Orthogonale Dünngitter-Teilraumzerlegungen

In der Simulation treten Häufg hochdimensionale partielle Differentialgleichungen auf. Das Lösen dieser wird für volle Gitter sehr schnell zu teuer. In dieser Arbeit wird ein Verfahren für das Lösen partieller Differentialgleichungen mit Hilfe von Dünnen Gittern, welche für mehrdimensionale Probleme besser skalieren, sowie dessen Implementierung in das Programmpaket SG++ vorgestellt. Durch Funktionsdarstellung in einem Erzeugendensystem wird die Verwendung einer L2-orthogonalen Teilraumzerlegung ermöglicht. Projektionsoperatoren ersetzen hierbei die explizite Transformation in eine Prewavelet-Basis. Diese Zerlegung erlaubt das Lumping der Steifgkeitsmatrix, also das Weglassen von großen Blöcken der Matrix. Hiermit wird ein Algorithmus zur Matrixmultiplikation, welcher dem von Schwab und Todor ähnelt implementiert. Dieser wird in einem konjugierten Gradienten-Verfahren verwendet und auch auf krummberandete Gebieten angewendet. Des Weiteren wird die Teilraumzerlegung durch L2-Projektion mit anderen Zerlegungen in Bezug auf Laufzeit und Fehlerentwicklung verglichen

Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond

Local and Dimension Adaptive Sparse Grid Interpolation and Quadrature

In this paper we present a locally and dimension-adaptive sparse grid method for interpolation and integration of high-dimensional functions with discontinuities. The proposed algorithm combines the strengths of the generalised sparse grid algorithm and hierarchical surplus-guided local adaptivity. A high-degree basis is used to obtain a high-order method which, given sufficient smoothness, performs significantly better than the piecewise-linear basis. The underlying generalised sparse grid algorithm greedily selects the dimensions and variable interactions that contribute most to the variability of a function. The hierarchical surplus of points within the sparse grid is used as an error criterion for local refinement with the aim of concentrating computational effort within rapidly varying or discontinuous regions. This approach limits the number of points that are invested in `unimportant' dimensions and regions within the high-dimensional domain. We show the utility of the proposed method for non-smooth functions with hundreds of variables

Erkennung von Silent Faults mit der Kombinationstechnik

Innerhalb dieser Arbeit werden zwei Verfahren vorgestellt, um bisher unerkannte Fehler während der Simulation von Diffusions- und Advektionsproblemen mit der Dünngitter Kombinationstechnik aufzudecken. Der Schwerpunkt liegt hierbei auf der Erkennung von Fehlern anhand der zeitlichen Entwicklung einer einzelnen Kombinationslösung, ohne Vergleiche zwischen anderen Kombinationslösungen herzustellen. Die Untersuchungen zeigen, dass die beiden Verfahren im Falle eines für die Lösung relevanten Fehlers zum Zeitpunkt der Korruption ein unnatürliches Verhalten aufweisen, welches ein hohes Potential besitzt, um von einem automatisierten Erkennugsverfahren detektiert werden zu können

Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids

International audienceAn approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function.The value function can be characterized as the solution of an evolutionary Hamilton-Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality.We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored.We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory