scholarly journals Aspects of parallel computing to solve Helmholtz equation by a direct solver with low-rank approximation and the HSS format of data storage

2015 ◽  
pp. 607-616
Author(s):  
Б.М. Глинский ◽  
В.И. Костин ◽  
Н.В. Кучин ◽  
С.А. Соловьев ◽  
В.А. Чеверда
Keyword(s):  
Helmholtz Equation ◽  
Heterogeneous Media ◽  
Gaussian Elimination ◽  
Full Rank ◽  
Low Rank ◽  
Approximation Technique ◽  
Low Rank Approximation ◽  
Direct Solvers ◽  
Linear Algebraic Equations ◽  
Rank Approximation

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).

scholarly journals Solution of the Helmholtz problem using the preconditioned low-rank approximation technique

2015 ◽  
pp. 268-280
Author(s):  
К.В. Воронин ◽  
С.А. Соловьев
Keyword(s):  
Convergence Rate ◽  
Heterogeneous Media ◽  
Iterative Refinement ◽  
Low Rank ◽  
Seismic Exploration ◽  
Approximation Technique ◽  
Low Rank Approximation ◽  
Iterative Processes ◽  
Helmholtz Problem ◽  
Rank Approximation

Предложен алгоритм решения задачи Гельмгольца в трехмерных неоднородных средах с использованием метода аппроксимации матрицами малого ранга. Рассматриваемый метод применяется в качестве предобусловливателя для двух итерационных процессов. Первый - простой в реализации и экономичный метод итерационного уточнения, второй - метод BiCGStab крыловского типа. Скорость сходимости обоих методов исследуется относительно качества предобусловливателя, которое определяется точностью малоранговой аппроксимации. Показано, что для типичных задач сейсморазведки скорость сходимости двух рассматриваемых методов примерно одинакова начиная с некоторой точности малоранговой аппроксимации. Вычислительные эксперименты показали, что при точности, достаточной для решения практических задач, предложенный метод более чем в 2 раза экономнее по использованию памяти и в 3 раза производительнее, чем прямой метод PARDISO библиотеки Intel MKL. An algorithm for solving the Helmholtz problem in 3D heterogeneous media using the low-rank approximation technique is proposed. This technique is applied as a preconditioner for two different iterative processes: an iterative refinement and BiCGStab. The iterative refinement approach is known to be very simple and straightforward but can suffer from the lack of convergence; BiCGStab is more stable and more sophisticated as well. A dependence of the convergence rate on low-rank approximation quality is studied for these iterative processes. For typical problems of seismic exploration, it is shown that, starting with some low-rank accuracy, the convergence rate of the iterative refinement is very similar to BiCGStab. Therefore, it is preferable to use the more efficient iterative refinement method. Numerical experiments also show that, for reasonable (from the practical standpoint) low-rank accuracy, the proposed method provides three times performance gain (for sequential code) and reduces the memory usage up to a factor of two in comparison with the Intel MKL PARDISO high performance direct solver.

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

2014 ◽  
Vol 50 (2) ◽  
pp. 517-520 ◽  
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

Improved MR image denoising via low- rank approximation and Laplacian-of-Gaussian edge detector

2020 ◽  
Vol 14 (12) ◽  
pp. 2791-2798
Author(s):  
Xiaoqun Qiu ◽  
Zhen Chen ◽  
Saifullah Adnan ◽  
Hongwei He
Keyword(s):  
Image Denoising ◽  
Low Rank ◽  
Edge Detector ◽  
Low Rank Approximation ◽  
Mr Image ◽  
Rank Approximation

scholarly journals Fast Joint Multiband Reconstruction From Wideband Images Based on Low-Rank Approximation

2020 ◽  
Vol 6 ◽  
pp. 922-933
Author(s):  
M. Amine Hadj-Youcef ◽  
Francois Orieux ◽  
Alain Abergel ◽  
Aurelia Fraysse
Keyword(s):  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Approximation

scholarly journals Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product

2021 ◽  
Vol 11 (10) ◽  
pp. 4582
Author(s):  
Kensuke Tanioka ◽  
Satoru Hiwa
Keyword(s):  
Dimensional Reduction ◽  
Low Rank ◽  
Inner Product ◽  
Low Rank Approximation ◽  
Correlation Matrices ◽  
Rank Matrix ◽  
Clustering Approach ◽  
Rank Approximation ◽  
The Difference ◽  
Low Rank Matrix

In the domain of functional magnetic resonance imaging (fMRI) data analysis, given two correlation matrices between regions of interest (ROIs) for the same subject, it is important to reveal relatively large differences to ensure accurate interpretation. However, clustering results based only on differences tend to be unsatisfactory and interpreting the features tends to be difficult because the differences likely suffer from noise. Therefore, to overcome these problems, we propose a new approach for dimensional reduction clustering. Methods: Our proposed dimensional reduction clustering approach consists of low-rank approximation and a clustering algorithm. The low-rank matrix, which reflects the difference, is estimated from the inner product of the difference matrix, not only from the difference. In addition, the low-rank matrix is calculated based on the majorize–minimization (MM) algorithm such that the difference is bounded within the range −1 to 1. For the clustering process, ordinal k-means is applied to the estimated low-rank matrix, which emphasizes the clustering structure. Results: Numerical simulations show that, compared with other approaches that are based only on differences, the proposed method provides superior performance in recovering the true clustering structure. Moreover, as demonstrated through a real-data example of brain activity measured via fMRI during the performance of a working memory task, the proposed method can visually provide interpretable community structures consisting of well-known brain functional networks, which can be associated with the human working memory system. Conclusions: The proposed dimensional reduction clustering approach is a very useful tool for revealing and interpreting the differences between correlation matrices, even when the true differences tend to be relatively small.

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