.
A Multigrid Multilevel Monte Carlo method for transport in the Darcy-Stokes flow
Prashant Kumar | TU Delft | Netherlands
Transport in a coupled Darcy-Stokes system can be used to describe a large number of dynamical scenarios. For instance, the model can be used for the risk assessment in case of an accidental discharge of radioactive contaminants or chemical spillage in the surface water bodies and the subsequent transport to the connected aquifers. Uncertainty Quantification (UQ) of contaminant transport in a coupled Darcy-Stokes flow is a computationally expensive task as this involves solving a high-dimensional UQ problem, primarily due to unknown permeability in the Darcy domain. In this work, we propose a multilevel Monte Carlo (MLMC) method for this problem.
Each sample of the quantity of interest essentially requires solving a discrete Darcy-Stokes system for the velocity field that is utilized for the unsteady transport. Thus, a careful consideration of numerical strategies to solve each of these sub problems is needed to obtain an efficient MLMC estimator. We focus on two aspects a) development of an efficient multigrid algorithm for solving the discrete Darcy-Stokes problem b) an optimal time-stepping technique for contaminant transport.
Multigrid solver for the Darcy-Stokes problem with highly heterogeneous permeability is based on a monolithic framework that treats the coupled problem as a single problem. This approach is highly effective as the two problems are strongly coupled by three interface conditions. To cover realistic cases, we also propose a multi-block version of this solver that uses the grid partitioning technique. The proposed solver is robust and also performs well on very coarse grids, thus making it a highly suitable solver for MLMC applications.
Finally, an optimal time-stepping method for transport is proposed that is based on the Alternating Direction Implicit scheme. This is an operator-splitting techniques that yields tridiagonal matrices which can be solved very efficiently.