Mario Bebendorf

Approximation of boundary element matrices

Summary. This article considers the problem of approximating a general asymptotically smooth function in two variables, typically arising in integral formulations of boundary value problems, by a sum of products of two functions in one variable. From these results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed. This algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance.

Adaptive low-rank approximation of collocation matrices

This article deals with the solution of integral equations using collocation methods with almost linear complexity. Methods such as fast multipole, panel clustering and ℋ-matrix methods gain their efficiency from approximating the kernel function. The proposed algorithm which uses the ℋ-matrix format, in contrast, is purely algebraic and relies on a small part of the collocation matrix for its blockwise approximation by low-rank matrices. Furthermore, a new algorithm for matrix partitioning that significantly reduces the number of blocks generated is presented.

Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems

Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.

Existence of ℋ-matrix approximants to the inverse FE-matrix of elliptic operators with L ∞ -coefficients

AbstractThis article deals with the existence of blockwise low-rank approximants — so-called ℋ-matrices — to inverses of FEM matrices in the case of uniformly elliptic operators with L∞-coefficients. Unlike operators arising from boundary element methods for which the ℋ-matrix theory has been extensively developed, the inverses of these operators do not benefit from the smoothness of the kernel function. However, it will be shown that the corresponding Green functions can be approximated by degenerate functions giving rise to the existence of blockwise low-rank approximants of FEM inverses. Numerical examples confirm the correctness of our estimates. As a side-product we analyse the ℋ-matrix property of the inverse of the FE mass matrix.

A Note on the Poincaré Inequality for Convex Domains

In this article a proof for the Poincaré inequality with explicit constant for convex domains is given. This proof is a modification of the original proof [5], which is valid only for the two-dimensional case.

Hierarchical LU Decomposition-based Preconditioners for BEM

Abstract.The adaptive cross approximation method can be used to efficiently approximate stiffness matrices arising from boundary element applications by hierarchical matrices. In this article an approximative LU decomposition in the same format is presented which can be used for preconditioning the resulting coefficient matrices efficiently. If the LU decomposition is computed with high precision, it may even be used as a direct yet efficient solver.

Recompression Techniques for Adaptive Cross Approximation

The adaptive cross approximation method (ACA) generates low-rank approximations to suitable m× n sub-blocks of discrete integral formulations of elliptic boundary value problems. A characteristic property is that the approximation, which requires k(m+ n), k ∼ | log e|∗, units of storage, is generated in an adaptive and purely algebraic manner using only few of the matrix entries. In this article we present further recompression techniques which are based on ACA and bring the required amount of storage down to sublinear order kk′, where k′ depends logarithmically on the accuracy of the approximation but is independent of the matrix size. The additional compression is due to a certain smoothness of the vectors generated by ACA.

Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients

This article deals with the efficient (approximate) inversion of finite element stiffness matrices of general second-order elliptic operators with L∞-coefficients. It will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices (H-matrices). Furthermore, numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity.

Why Finite Element Discretizations Can Be Factored by Triangular Hierarchical Matrices

Although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second-order elliptic partial differential operators is far from being optimal, these methods are often preferred over modern iterative methods. This is mainly due to their robustness. In this article it is shown that an approximate

Constructing nested bases approximations from the entries of non-local operators

LU