Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners

IF 1.8 3区 数学 Q1 MATHEMATICS
B. Khoromskij, V. Khoromskaia
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引用次数: 1

Abstract

In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {\mathbb{R}}^d $$ , d=2,3$$ d=2,3 $$ . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large L×L$$ L\times L $$ , or L×L×L$$ L\times L\times L $$ lattice, respectively. The finite element method discretization procedure on a 3D n×n×n$$ n\times n\times n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size n×n×n$$ n\times n\times n $$ . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$ , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.
用张量结构预条件快速求解具有随机跳跃系数的三维椭圆方程
在本文中,我们提出并分析了随机介质中周期椭圆问题的快速求解的数值算法ℝd$${\mathbb{R}}^d$$,d=2,3$$d=2,3$$。对于跳跃方程系数,分别考虑了二维(2D)和三维(3D)椭圆问题,该跳跃方程系数是随机分布在大L×L$$L\times L$$或L×L×L$L\timesL$$格上的凸块的棋盘型配置。详细描述了三维n×n×n$n\times\timesn$$均匀张量网格上的有限元离散化过程,并提出了Kronecker张量积方法来快速生成刚度矩阵。我们介绍了张量技术,用于在周期设置中构造低Kronecker秩谱等价预条件器,用于预条件共轭梯度迭代的框架中。离散的三维周期拉普拉斯伪逆首先在傅立叶基中对角化,然后将对角矩阵重塑为大小为n×n×n$$n\times\timesn$$的完全填充三阶张量。后者通过使用多重网格Tucker到正则张量变换由低阶正则张量近似。例如,我们将所提出的求解器应用于随机均匀化方法的数值分析中,其中三维椭圆方程应求解数百次,并且对于方程系数的每次随机采样,都必须构造新的刚度矩阵和右手边。在2D和3D情况下,所提出的求解器在晶格参数L$$L$$和网格大小nd$${n}^d$$方面的计算特性在数值测试中得到了说明。我们的求解器可用于各种应用,其中椭圆问题应针对许多不同的系数求解,例如,在多粒子动力学、蛋白质对接问题或随机建模中。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
3.40
自引率
2.30%
发文量
50
审稿时长
12 months
期刊介绍: Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review. Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects. Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.
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