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In this article we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log-normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

We investigate approximations by finite sums of products of functions with separated variables to eigenfunctions of multivariate elliptic operators, and especially conditions providing an exponential decrease of the error with respect to the number of terms. The results of the consistent use of tensor formats can be regarded as a base for a new class of iterative eigensolvers with almost linear complexity in the univariate problem size. The results of numerical experiments clearly indicate the linear-logarithmic scaling of low-rank tensor method in the univariate problem size. The algorithms work equally well for the computation of both, minimal and maximal eigenvalues of the discrete elliptic operators.

SUMMARY A population balance system that models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain, while the equation for the particle size distribution is given in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system, which is based on finite element schemes, a finite difference method and a modern method to evaluate convolution integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel. Copyright © 2011 John Wiley & Sons, Ltd.

In this article, we propose a new multigrid-based algorithm for solving integral equations of the reference interactions site model (RISM). We also investigate the relationship between the parameters of the algorithm and the numerical accuracy of the hydration free energy calculations by RISM. For this purpose, we analyzed the performance of the method for several numerical tests with polar and nonpolar compounds. The results of this analysis provide some guidelines for choosing an optimal set of parameters to minimize computational expenses. We compared the performance of the proposed multigrid-based method with the one-grid Picard iteration and nested Picard iteration methods. We show that the proposed method is over 30 times faster than the one-grid iteration method, and in the high accuracy regime, it is almost seven times faster than the nested Picard iteration method.

The efficiency of hierarchical matrices is based on the approximate evaluation of usual matrix operations. The introduced approximation error may, however, lead to a loss of important matrix properties. In this article we present a technique which preserves the positive definiteness of a matrix independently of the approximation quality. The importance of this technique is illustrated by an elliptic mixed boundary value problem with tiny Dirichlet part. Copyright © 2007 John Wiley & Sons, Ltd.

This article deals with the existence of blockwise low-rank approximants, so called ℋ-matrices, to inverses of FE 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 this does not affect the existence of low-rank approximants. Numerical examples show the correctness of our estimates.

ℋ2-matrices are a refined version of hierarchical matrices, which in turn are based on the panel-clustering approach for approximating certain integral operators. Their main advantage, as compared with their predecessors, is that the associated algorithms are of significantly lower complexity while providing the same order of accuracy. Under certain conditions, it is even possible to reach optimal complexity.

In previous papers a class of ℋ-matrices was introduced which are data-sparse and allow an approximate matrix arithmetic of nearly optimal complexity. The complexity analysis for the (approximate) matrix arithmetics in the class of ℋ-matrices is based on two criteria, the sparsity and the idempotency. We describe a general strategy for the construction of the ℋ-matrices where the two criteria are fulfilled.

Several types of ℋ-matrices were shown to provide a data-sparse approximation of non-local (integral) operators in FEM and BEM applications. The general construction is applied to the operators with asymptotically smooth kernel function provided that the Galerkin ansatz space has a hierarchical structure. The new class of ℋ-matrices is based on the so-called blended FE and polynomial approximations of the kernel function and leads to matrix blocks with a tensor-product of block-Toeplitz (block-circulant) and rank-k matrices. This requires the translation (rotation) invariance of the kernel combined with the corresponding tensor-product grids. The approach allows for the fast evaluation of volume/boundary integral operators with possibly non-smooth kernels defined on canonical domains/manifolds in the FEM/BEM applications. (Here and in the following, we call domains canonical if they are obtained by translation or rotation of one of their parts, e.g. parallelepiped, cylinder, sphere, etc.) In particular, we provide the error and complexity analysis for blended expansions to the Helmholtz kernel. Copyright © 2002 John Wiley & Sons, Ltd.