FEM for elliptic eigenvalue problems: how coarse can the coarsest mesh be chosen? An experimental study

In this paper, we consider the numerical discretization of elliptic eigenvalue problems by Finite Element Methods and its solution by a multigrid method. From the general theory of finite element and multigrid methods, it is well known that the asymptotic convergence rates become visible only if the mesh width h is sufficiently small, h≤h 0. We investigate the dependence of the maximal mesh width h 0 on various problem parameters such as the size of the eigenvalue and its isolation distance. In a recent paper (Sauter in Finite elements for elliptic eigenvalue problems in the preasymptotic regime. Technical Report. Math. Inst., Univ. Zürich, 2007), the dependence of h 0 on these and other parameters has been investigated theoretically. The main focus of this paper is to perform systematic experimental studies to validate the sharpness of the theoretical estimates and to get more insights in the convergence of the eigenfunctions and -values in the preasymptotic regim

Complexity Analysis of a Fast Directional Matrix-Vector Multiplication

We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered directional multi-level approximation of the Helmholtz kernel can be applied even on high-frequency levels efficiently. We provide a detailed analysis of the almost linear asymptotic complexity of the presented method. Our numerical experiments are in good agreement with the provided theory.Comment: 20 pages, 2 figures, 1 tabl

On thin plate spline interpolation

We present a simple, PDE-based proof of the result [M. Johnson, 2001] that the error estimates of [J. Duchon, 1978] for thin plate spline interpolation can be improved by $h^{1/2}$. We illustrate that ${\mathcal H}$-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problem

Comparison of some Reduced Representation Approximations

In the field of numerical approximation, specialists considering highly complex problems have recently proposed various ways to simplify their underlying problems. In this field, depending on the problem they were tackling and the community that are at work, different approaches have been developed with some success and have even gained some maturity, the applications can now be applied to information analysis or for numerical simulation of PDE's. At this point, a crossed analysis and effort for understanding the similarities and the differences between these approaches that found their starting points in different backgrounds is of interest. It is the purpose of this paper to contribute to this effort by comparing some constructive reduced representations of complex functions. We present here in full details the Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category

Application of hierarchical matrices for computing the Karhunen-Loève expansion

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented

Approximation of integral operators using product-convolution expansions

International audienceWe consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices