Trading performance for memory in sparse direct solvers using low-rank compression

Предложен алгоритм решения систем линейных алгебраических уравнений (СЛАУ), основанный на методе исключении Гаусса и предназначенный для решения уравнения Гельмгольца в трехмерных неоднородных средах. Для решения СЛАУ, возникающих в геофизических приложениях, разработана параллельная версия алгоритма, направленная на использование гетерогенных высокопроизводительных вычислительных систем, содержащих узлы с MPP- и SMP-архитектурой. Малоранговая аппроксимация, HSS-формат и динамическое распределение промежуточных результатов среди кластерных узлов позволяют решать задачи в разы большие, чем при использовании традиционных прямых методов, сохраняющих блоки $L$-фактора в полном ранге (Full-Rank, FR). Использование предложенного алгоритма позволяет сократить время расчетов, что актуально для решения трехмерных задач геофизики. Численные эксперименты подтверждают упомянутые преимущества предложенного малорангового прямого метода (Low-Rank, LR) по сравнению с прямыми FR-методами. На модельных геофизических задачах показана жизнеспособность реализованного алгоритма. An algorithm for solving systems of linear algebraic equations based on the Gaussian elimination method is proposed. The algorithm is aimed to solve boundary value problems for the Helmholtz equation in 3D heterogeneous media. In order to solve linear systems raised from geophysical applications, we developed a parallel version targeted on heterogeneous high-performance computing clusters (MPP and SMP architecture). Using the low-rank approximation technique and the HSS format allows us to solve problems larger than by the use of traditional direct solvers with saving the L-factor in full rank (FR). Using the proposed approach reduces computation time; it is the key-point of 3D geophysical problems. Numerical experiments demonstrate a number of advantages of the proposed low-rank approach in comparison with direct solvers (FR-approaches).

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BLAS IV: A BLAS for Rk Matrix Algebra

Applied Computational Electromagnetics Society ◽

10.47037/2020.aces.j.351102 ◽

2021 ◽

Vol 35 (11) ◽

pp. 1266-1267

Author(s):

John Shaeffer

Keyword(s):

Linear Algebra ◽

Matrix Algebra ◽

Full Rank ◽

Low Rank ◽

System Matrix ◽

Low Level ◽

Matrix Operations ◽

Full Wave ◽

Direct Solvers ◽

Alternative Approach

Basic Linear Algebra Subroutines (BLAS) are well-known low-level workhorse subroutines for linear algebra vector-vector, matrixvector and matrix-matrix operations for full rank matrices. The advent of block low rank (Rk) full wave direct solvers, where most blocks of the system matrix are Rk, an extension to the BLAS III matrix-matrix work horse routine is needed due to the agony of Rk addition. This note outlines the problem of BLAS III for Rk LU and solve operations and then outlines an alternative approach, which we will call BLAS IV. This approach utilizes the thrill of Rk matrix-matrix multiply and uses the Adaptive Cross Approximation (ACA) as a methodology to evaluate sums of Rk terms to circumvent the agony of low rank addition.

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Shared-Memory Parallelism and Low-Rank Approximation Techniques Applied to Direct Solvers in FEM Simulation

IEEE Transactions on Magnetics ◽

10.1109/tmag.2013.2284024 ◽

2014 ◽

Vol 50 (2) ◽

pp. 517-520 ◽

Cited By ~ 5

Author(s):

Patrick Amestoy ◽

Alfredo Buttari ◽

Guillaume Joslin ◽

Jean-Yves L'Excellent ◽

Mohamed Sid-Lakhdar ◽

...

Keyword(s):

Shared Memory ◽

Fem Simulation ◽

Low Rank ◽

Low Rank Approximation ◽

Approximation Techniques ◽

Direct Solvers ◽

Rank Approximation

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Tensor Recovery via Nonconvex Low-Rank Approximation

2020 28th European Signal Processing Conference (EUSIPCO) ◽

10.23919/eusipco47968.2020.9287404 ◽

2021 ◽

Author(s):

Lin Chen ◽

Xue Jiang ◽

Xingzhao Liu ◽

Zhixin Zhou

Keyword(s):

Low Rank ◽

Low Rank Approximation ◽

Tensor Recovery ◽

Rank Approximation

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The Influences of the Rank of Coal on Methane Sorption Capacity in Coals/Wpływ Rzędu Węgla Na Pojemność Sorpcyjną Metanu W Węglach

Archives of Mining Sciences ◽

10.2478/amsc-2014-0037 ◽

2014 ◽

Vol 59 (2) ◽

pp. 509-516

Author(s):

Andrzej Olajossy

Keyword(s):

Sorption Capacity ◽

Coalbed Methane ◽

Vitrinite Reflectance ◽

Sorption Isotherms ◽

Volatile Matter ◽

Low Rank ◽

Hard Coal ◽

Petrographic Composition ◽

Methane Sorption ◽

Indian Coals

Abstract Methane sorption capacity is of significance in the issues of coalbed methane (CBM) and depends on various parameters, including mainly, on rank of coal and the maceral content in coals. However, in some of the World coals basins the influences of those parameters on methane sorption capacity is various and sometimes complicated. Usually the rank of coal is expressed by its vitrinite reflectance Ro. Moreover, in coals for which there is a high correlation between vitrinite reflectance and volatile matter Vdaf the rank of coal may also be represented by Vdaf. The influence of the rank of coal on methane sorption capacity for Polish coals is not well understood, hence the examination in the presented paper was undertaken. For the purpose of analysis there were chosen fourteen samples of hard coal originating from the Upper Silesian Basin and Lower Silesian Basin. The scope of the sorption capacity is: 15-42 cm3/g and the scope of vitrinite reflectance: 0,6-2,2%. Majority of those coals were of low rank, high volatile matter (HV), some were of middle rank, middle volatile matter (MV) and among them there was a small number of high rank, low volatile matter (LV) coals. The analysis was conducted on the basis of available from the literature results of research of petrographic composition and methane sorption isotherms. Some of those samples were in the form (shape) of grains and others - as cut out plates of coal. The high pressure isotherms previously obtained in the cited studies were analyzed here for the purpose of establishing their sorption capacity on the basis of Langmuire equation. As a result of this paper, it turned out that for low rank, HV coals the Langmuire volume VL slightly decreases with the increase of rank, reaching its minimum for the middle rank (MV) coal and then increases with the rise of the rank (LV). From the graphic illustrations presented with respect to this relation follows the similarity to the Indian coals and partially to the Australian coals.

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