Approximate Approximations and their Applications

This paper gives a survey of an approximation method which was proposed by V. Maz'ya as underlying procedure for numerical algorithms to solve initial and boundary value problems of mathematical physics. Due to a greater flexibility in the choice of approximating functions it allows efficient approximations of multi-dimensional integral operators often occuring in applied problems. Its application especially in connection with integral equation methods is very promising, which has been proved already for different classes of evolution equations. The survey describes some basic results concerning error estimates for quasi-interpolation and cubature of integral operators with singular kernels as well as a multiscale and wavelet approach to approximate those operators over bounded domains. Finally a general numerical method for solving nonlocal nonlinear evolution equations is presented

Computation of volume potentials over bounded domains via approximate approximations

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.Comment: 18 page

Approximate Approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.Comment: 29 pages, 17 figure

Potentials of Gaussians and approximate wavelets

We derive new formulas for harmonic, diffraction, elastic, and hydrodynamic potentials acting on anisotropic Gaussians and approximate wavelets. These formulas can be used to construct accurate cubature formulas for these potentials

On Quasi-interpolation with non-uniformly distributed centers on Domains and Manifolds

The paper studies quasi-interpolation by scaled shifts of a smooth and rapidly decaying function. The centers are images of a smooth mapping of the hZn-lattice in Rs, s ≥ n, and the scaling parameters are proportional to h. We show that for a large class of generating functions the quasi-interpolants provide high order approximations up to some prescribed accuracy. Although the approximants do not converge as h tends to zero, this is not feasible in computations if a scalar parameter is suitably chosen. The lack of convergence is compensated for by more flexibility in the choice of generating functions used in numerical methods for solving operator equations

Fast boundary element methods for the simulation of wave phenomena

This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) für ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nützliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten führt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren Einträge die Integration singulärer Kernfunktionen über Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln für die schwierigsten Fälle. Dies ermöglicht die genaue Berechnung der Matrixeinträge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation präsentiert, der die Matrizen komprimiert und die Komplexität der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der räumlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die überlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

Approximate approximations from scattered data

AbstractThe aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators

APPROXIMATE APPROXIMATIONS ON NONUNIFORM GRIDS

We present an extension of approximate quasi-interpolation on uniformly distributed nodes, to functions given on a set of nodes close to an uniform, not necessarily cubic, grid.We present an extension of approximate quasi-interpolation on uniformly distributed nodes, to functions given on a set of nodes close to an uniform, not necessarily cubic, grid

Potentials of Gaussians and approximate wavelets

We derive new formulas for harmonic, diffraction, elastic, and hydrodynamic potentials acting on anisotropic Gaussians and approximate wavelets. These formulas can be used to construct accurate cubature formulas for these potentials

Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique