Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents for better readability, part on wavelet preconditioner adde

New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years

Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

The ANOVA decomposition and generalized sparse grid methods for the high-dimensional backward Kolmogorov equation

In this thesis, we discuss numerical methods for the solution of the high-dimensional backward Kolmogorov equation, which arises in the pricing of options on multi-dimensional jump-diffusion processes. First, we apply the ANOVA decomposition and approximate the high-dimensional problem by a sum of lower-dimensional ones, which we then discretize by a θ-scheme and generalized sparse grids in time and space, respectively. We solve the resultant systems of linear equations by iterative methods, which requires both preconditioning and fast matrix-vector multiplication algorithms. We make use of a Linear Program and an algebraic formula to compute optimal diagonal scaling parameters. Furthermore, we employ the OptiCom as non-linear preconditioner. We generalize the unidirectional principle to non-local operators and develop a new matrix-vector multiplication algorithm for the OptiCom. As application we focus on the Kou model. Using a new recurrence formula, the computational complexity of the operator application remains linear in the number of degrees of freedom. The combination of the above-mentioned methods allows us to efficiently approximate the solution of the backward Kolmogorov equation for a ten-dimensional Kou model.Die ANOVA-Zerlegung und verallgemeinerte dünne Gitter für die hochdimensionale Kolmogorov-Rückwärtsgleichung In der vorliegenden Arbeit betrachten wir numerische Verfahren zur Lösung der hochdimensionalen Kolmogorov-Rückwärtsgleichung, die beispielsweise bei der Bewertung von Optionen auf mehrdimensionalen Sprung-Diffusionsprozessen auftritt. Zuerst wenden wir eine ANOVA-Zerlegung an und approximieren das hochdimensionale Problem mit einer Summe von niederdimensionalen Problemen, die wir mit einem θ-Verfahren in der Zeit und mit verallgemeinerten dünnen Gittern im Ort diskretisieren. Wir lösen die entstehenden linearen Gleichungssysteme mit iterativen Verfahren, wofür eine Vorkonditionierung als auch schnelle Matrix-Vektor-Multiplikationsalgorithmen nötig sind. Wir entwickeln ein Lineares Programm und eine algebraische Formel, um optimale Diagonalskalierungen zu finden. Des Weiteren setzen wir die OptiCom als nicht-lineares Vorkonditionierungsverfahren ein. Wir verallgemeinern das unidirektionale Prinzip auf nicht-lokale Operatoren und entwickeln einen für die OptiCom optimierten Matrix-Vektor-Multiplikationsalgorithmus. Als Anwendungsbeispiel betrachten wir das Kou-Modell. Mit einer neuen Rekurrenzformel bleibt die Gesamtkomplexität der Operatoranwendung linear in der Anzahl der Freiheitsgrade. Unter Einbeziehung aller genannten Methoden ist es nun möglich, die Lösung der Kolmogorov-Rückwärtsgleichung für ein zehndimensionales Kou-Modell effizient zu approximieren

OPTIMAL DOMAIN DECOMPOSITION METHOD FOR NUMERICAL SOLUTION OF ELLIPTIC EQUATION WITH DISCONTINUOUS

Рассматривается эллиптическое уравнение второго порядка в области, составленной из конечного числа ячеек произвольной неравномерной ортогональной сетки, являющихся подобластями декомпозиции. В качестве модельного взято уравнение в дивиргентной форме с диагональной матрицей коэффициентов, которые принимают произвольные положительные конечные значения в каждой ячейке этой сетки. Переменная ортогональная дискретизационная конечно-элементная сетка удовлетворяет только одному условию: на каждой ячейке декомпозиционной сетки она равномерная. Для решения конечно-элементной задачи предлагается итерационный метод декомпозиции области типа Дирихле-Дирихле, имеющий линейную сложность. Наиболее трудной проблемой при его создании является получение эффективного предобусловливателя-солвера для интерфейсного дополнения Шура. Она тесно связана с получением граничных норм для дискретно-гармонических конечно-элементных функций в узких прямоугольниках.Second order elliptic equation is considered in the domain, which is the union of a finite number of cells of an arbitrary nonuniform orthogonal decomposition grid. For a model problem is taken the equation in the divergent form and the diagonal matrix of coefficients, which are arbitrary positive finite numbers in each cell. The variable orthogonal finite element discretization mesh has to satisfy only one condition: it is uniform in each cell. No other conditions on the coefficients of the elliptic equation and step sizes of the discretization and decomposition meshes are imposed. For the resulting finite element problem, we suggest the domain decomposition algorithm of linear total arithmetical complexity, not depending on any of the three factors contributing to the orthotropism of the discretization on subdomains. The main problem at designing such an algorithm is preconditioning of the inter-subdomain Schur complement. It is closely related to the derivation of boundary norms for discrete harmonic finite element functions on the shape irregular rectangles.418-42

Two sides tangential filtering decomposition

AbstractIn this paper we study a class of preconditioners that satisfy the so-called left and/or right filtering conditions. For practical applications, we use a multiplicative combination of filtering based preconditioners with the classical ILU(0) preconditioner, which is known to be efficient. Although the left filtering condition has a more sound theoretical motivation than the right one, extensive tests on convection–diffusion equations with heterogeneous and anisotropic diffusion tensors reveal that satisfying left or right filtering conditions lead to comparable results. On the filtering vector, these numerical tests reveal that e=[1,…,1]T is a reasonable choice, which is effective and can avoid the preprocessing needed in other methods to build the filtering vector. Numerical tests show that the composite preconditioners are rather robust and efficient for these problems with strongly varying coefficients

A new approach to energy-based sparse finite-element spaces

We show that the logarithmic factor in the standard error estimate for sparse finite element (FE) spaces in arbitrary dimension d is removable in the energy (H1) norm. Via a penalized sparse grid condition, we then propose and analyse a new version of the energy-based sparse FE spaces introduced first in Bungartz (1992, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation. Munich, Germany: TU München) and known to satisfy an optimal approximation property in the energy nor