Improving multifrontal methods by means of block low-rank representations

Submitted for publication to SIAMMatrices coming from elliptic Partial Differential Equations (PDEs) have been shown to have a low-rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives to the blocks a geometrical meaning, such approximations can be computed using an SVD or a rank-revealing QR factorization. The resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations. Several strategies have been proposed to exploit this property. We propose a low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method. We present experimental results that show how the BLR format delivers gains that are comparable to those obtained with hierarchical formats such as Hierarchical matrices (H matrices) and Hierarchically Semi-Separable (HSS matrices) but provides much greater flexibility and ease of use which are essential in the context of a general purpose, algebraic solver

Grouping variables in Frontal Matrices to improve Low-Rank Approximations in a Multifrontal Solver

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Investigation of N-carbamoylamino acid nitrosation by {NO+O2} in solid-gas phase. Effects of NOx speciation and kinetic evidence of a multiple-stage process

Nitrosation of N-carbamoylamino acids (CAA) by gaseous NO+O2, an interesting synthetic pathway to amino acid N-carboxyanhydrides (NCA), alternative to the phosgene route, was investigated on N-carbamoyl-valine either in acetonitrile suspension or solventless conditions, and compared to the classical nitrosating system NaNO2+CF3COOH (TFA), the latter being quite less efficient in terms of either rate, stoichiometric demand, or further tractability of the product. The rate and efficiency of the NO+O2 reaction mainly depends on the O2/NO ratio. Evaluation of the contribution of various nitrosating species (N2O3, N2O4, HNO2) through stoichiometric balance showed the reaction to be effected mostly by N2O3 for O2/NO ratios below 0.3, and by N2O4 for O2/NO ratios above 0.4. The relative contribution of (subsequently formed) HNO2 always remains minor. Differential scanning calorimetry (DSC) monitoring of the reaction in the solid phase by either HNO2 (from NaNO2+TFA), gaseous N2O4 or gaseous N2O3 , provides the associated rate constants (ca. 0.1, 2 and 10^8 s-1 at 25°C, respectively), showing that N2O3 is by far the most reactive of these nitrosating species. From the DSC measurement, the latent heat of fusion of N2O3, 2.74 kJ.mol-1 at -105°C is also obtained for the first time. The kinetics was investigated under solventless conditions at 0°C, by either quenching experiments or less tedious, rough calorimetric techniques. Auto-accelerated, parabolic-shaped kinetics was observed in the first half of the reaction course, together with substantial heat release (temperature increase of ca. 20°C within 1-2 min in a 20-mg sample), followed by pseudo-zero-order kinetics after a sudden, important decrease in apparent rate. This kinetic break is possibly due to the transition between the initial solid-gas system and a solid-liquid-gas system resulting from water formation. Overall rate constants increased with parameters such as the specific surface of the solid, the O2/NO ratio, or the presence of moisture (or equivalently the hydrophilicity of the involved CAA), however without precise relationship, while the last two parameters may directly correlate to the increasing acidity of the medium

Space-time residual-based a posteriori estimator for the A-phi formulation in eddy current problems

International audienceIn this paper, an a posteriori residual error estimator is presented for the 3-D eddy current problem modeled by the space-time A - φ potential formulation. It is solved by the finite element method in space and the backward Euler scheme in time. Once the reliability as well as the efficiency of the estimator is established, two numerical tests are proposed: 1) an analytical one to validate the theoretical results and 2) a physical one to illustrate the performance for a real eddy current problem

Process improvement in amino acid N-carboxyanhydride synthesis by N-carbamoyl amino acid nitrosation

International audienc

Testing and comparing the Augmented Block Cimmino Distributed solver on some class of applications

International audienc

Improving Multifrontal Methods by Means of Block Low-Rank Representations

International audienceMatrices coming from elliptic partial differential equations have been shown to have alow-rank property: well-defined off-diagonal blocks of their Schur complements can be approximatedby low-rank products. Given a suitable ordering of the matrix which gives the blocks a geometricalmeaning, such approximations can be computed using an SVD or a rank-revealing QR factorization.The resulting representation offers a substantial reduction of the memory requirement and givesefficient ways to perform many of the basic dense linear algebra operations. Several strategies,mostly based on hierarchical formats, have been proposed to exploit this property. We study asimple, nonhierarchical, low-rank format called block low-rank (BLR) and explain how it can be usedto reduce the memory footprint and the complexity of sparse direct solvers based on the multifrontalmethod. We present experimental results on matrices coming from elliptic PDEs and from variousother applications. We show that even if BLR-based factorizations are asymptotically less efficientthan hierarchical approaches, they still deliver considerable gains. The BLR format is compatiblewith numerical pivoting, and its simplicity and flexibility make it easy to use in the context of ageneral purpose, algebraic solver