Boris N. Khoromskij

On H2-Matrices

A class of matrices (H-matrices) has recently been introduced by one of the authors. 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 almost linear complexity, (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the H-matrix format are again of almost linear complexity, (iv) The same statement holds for the inverse of an H-matrix.

A Sparse ℋ-Matrix Arithmetic.

The preceding Part I of this paper has introduced a class of matrices (ℋ-matrices) which are data-sparse and allow an approximate matrix arithmetic of almost linear complexity. The matrices discussed in Part I are able to approximate discrete integral operators in the case of one spatial dimension.Abstract The preceding Part I of this paper has introduced a class of matrices (ℋ-matrices) which are data-sparse and allow an approximate matrix arithmetic of almost linear complexity. The matrices discussed in Part I are able to approximate discrete integral operators in the case of one spatial dimension.In the present Part II, the construction of ℋ-matrices is explained for FEM and BEM applications in two and three spatial dimensions. The orders of complexity of the various matrix operations are exactly the same as in Part I. In particular, it is shown that the applicability of ℋ-matrices does not require a regular mesh. We discuss quasi-uniform unstructured meshes and the case of composed surfaces as well.

Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs

We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the

Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices

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Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays

-term truncated Karhunen-Loeve expansion. Our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high-dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension

Approximate iterations for structured matrices

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A sparse H -matrix arithmetic: general complexity estimates

of the parameter space. It is based on rank-reduced, tensor-formatted separable approximations of the high-dimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent low-rank tensor-structured preconditioners to the parametric matrices resulting from a finite element discretization of the high-dimensional parametric, deterministic problems. Numerical illustrations for the

Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format

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H-matrix approximation for the operator exponential with applications

-dimensional parametric elliptic PDEs resulting from sPDEs on parameter spaces of dimensions

Hierarchical Matrices based on a Weak Admissibility Criterion

M\leq100