Linear-time CUR approximation of BEM matrices

International audienceIn this paper we propose linear-time CUR approximation algorithms for admissible matrices obtained from the hierarchical form of Boundary Element matrices. We propose a new approach called geometric sampling to obtain indices of most significant rows and columns usinginformation from the domains where the problem is posed. Our strategy is tailored to Boundary Element Methods (BEM) since it uses directly and explicitly the cluster tree containing information from the problem geometry. Our CUR algorithm has precision comparable with low-rankapproximations created with the truncated QR factorization with column pivoting (QRCP) and the Adaptive Cross Approximation (ACA) with full pivoting, which are quadratic-cost methods. When compared to the well-known linear-time algorithm ACA with partial pivoting, we show that our algorithm improves, in general, the convergence error and overcomes some cases where ACA fails. We provide a general relative error bound for CUR approximations created with geometrical sampling. Finally, we evaluate the performance of our algorithms on traditional BEM problemsdefined over different geometries.Dans cet article, nous présentons des algorithmes pour créer une approximation de rang faible de type CUR pour des matrices résultant de la discrétisation des équations intégrales par la méthode des éléments de frontière (BEM). Notre approche consiste à utiliser l’information sur la géométrie du problème pour choisir des colonnes et des lignes les plus représentatives de la matrice. Nous montrons que notre algorithme principal, dont le coût est linéaire, a la même précision que des méthodes, ayant coût quadratique, comme QRCP et Approximation Adaptative Croisée (ACA) avec pivotage complet. Nous présentons des expériences numériques sur des domaines complexes en utilisant des noyaux intégrales fréquemment utilisés dans la littérature

A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization

We present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices appear in many applications, e.g., finite element methods, boundary element methods, etc. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The compression algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. This work is part of a more global effort, the STRUMPACK (STRUctured Matrices PACKage) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver

Efficient Multikernel Hierarchical Compression for Boundary Element Matrices

We present a new scheme that allows for the compression of operators of the combined field integral equation in the low-frequency regime using a hierarchical decomposition that leverages a unified pseudoskeleton approximation of both the single- and the double-layer Green’s function at the quadrature points. Compared with a standard adaptive cross approximation, the numerical results show a reduced memory consump- tion without sacrificing computational run time

Fluctuating surface-current formulation of radiative heat transfer: theory and applications

We describe a novel fluctuating-surface current formulation of radiative heat transfer between bodies of arbitrary shape that exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering. Unlike previous approaches to non-equilibrium fluctuations that involve scattering matrices---relating "incoming" and "outgoing" waves from each body---our approach is formulated in terms of "unknown" surface currents, laying at the surfaces of the bodies, that need not satisfy any wave equation. We show that our formulation can be applied as a spectral method to obtain fast-converging semi-analytical formulas in high-symmetry geometries using specialized spectral bases that conform to the surfaces of the bodies (e.g. Fourier series for planar bodies or spherical harmonics for spherical bodies), and can also be employed as a numerical method by exploiting the generality of surface meshes/grids to obtain results in more complicated geometries (e.g. interleaved bodies as well as bodies with sharp corners). In particular, our formalism allows direct application of the boundary-element method, a robust and powerful numerical implementation of the surface-integral formulation of classical electromagnetism, which we use to obtain results in new geometries, including the heat transfer between finite slabs, cylinders, and cones