Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner

Fast Numerical Methods for Non-local Operators

[no abstract available

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Applications of nonlinear approximation for problems in learning theory and applied mathematics

A major pillar of approximation theory in establishing the ability of one class of functions to be represented by another. Establishing such a relationship often leads to efficient numerical approximation methods. In this work, several expressibility theorems are established and several novel numerical approximation techniques are also presented. Not only are these novel methods supported by the presented theory, but also, provided numerical experiments show that these novel methods may be applied to a wide range of applications from image compression to the solutions of high-dimensional PDE

Innovative Approaches to the Numerical Approximation of PDEs

This workshop was about the numerical solution of PDEs for which classical approaches, such as the finite element method, are not well suited or need further (theoretical) underpinnings. A prominent example of PDEs for which classical methods are not well suited are PDEs posed in high space dimensions. New results on low rank tensor approximation for those problems were presented. Other presentations dealt with regularity of PDEs, the numerical solution of PDEs on surfaces, PDEs of fractional order, numerical solvers for PDEs that converge with exponential rates, and the application of deep neural networks for solving PDEs

Wavelet and Multiscale Methods

[no abstract available

Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDES on Tensor-Product Domains

This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by semilinear elliptic partial differential equations (PDEs). Semilinearity here refers to a special case of nonlinearity, i.e., the case of a linear operator combined with a nonlinear operator acting as a perturbation. In general, such BVPs are solved in an iterative fashion. It is, therefore, of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. Unlike the typical finite element method (FEM) theory for the numerical solution of the nonlinear operators, the new adaptive wavelet theory proposed in [Cohen.Dahmen.DeVore:2003:a, Cohen.Dahmen.DeVore:2003:b] guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets are the ideal candidate for this purpose since they allow to represent functions in infinite-dimensional general Banach or Hilbert spaces and operators on these. The purpose of adaptivity in the solution process of nonlinear PDEs is to invest extra degrees of freedom (DOFs) only where necessary, i.e., where the exact solution requires a higher number of function coefficients to represent it accurately. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the l_2 sequence spaces of expansion coefficients exist. This new paradigm presents nevertheless some problems in the design of practical algorithms. Imposing a certain structure, a tree structure, remedies these problems completely while restricting the applicability of the theoretical scheme only very slightly. It turns out that the considered approach naturally fits the theoretical background of nonlinear PDEs. The practical realization on a computer, however, requires to reduce the relevant ingredients to finite-dimensional quantities. It is this particular aspect that is the guiding principle of this thesis. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. In the implementation, great emphasis is put on speed, i.e., overall execution speed and convergence speed, while not sacrificing on the freedom to adapt many important numerical details at runtime and not at the compilation stage. This means that the user can test and choose wavelets perfectly suitable for any specific task without having to rebuild the software. The computational overhead of these freedoms is minimized by caching any interim data, e.g., values for the preconditioners and polynomial representations of wavelets in multiple dimensions. Exploiting the structure in the construction of wavelet spaces prevents this step from becoming a burden on the memory requirements while at the same time providing a huge performance boost because necessary computations are only executed as needed and then only once. The essential BVP boundary conditions are enforced using trace operators, which leads to a saddle point problem formulation. This particular treatment of boundary conditions is very flexible, which especially useful if changing boundary conditions have to be accommodated, e.g., when iteratively solving control problems with Dirichlet boundary control based upon the herein considered PDE operators. Another particular feature is that saddle point problems allow for a variety of different geometrical setups, including fictitious domain approaches. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. Local transformations of the wavelet basis are employed to lower the absolute condition number of the already optimally preconditioned operators. The effect of these basis transformations can be seen in the absolute runtimes of solution processes, where the semilinear PDEs are solved as fast as in fractions of a second. This task can be accomplished using simple Richardson-style solvers, e.g., the method of steepest descent, or more involved solvers like the Newton's method. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of semilinear PDE sub-problems. The efficiency of different numerical methods is compared and the theoretical optimal convergence rates and complexity estimates are verified. In summary, this thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve semilinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

We study tensor product multiscale many-particle spaces with finite-order weights and their application for the electronic Schrödinger equation. Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, typically Monte Carlo methods (VMC/DMC) or nonlinear model approximations like Hartree-Fock (HF), coupled cluster (CC) or density functional theory (DFT) are used. In this work we develop and implement in parallel a numerical method based on adaptive sparse grids and a particle-wise subspace splitting with respect to one-particle functions which stem from a nonlinear rank-1 approximation. Sparse grids allow to overcome the exponential complexity exhibited by conventional discretization procedures and deliver a convergent numerical approach with guaranteed convergence rates. In particular, the introduced weighted many-particle tensor product multiscale approximation spaces include the common configuration interaction (CI) spaces as a special case. To realize our new approach, we first introduce general many-particle Sobolev spaces, which particularly include the standard Sobolev spaces as well as Sobolev spaces of dominated mixed smoothness. For this novel variant of sparse grid spaces we show estimates for the approximation and complexity orders with respect to the smoothness and decay parameters. With known regularity properties of the electronic wave function it follows that, up to logarithmic terms, the convergence rate is independent of the number of electrons and almost the same as in the two-electron case. However, besides the rate, also the dependence of the complexity constants on the number of electrons plays an important role for a truly practical method. Based on a splitting of the one-particle space we construct a subspace splitting of the many-particle space, which particularly includes the known ANOVA decomposition, the HDMR decomposition and the CI decomposition as special cases. Additionally, we introduce weights for a restriction of this subspace splitting. In this way weights of finite order q lead to many-particle spaces in which the problem of an approximation of an N-particle function reduces to the problem of the approximation of q-particle functions. To obtain as small as possible constants with respect to the cost complexity, we introduce a heuristic adaptive scheme to build a sequence of finite-dimensional subspaces of a weighted tensor product multiscale many-particle approximation space. Furthermore, we construct a multiscale Gaussian frame and apply Gaussians and modulated Gaussians for the nonlinear rank-1 approximation. In this way, all matrix entries of the corresponding discrete eigenvalue problem can be computed in terms of analytic formulae for the one and two particle operator integrals. Finally, we apply our novel approach to small atomic and diatomic systems with up to 6 electrons (18 space dimensions). The numerical results demonstrate that our new method indeed allows for convergence with expected rates.Tensorprodukt-Multiskalen-Mehrteilchenräume mit Gewichten endlicher Ordnung für die elektronische Schrödingergleichung In der vorliegenden Arbeit beschäftigen wir uns mit gewichteten Tensorprodukt-Multiskalen-Mehrteilchen-Approximationsräumen und deren Anwendung zur numerischen Lösung der elektronischen Schrödinger-Gleichung. Aufgrund der hohen Problemdimension ist eine direkte numerische Lösung der elektronischen Schrödinger-Gleichung mit Standard-Diskretisierungsverfahren zur linearen Approximation unmöglich, weshalb üblicherweise Monte Carlo Methoden (VMC/DMC) oder nichtlineare Modellapproximationen wie Hartree-Fock (HF), Coupled Cluster (CC) oder Dichtefunktionaltheorie (DFT) verwendet werden. In dieser Arbeit wird eine numerische Methode auf Basis von adaptiven dünnen Gittern und einer teilchenweisen Unterraumzerlegung bezüglich Einteilchenfunktionen aus einer nichtlinearen Rang-1 Approximation entwickelt und für parallele Rechnersysteme implementiert. Dünne Gitter vermeiden die in der Dimension exponentielle Komplexität üblicher Diskretisierungsmethoden und führen zu einem konvergenten numerischen Ansatz mit garantierter Konvergenzrate. Zudem enthalten unsere zugrunde liegenden gewichteten Mehrteilchen Tensorprodukt-Multiskalen-Approximationsräume die bekannten Configuration Interaction (CI) Räume als Spezialfall. Zur Konstruktion unseres Verfahrens führen wir zunächst allgemeine Mehrteilchen-Sobolevräume ein, welche die Standard-Sobolevräume sowie Sobolevräume mit dominierender gemischter Glattheit beinhalten. Wir analysieren die Approximationseigenschaften und schätzen Konvergenzraten und Kostenkomplexitätsordnungen in Abhängigkeit der Glattheitsparameter und Abfalleigenschaften ab. Mit Hilfe bekannter Regularitätseigenschaften der elektronischen Wellenfunktion ergibt sich, dass die Konvergenzrate bis auf logarithmische Terme unabhängig von der Zahl der Elektronen und fast identisch mit der Konvergenzrate im Fall von zwei Elektronen ist. Neben der Rate spielt allerdings die Abhängigkeit der Konstanten in der Kostenkomplexität von der Teilchenzahl eine wichtige Rolle. Basierend auf Zerlegungen des Einteilchenraumes konstruieren wir eine Unterraumzerlegung des Mehrteilchenraumes, welche insbesondere die bekannte ANOVA-Zerlegung, die HDMR-Zerlegung sowie die CI-Zerlegung als Spezialfälle beinhaltet. Eine zusätzliche Gewichtung der entsprechenden Unterräume mit Gewichten von endlicher Ordnung q führt zu Mehrteilchenräumen, in denen sich das Approximationsproblem einer N-Teilchenfunktion zu Approximationsproblemen von q-Teilchenfunktionen reduziert. Mit dem Ziel, Konstanten möglichst kleiner Größe bezüglich der Kostenkomplexität zu erhalten, stellen wir ein heuristisches adaptives Verfahren zur Konstruktion einer Sequenz von endlich-dimensionalen Unterräumen eines gewichteten Mehrteilchen-Tensorprodukt-Multiskalen-Approximationsraumes vor. Außerdem konstruieren wir einen Frame aus Multiskalen-Gauss-Funktionen und verwenden Einteilchenfunktionen im Rahmen der Rang-1 Approximation in der Form von Gauss- und modulierten-Gauss-Funktionen. Somit können die zur Aufstellung der Matrizen des zugehörigen diskreten Eigenwertproblems benötigten Ein- und Zweiteilchenintegrale analytisch berechnet werden. Schließlich wenden wir unsere Methode auf kleine Atome und Moleküle mit bis zu sechs Elektronen (18 Raumdimensionen) an. Die numerischen Resultate zeigen, dass sich die aus der Theorie zu erwartenden Konvergenzraten auch praktisch ergeben