Sergej Rjasanow

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.

The adaptive cross-approximation technique for the 3D boundary-element method

It is well known that the classical boundary-element method (BEM) yields fully populated matrices. Their manipulation is cumbersome with respect to memory consumption and computational costs. This paper describes a novel approach where the matrices are split into collections of blocks of various sizes. Those blocks which describe remote interactions are adaptively approximated by low rank submatrices. This procedure reduces the algorithmic complexity for matrix setup and matrix-by-vector products to approximately O(N). The proposed method has been examined in a testing environment and implemented into an existing BEM-finite-element method (FEM) code for electromagnetic and electromechanical problems. The advantages of the new method are demonstrated by means of several examples.

Numerical solution of the Boltzmann equation on the uniform grid

Abstract In the present paper a new numerical method for the Boltzmann equation is developed. The gain part of the collision integral is written in a form which allows its numerical computation on the uniform grid to be carried out efficiently. The amount of numerical work is shown to be of the order O(n6log(n)) for the most general model of interaction and of the order O(n6) for the Variable Hard Spheres (VHS) interaction model, while the formal accuracy is of the order O(n−2). Here n denotes the number of discretisation points in one direction of the velocity space. Some numerical examples for Maxwell pseudo-molecules and for the hard spheres model illustrate the accuracy and the efficiency of the method in comparison with DSMC computations in the spatially homogeneous case.

Novel formulation of nonlocal electrostatics

The accurate modeling of the dielectric properties of water is crucial for many applications in physics, computational chemistry, and molecular biology. This becomes possible in the framework of nonlocal electrostatics, for which we propose a novel formulation allowing for numerical solutions for the nontrivial molecular geometries arising in the applications mentioned before. Our approach is based on the introduction of a secondary field psi, which acts as the potential for the rotation free part of the dielectric displacement field D. For many relevant models, the dielectric function of the medium can be expressed as the Greens function of a local differential operator. In this case, the resulting coupled Poisson (-Boltzmann) equations for psi and the electrostatic potential phi reduce to a system of coupled partial differential equations. The approach is illustrated by its application to simple geometries.

Higher Order BEM-Based FEM on Polygonal Meshes

The BEM-based finite element method is reviewed and extended with higher order basis functions on general polygonal meshes. These functions are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are treated by means of boundary integral formulations and are approximated using the boundary element method in the numerics. To obtain higher order convergence, a new approximation of the material coefficient is proposed since previous strategies are not sufficient. Following recent ideas, error estimates are proved which guarantee quadratic convergence in the

Electrostatic potentials of proteins in water: a structured continuum approach

H^1

Comparison between different approaches for fast and efficient 3-D BEM computations

-norm and cubic convergence in the

Direct simulation of the uniformly heated granular boltzmann equation

L_2

FEM with Trefftz trial functions on polyhedral elements

-norm. The numerical realization is discussed and several experiments confirm the theoretical results.

Matrix Compression for the Radiation Heat Transfer in Exhaust Pipes

Electrostatic interactions play a crucial role in many biomolecular processes, including molecular recognition and binding. Biomolecular electrostatics is modulated to a large extent by the water surrounding the molecules. Here, we present a novel approach to the computation of electrostatic potentials which allows the inclusion of water structure into the classical theory of continuum electrostatics. Based on our recent purely differential formulation of nonlocal electrostatics [Hildebrandt, et al. (2004) Phys. Rev. Lett., 93, 108104] we have developed a new algorithm for its efficient numerical solution. The key component of this algorithm is a boundary element solver, having the same computational complexity as established boundary element methods for local continuum electrostatics. This allows, for the first time, the computation of electrostatic potentials and interactions of large biomolecular systems immersed in water including effects of the solvents structure in a continuum description. We illustrate the applicability of our approach with two examples, the enzymes trypsin and acetylcholinesterase. The approach is applicable to all problems requiring precise prediction of electrostatic interactions in water, such as protein-ligand and protein-protein docking, folding and chromatin regulation. Initial results indicate that this approach may shed new light on biomolecular electrostatics and on aspects of molecular recognition that classical local electrostatics cannot reveal.