Construction of Piecewise Linear Wavelets.

It is well known that in many areas of computational mathematics, wavelet based algorithms are becoming popular for modeling and analyzing data and for providing efficient means for hierarchical data decomposition of function spaces into mutually orthogonal wavelet spaces. Wavelet construction in more than one-dimensional setting is a very challenging and important research topic. In this thesis, we first introduce the method of construction wavelets by using semi-wavelets. Second, we construct piecewise linear wavelets with smaller support over type-2 triangulations. Then, parameterized wavelets are constructed using the orthogonality conditions

Bivariate C1 cubic spline space over a nonuniform type-2 triangulation and its subspaces with boundary conditions

AbstractIn this paper, we discuss the algebraic structure of bivariate C1 cubic spline spaces over nonuniform type-2 triangulation and its subspaces with boundary conditions. The dimensions of these spaces are determined and their local support bases are constructed

A quadratic finite element wavelet Riesz basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in $\mathbb{R}^2$. The wavelets are stable in $H^s$ for $|s|<\frac{3}{2}$ and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for $s \in \{-1,0,1\}$ are provided for the unit square.Comment: 13 page

Final report on the Copper Mountain conference on multigrid methods

The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners

Piecewise Linear Wavelet Collocation on Triangular Grids, Approximation of the Boundary Manifold and Quadrature

In this paper we consider a piecewise linear collocation method for the solution of a pseudo-differential equations of order r = 0,-1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose three, four, and six term linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. Though not all wavelets have vanishing moments, we derive the usual compression results, i.e. we prove that, for N degrees of freedom, the fully populated stiffness matrix of N2 entries can be approximated by a sparse matrix with no more than O(N [log N]2.25) non-zero entries. The main topic of the present paper, however, is to show that the parametrization can be approximated by low order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are combinations of product integration applied to non analytic factors of the integrand and of high order Gau{\ss} rules applied to the analytic parts. The whole algorithm for the assembling of the matrix requires no more than O(N [log N]4.25) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N-1[log N]2). Note that, in contrast to well-known algorithms by v.Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required

Multiskalen-Verfahren für Konvektions-Diffusions Probleme

In dieser Arbeit werden erstmalig über einen zur Nichtstandardform gehörenden Erzeugendensystem-Ansatz robuste Wavelet-basierte Multiskalen-Löser für allgemeine zweidimensionale stationäre Konvektions-Diffusions-Probleme entworfen und praktisch umgesetzt. Für Multiskalen-Verfahren, die lediglich direkte Unterraumzerlegungen verwenden, ist es im allgemeinen nicht mehr möglich, zugehörige Multiskalen-Glätter zu konstruieren, die im Grenzfall sehr starker Konvektion auf jeder Skala zu einem direkten Löser entarten. Als eine Möglichkeit zur Konstruktion robuster Multiskalen-Methoden bleibt die Wahl der Multiskalen-Zerlegungen selbst. Es ist sicherzustellen, dass man sowohl hinsichtlich der singulären Störung stabile Grobgitter- probleme als auch bezüglich der Maschenweite stabile Unterraum- zerlegungen erhält. Gleichzeitig muss der Aspekt der approximativen Gauss-Elimination beachtet werden, der durch das Zusammenspiel matrixabhängiger Prolongationen und Restriktionen mit einer hierarchischen Basis Zerlegung gegeben ist. Um alle diese Forderungen zu erfüllen, wird zunächst ausgehend von geometrischen Vergröberungen ein allgemeines Petrov--Galerkin Multiskalen-Konzept entwickelt, bei dem die Zerlegungen auf der Ansatz- und Testseite unterschiedlich sind. Es werden matrixabhängige Prolongationen, die von robusten Mehrgitter-Techniken her bekannt sind, verwendet, zusammen mit Wavelet-artigen und hierarchischen Multiskalen-Zerlegungen der Ansatz- und Testräume bezüglich des feinsten Gitters. Die Kernidee bei den vorgeschlagenen Verfahren ist, jeweils einen der Komplementräume auf der Ansatz- oder Testseite hierarchisch zu wählen, um zusammen mit einer problemabhängigen Vergröberung auf der anderen Seite physikalisch sinnvolle Grobgitter- diskretisierungen und gleichzeitig einen approximativen Eliminations- effekt zu erreichen. Die Komplementräume auf der entsprechend anderen Seite werden hingegen Wavelet-artig aufgespannt, was insbesondere zu einer Stabilisierung des Verfahrens bezüglich der Abhängigkeit von der Maschenweite der Diskretisierung führt. Mit den weiterhin entwickelten AMGlet-Zerlegungen, die auf rein algebraischen Prinzipien beruhen, gelingt es, geometrisch orientierte Tensorprodukt- Konstruktionen, die für separable Probleme erfolgreich sind, zu verlassen, um schwierige nichtseparable Aufgaben in unter Umständen kompliziert berandeten Gebieten behandeln zu können. Dies eröffnet darüberhinaus auch den Übergang von Modellproblemen hin zu praxisnahen Fragestellungen. Unterschiedliche numerische Beispiele zeigen, dass man durch die vorgeschlagenen Konstruktionen zu verallgemeinerten Hierarchische Basis Mehrgitter-Verfahren mit robusten Konvergenzeigenschaften gelangt

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines