Steffen Börm

Introduction to hierarchical matrices with applications

We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix – matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity. We give a review of specialised variants of H-matrices, especially of H 2 -matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory. q 2003 Elsevier Science Ltd. All rights reserved.

Data-sparse approximation by adaptive H 2 -matrices

Abstract A class of matrices (ℋ2-matrices) has recently been introduced for storing discretisations of elliptic problems and integral operators from the BEM. These matrices have the following properties: (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but after truncation to the ℋ2-matrix format these operations are again of quasi-linear complexity.We introduce the basic ideas of ℋ- and ℋ2-matrices and present an algorithm that adaptively computes approximations of general matrices in the latter format.

H 2 -matrix approximation of integral operators by interpolation

Typical panel clustering methods for the fast evaluation of integral operators are based on the Taylor expansion of the kernel function and therefore usually require the user to implement the evaluation of the derivatives of this function up to an arbitrary degree.We propose an alternative approach that replaces the Taylor expansion by simple polynomial interpolation. By applying the interpolation idea to the approximating polynomials on different levels of the cluster tree, the matrix vector multiplication can be performed in only O(npd) operations for a polynomial order of p and an n-dimensional trial space.The main advantage of our method, compared to other methods, is its simplicity: Only pointwise evaluations of the kernel and of simple polynomials have to be implemented.

Hybrid cross approximation of integral operators

The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the -matrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by low-rank matrices. The low-rank matrices are assembled by a new hybrid algorithm (HCA) that has the same proven convergence as standard interpolation but also the same efficiency as the (heuristic) adaptive cross approximation (ACA).

Low-rank approximation of integral operators by interpolation

A central component of the analysis of panel clustering techniques for the approximation of integral operators is the so-called η -admissibility condition “ min {diam(τ),diam(σ)} ≤ 2ηdist(τ,σ)” that ensures that the kernel function is approximated only on those parts of the domain that are far from the singularity. Typical techniques based on a Taylor expansion of the kernel function require a subdomain to be “far enough” from the singularity such that the parameter η has to be smaller than a given constant depending on properties of the kernel function. In this paper, we demonstrate that any η is sufficient if interpolation instead of Taylor expansion␣is␣used for the kernel approximation, which paves the way for grey-box panel clustering algorithms.

Approximation of Integral Operators by Variable-Order Interpolation

Summary.We employ a data-sparse, recursive matrix representation, so-called -matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lower-order ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence (h) is retained for the classical double layer potential discretized with piecewise constant functions.

Analysis of tensor product multigrid

We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced.The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semi-coarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at one-dimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid V-cycle are bounded independently of the number of grid levels involved. In addition, the estimates reveal that convergence is also robust with respect to a singular perturbation in one coordinate direction.Finally, we supply numerical evidence that the algorithm performs satisfactorily in settings more general than those covered by the proof.

Approximation of Integral Operators by **-Matrices with Adaptive Bases

Abstract-matrices can be used to construct efficient approximations of discretized integral operators. The -matrix approximation can be constructed efficiently by interpolation, Taylor or multipole expansion of the integral kernel function, but the resulting representation requires a large amount of storage.In order to improve the efficiency, local Schur decompositions can be used to eliminate redundant functions from an original approximation, which leads to a significant reduction of storage requirements and algorithmic complexity.

Construction of Data-Sparse

Discretizing an integral operator by a standard finite element or boundary element method typically leads to a dense matrix. Since its storage complexity grows quadratically with the number of degrees of freedom, the standard representation of the matrix as a two-dimensional array cannot be applied to large problem sizes. H2-matrix techniques use a multilevel approach to represent the dense matrix in a more efficient data-sparse format. We consider the challenging task of finding a good multilevel representation of the matrix without relying on a priori information of its contents. This paper presents a relatively simple algorithm that can use any of the popular low-rank approximation schemes (e.g., cross approximation) to find an “initial guess” and constructs a matching multilevel structure on the fly. Numerical experiments show that the resulting technique is as fast as competing methods and requires far less storage for large problem dimensions.

\mathcal{H}^2

We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse