In this paper, we introduce a novel matrix decomposition method, referred to as the ( D )-decomposition, designed to improve computational efficiency and stability for solving high-dimensional linear systems. The decomposition factorizes a matrix ( A in mathbb{R}^{n times n} ) into three matrices ( A = PDQ ), where ( P ), ( D ), and ( Q ) are structured to exploit sparsity, low rank, and other matrix properties. We provide rigorous proofs for the existence, uniqueness, and stability of the decomposition under various conditions, including noise perturbations and rank constraints. The ( D )-decomposition offers significant computational advantages, particularly for sparse or low-rank matrices, reducing the complexity from ( O(n^3) ) for traditional decompositions to ( O(n^2 k) ) or better, depending on the structure of the matrix. This method is particularly suited for large-scale applications in machine learning, signal processing, and data science. Numerical examples demonstrate the method's superior performance over traditional LU and QR decompositions, particularly in the context of dimensionality reduction and large-scale matrix factorization.
在本文中，我们介绍了一种新颖的矩阵分解方法，称为 ( D )-分解，旨在提高求解高维线性系统的计算效率和稳定性。该分解法将矩阵 A 分解为三个矩阵 A = PDQ，其中 P、D 和 Q 的结构利用了稀疏性、低秩和其他矩阵特性。我们提供了在各种条件（包括噪声扰动和秩约束）下分解的存在性、唯一性和稳定性的严格证明。D) 分解具有显著的计算优势，特别是对于稀疏或低秩矩阵，根据矩阵的结构，复杂度从传统分解的( O(n^3) )降低到( O(n^2 k) )或更高。数值示例证明了该方法优于传统 LU 和 QR 分解的性能，尤其是在降维和大规模矩阵因式分解方面。
{"title":"Efficient Matrix Decomposition for High-Dimensional Structured Systems: Theory and Applications","authors":"Ronald Katende","doi":"arxiv-2409.06321","DOIUrl":"https://doi.org/arxiv-2409.06321","url":null,"abstract":"In this paper, we introduce a novel matrix decomposition method, referred to\u0000as the ( D )-decomposition, designed to improve computational efficiency and\u0000stability for solving high-dimensional linear systems. The decomposition\u0000factorizes a matrix ( A in mathbb{R}^{n times n} ) into three matrices (\u0000A = PDQ ), where ( P ), ( D ), and ( Q ) are structured to exploit\u0000sparsity, low rank, and other matrix properties. We provide rigorous proofs for\u0000the existence, uniqueness, and stability of the decomposition under various\u0000conditions, including noise perturbations and rank constraints. The ( D\u0000)-decomposition offers significant computational advantages, particularly for\u0000sparse or low-rank matrices, reducing the complexity from ( O(n^3) ) for\u0000traditional decompositions to ( O(n^2 k) ) or better, depending on the\u0000structure of the matrix. This method is particularly suited for large-scale\u0000applications in machine learning, signal processing, and data science.\u0000Numerical examples demonstrate the method's superior performance over\u0000traditional LU and QR decompositions, particularly in the context of\u0000dimensionality reduction and large-scale matrix factorization.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Isabella Furci, Andrea Adriani, Stefano Serra-Capizzano
In recent years more and more involved block structures appeared in the literature in the context of numerical approximations of complex infinite dimensional operators modeling real-world applications. In various settings, thanks the theory of generalized locally Toeplitz matrix-sequences, the asymptotic distributional analysis is well understood, but a general theory is missing when general block structures are involved. The central part of the current work deals with such a delicate generalization when blocks are of (block) unilevel Toeplitz type, starting from a problem of recovery with missing data. Visualizations, numerical tests, and few open problems are presented and critically discussed.
{"title":"Block structured matrix-sequences and their spectral and singular value canonical distributions: a general theory","authors":"Isabella Furci, Andrea Adriani, Stefano Serra-Capizzano","doi":"arxiv-2409.06465","DOIUrl":"https://doi.org/arxiv-2409.06465","url":null,"abstract":"In recent years more and more involved block structures appeared in the\u0000literature in the context of numerical approximations of complex infinite\u0000dimensional operators modeling real-world applications. In various settings,\u0000thanks the theory of generalized locally Toeplitz matrix-sequences, the\u0000asymptotic distributional analysis is well understood, but a general theory is\u0000missing when general block structures are involved. The central part of the\u0000current work deals with such a delicate generalization when blocks are of\u0000(block) unilevel Toeplitz type, starting from a problem of recovery with\u0000missing data. Visualizations, numerical tests, and few open problems are\u0000presented and critically discussed.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of $Lambda$ under some reasonable conditions, where $Lambda$ denotes the error of the underlying MS-GFEM rs{approximation}. Notably, the convergence proof of GMRES does not rely on the `Elman theory'. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. Moreover, the convergence rate $Lambda$ is not only independent of the fine-mesh size $h$ and the number of subdomains, but decays with increasing wavenumber $k$. In particular, in the constant-coefficient case, with $hsim k^{-1-gamma}$ for some $gammain (0,1]$, it holds that $Lambda sim k^{-1+frac{gamma}{2}}$.
我们介绍并分析了基于多尺度谱广义有限元法（MS-GFEM）的异质亥姆霍兹（Helmholtz）问题的两级受限加法施瓦茨（RAS）预处理器 [C. Ma, C. Alber and R. Scheichl, SIAM.Ma, C.Alber, and R. Scheichl, SIAM.J. Numer.Anal., 61 (2023), pp.］该预处理使用带有阻抗边界条件的局部求解，以及基于局部特征问题构建的 MS-GFEM 近似空间的全局粗求解。它首先将 MS-GFEM 表述为 Richardson 迭代法，在不使用超采样技术的情况下，简化为最近在 [Q. Hu and Z. Li, arXiv2402.06905] 中提出并分析的预处理方法。我们证明，在一些合理的条件下，Richardson 迭代方法和 GMRES 中使用的预处理都能以 $Lambda$ 的速度收敛，其中 $Lambda$ 表示底层 MS-GFEM （rs{approximation}）的误差。值得注意的是，GMRES 的收敛证明并不依赖于 "埃尔曼理论"。超采样产生的 MS-GFEM 指数收敛特性确保了只需少量迭代就能在较小的粗空间内实现收敛。此外，收敛速率 $Lambda$ 不仅与细网格大小 $h$ 和子域数量无关，而且随着波长数 $k$ 的增加而衰减。特别是，在实体-系数情况下，当某个 $gammain(0,1]$ 为 $hsim k^{-1-gamma}$ 时，$Lambda sim k^{-1+frac{gamma}{2}}$ 是成立的。
{"title":"Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems","authors":"Chupeng Ma, Christian Alber, Robert Scheichl","doi":"arxiv-2409.06533","DOIUrl":"https://doi.org/arxiv-2409.06533","url":null,"abstract":"We present and analyze a two-level restricted additive Schwarz (RAS)\u0000preconditioner for heterogeneous Helmholtz problems, based on a multiscale\u0000spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C.\u0000Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The\u0000preconditioner uses local solves with impedance boundary conditions, and a\u0000global coarse solve based on the MS-GFEM approximation space constructed from\u0000local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson\u0000iterative method, and without using an oversampling technique, reduces to the\u0000preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv\u00002402.06905]. We prove that both the Richardson iterative method and the preconditioner\u0000used within GMRES converge at a rate of $Lambda$ under some reasonable\u0000conditions, where $Lambda$ denotes the error of the underlying MS-GFEM\u0000rs{approximation}. Notably, the convergence proof of GMRES does not rely on\u0000the `Elman theory'. An exponential convergence property of MS-GFEM, resulting\u0000from oversampling, ensures that only a few iterations are needed for\u0000convergence with a small coarse space. Moreover, the convergence rate $Lambda$\u0000is not only independent of the fine-mesh size $h$ and the number of subdomains,\u0000but decays with increasing wavenumber $k$. In particular, in the\u0000constant-coefficient case, with $hsim k^{-1-gamma}$ for some $gammain\u0000(0,1]$, it holds that $Lambda sim k^{-1+frac{gamma}{2}}$.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The second moment method is a linear acceleration technique which couples the transport equation to a diffusion equation with transport-dependent additive closures. The resulting low-order diffusion equation can be discretized independent of the transport discretization, unlike diffusion synthetic acceleration, and is symmetric positive definite, unlike quasi-diffusion. While this method has been shown to be comparable to quasi-diffusion in iterative performance for fixed source and time-dependent problems, it is largely unexplored as an eigenvalue problem acceleration scheme due to thought that the resulting inhomogeneous source makes the problem ill-posed. Recently, a preliminary feasibility study was performed on the second moment method for eigenvalue problems. The results suggested comparable performance to quasi-diffusion and more robust performance than diffusion synthetic acceleration. This work extends the initial study to more realistic reactor problems using state-of-the-art discretization techniques. Results in this paper show that the second moment method is more computationally efficient than its alternatives on complex reactor problems with unstructured meshes.
{"title":"A Second Moment Method for k-Eigenvalue Acceleration with Continuous Diffusion and Discontinuous Transport Discretizations","authors":"Zachary K. Hardy, Jim E. Morel, Jan I. C. Vermaak","doi":"arxiv-2409.06162","DOIUrl":"https://doi.org/arxiv-2409.06162","url":null,"abstract":"The second moment method is a linear acceleration technique which couples the\u0000transport equation to a diffusion equation with transport-dependent additive\u0000closures. The resulting low-order diffusion equation can be discretized\u0000independent of the transport discretization, unlike diffusion synthetic\u0000acceleration, and is symmetric positive definite, unlike quasi-diffusion. While\u0000this method has been shown to be comparable to quasi-diffusion in iterative\u0000performance for fixed source and time-dependent problems, it is largely\u0000unexplored as an eigenvalue problem acceleration scheme due to thought that the\u0000resulting inhomogeneous source makes the problem ill-posed. Recently, a\u0000preliminary feasibility study was performed on the second moment method for\u0000eigenvalue problems. The results suggested comparable performance to\u0000quasi-diffusion and more robust performance than diffusion synthetic\u0000acceleration. This work extends the initial study to more realistic reactor\u0000problems using state-of-the-art discretization techniques. Results in this\u0000paper show that the second moment method is more computationally efficient than\u0000its alternatives on complex reactor problems with unstructured meshes.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the eigenvalue problem for the magnetic Schr"odinger operator and take advantage of a property called gauge invariance to transform the given problem into an equivalent problem that is more amenable to numerical approximation. More specifically, we propose a canonical magnetic gauge that can be computed by solving a Poisson problem, that yields a new operator having the same spectrum but eigenvectors that are less oscillatory. Extensive numerical tests demonstrate that accurate computation of eigenpairs can be done more efficiently and stably with the canonical magnetic gauge.
{"title":"A Canonical Gauge for Computing of Eigenpairs of the Magnetic Schrödinger Operator","authors":"Jeffrey S. Ovall, Li Zhu","doi":"arxiv-2409.06023","DOIUrl":"https://doi.org/arxiv-2409.06023","url":null,"abstract":"We consider the eigenvalue problem for the magnetic Schr\"odinger operator\u0000and take advantage of a property called gauge invariance to transform the given\u0000problem into an equivalent problem that is more amenable to numerical\u0000approximation. More specifically, we propose a canonical magnetic gauge that\u0000can be computed by solving a Poisson problem, that yields a new operator having\u0000the same spectrum but eigenvectors that are less oscillatory. Extensive\u0000numerical tests demonstrate that accurate computation of eigenpairs can be done\u0000more efficiently and stably with the canonical magnetic gauge.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Complex networks are made up of vertices and edges. The edges, which may be directed or undirected, are equipped with positive weights. Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices in distinct layers represent different kinds of objects. Multiplex networks are special vertex-aligned multilayer networks, in which vertices in distinct layers are identified with each other and inter-layer edges connect each vertex with its copy in other layers and have a fixed weight $gamma>0$ associated with the ease of communication between layers. This paper discusses two different approaches to analyze communication in a multiplex. One approach focuses on the multiplex global efficiency by using the multiplex path length matrix, the other approach considers the multiplex total communicability. The sensitivity of both the multiplex global efficiency and the multiplex total communicability to structural perturbations in the network is investigated to help to identify intra-layer edges that should be strengthened to enhance communicability.
{"title":"Communication in Multiplex Transportation Networks","authors":"Silvia Noschese, Lothar Reichel","doi":"arxiv-2409.05575","DOIUrl":"https://doi.org/arxiv-2409.05575","url":null,"abstract":"Complex networks are made up of vertices and edges. The edges, which may be\u0000directed or undirected, are equipped with positive weights. Modeling complex\u0000systems that consist of different types of objects leads to multilayer\u0000networks, in which vertices in distinct layers represent different kinds of\u0000objects. Multiplex networks are special vertex-aligned multilayer networks, in\u0000which vertices in distinct layers are identified with each other and\u0000inter-layer edges connect each vertex with its copy in other layers and have a\u0000fixed weight $gamma>0$ associated with the ease of communication between\u0000layers. This paper discusses two different approaches to analyze communication\u0000in a multiplex. One approach focuses on the multiplex global efficiency by\u0000using the multiplex path length matrix, the other approach considers the\u0000multiplex total communicability. The sensitivity of both the multiplex global\u0000efficiency and the multiplex total communicability to structural perturbations\u0000in the network is investigated to help to identify intra-layer edges that\u0000should be strengthened to enhance communicability.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a point-wise divergence-free projection method for numerical approximations of photonic quasicrystals problems. The original three-dimensional quasiperiodic Maxwell's system is transformed into a periodic one in higher dimensions through a variable substitution involving the projection matrix, such that periodic boundary condition can be readily applied. To deal with the intrinsic divergence-free constraint of the Maxwell's equations, we present a quasiperiodic de Rham complex and its associated commuting diagram, based on which a point-wise divergence-free quasiperiodic Fourier spectral basis is proposed. With the help of this basis, we then propose an efficient solution algorithm for the quasiperiodic source problem and conduct its rigorous error estimate. Moreover, by analyzing the decay rate of the Fourier coefficients of the eigenfunctions, we further propose a divergence-free reduced projection method for the quasiperiodic Maxwell eigenvalue problem, which significantly alleviates the computational cost. Several numerical experiments are presented to validate the efficiency and accuracy of the proposed method.
{"title":"A divergence-free projection method for quasiperiodic photonic crystals in three dimensions","authors":"Zixuan Gao, Zhenli Xu, Zhiguo Yang","doi":"arxiv-2409.05528","DOIUrl":"https://doi.org/arxiv-2409.05528","url":null,"abstract":"This paper presents a point-wise divergence-free projection method for\u0000numerical approximations of photonic quasicrystals problems. The original\u0000three-dimensional quasiperiodic Maxwell's system is transformed into a periodic\u0000one in higher dimensions through a variable substitution involving the\u0000projection matrix, such that periodic boundary condition can be readily\u0000applied. To deal with the intrinsic divergence-free constraint of the Maxwell's\u0000equations, we present a quasiperiodic de Rham complex and its associated\u0000commuting diagram, based on which a point-wise divergence-free quasiperiodic\u0000Fourier spectral basis is proposed. With the help of this basis, we then\u0000propose an efficient solution algorithm for the quasiperiodic source problem\u0000and conduct its rigorous error estimate. Moreover, by analyzing the decay rate\u0000of the Fourier coefficients of the eigenfunctions, we further propose a\u0000divergence-free reduced projection method for the quasiperiodic Maxwell\u0000eigenvalue problem, which significantly alleviates the computational cost.\u0000Several numerical experiments are presented to validate the efficiency and\u0000accuracy of the proposed method.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sarah Perez, Jean-Matthieu Etancelin, Philippe Poncet
This article introduces a new efficient particle method for the numerical simulation of crystallization and precipitation at the pore scale of real rock geometries extracted by X-Ray tomography. It is based on the coupling between superficial velocity models of porous media, Lagrangian description of chemistry using Transition-State-Theory, involving underlying grids. Its ability to successfully compute dissolution process has been established in the past and is presently generalized to precipitation and crystallization by means of adsorption modeling. Numerical simulations of mineral CO2 trapping are provided, showing evidence of clogging/non-clogging regimes, and one of the main results is the introduction of a new non-dimensional number needed for this characterization.
本文介绍了一种新的高效粒子方法，用于对通过 X 射线断层扫描提取的真实岩石几何结构的孔隙尺度上的结晶和沉淀进行数值模拟。该方法基于多孔介质表层速度模型和使用过渡态理论的拉格朗日化学描述之间的耦合，涉及底层网格。该模型成功计算溶解过程的能力已在过去得到证实，目前正通过吸附建模将其推广到沉淀和结晶。提供了矿物二氧化碳捕集的数值模拟，显示了堵塞/非堵塞机制的证据，主要成果之一是引入了表征该特征所需的新的非维数。
{"title":"A semi-Lagrangian method for the direct numerical simulation of crystallization and precipitation at the pore scale","authors":"Sarah Perez, Jean-Matthieu Etancelin, Philippe Poncet","doi":"arxiv-2409.05449","DOIUrl":"https://doi.org/arxiv-2409.05449","url":null,"abstract":"This article introduces a new efficient particle method for the numerical\u0000simulation of crystallization and precipitation at the pore scale of real rock\u0000geometries extracted by X-Ray tomography. It is based on the coupling between\u0000superficial velocity models of porous media, Lagrangian description of\u0000chemistry using Transition-State-Theory, involving underlying grids. Its\u0000ability to successfully compute dissolution process has been established in the\u0000past and is presently generalized to precipitation and crystallization by means\u0000of adsorption modeling. Numerical simulations of mineral CO2 trapping are\u0000provided, showing evidence of clogging/non-clogging regimes, and one of the\u0000main results is the introduction of a new non-dimensional number needed for\u0000this characterization.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In block eigenvalue algorithms, such as the subspace iteration algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm, a large block size is often employed to achieve robustness and rapid convergence. However, using a large block size also increases the computational cost. Traditionally, the block size is typically reduced after convergence of some eigenpairs, known as deflation. In this work, we propose a non-deflation-based, more aggressive technique, where the block size is adjusted dynamically during the algorithm. This technique can be applied to a wide range of block eigensolvers, reducing computational cost without compromising convergence speed. We present three adaptive strategies for adjusting the block size, and apply them to four well-known eigensolvers as examples. Theoretical analysis and numerical experiments are provided to illustrate the efficiency of the proposed technique. In practice, an overall acceleration of 20% to 30% is observed.
{"title":"On a shrink-and-expand technique for block eigensolvers","authors":"Yuqi Liu, Yuxin Ma, Meiyue Shao","doi":"arxiv-2409.05572","DOIUrl":"https://doi.org/arxiv-2409.05572","url":null,"abstract":"In block eigenvalue algorithms, such as the subspace iteration algorithm and\u0000the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm,\u0000a large block size is often employed to achieve robustness and rapid\u0000convergence. However, using a large block size also increases the computational\u0000cost. Traditionally, the block size is typically reduced after convergence of\u0000some eigenpairs, known as deflation. In this work, we propose a\u0000non-deflation-based, more aggressive technique, where the block size is\u0000adjusted dynamically during the algorithm. This technique can be applied to a\u0000wide range of block eigensolvers, reducing computational cost without\u0000compromising convergence speed. We present three adaptive strategies for\u0000adjusting the block size, and apply them to four well-known eigensolvers as\u0000examples. Theoretical analysis and numerical experiments are provided to\u0000illustrate the efficiency of the proposed technique. In practice, an overall\u0000acceleration of 20% to 30% is observed.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the compressible Euler equations of gas dynamics with isentropic equation of state. In the low Mach number regime i.e. when the fluid velocity is very very small in comparison to the sound speed in the medium, the solution of the compressible system converges to the solution of its incompressible counter part. Standard numerical schemes fail to respect this transition property and hence are plagued with inaccuracies as well as instabilities. In this paper we introduce an extra flux term to the momentum flux. This extra term is brought to fore by looking at the incompressibility constraints of the asymptotic limit system. This extra flux term enables us to get a suitable flux splitting, so that an additive IMEX-RK scheme could be applied. Using an elliptic reformulation the scheme boils down to just solving a linear elliptic problem for the density and then explicit updates for the momentum. The IMEX schemes developed are shown to be formally asymptotically consistent with the low Mach number limit of the Euler equations. A second order space time fully discrete scheme is obtained in the finite volume framework using a combination of Rusanov flux for the explicit part and simple central differences for the implicit part. Numerical results are reported which elucidate the theoretical assertions regarding the scheme and its robustness.
{"title":"Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations","authors":"Saurav Samantaray","doi":"arxiv-2409.05854","DOIUrl":"https://doi.org/arxiv-2409.05854","url":null,"abstract":"We consider the compressible Euler equations of gas dynamics with isentropic\u0000equation of state. In the low Mach number regime i.e. when the fluid velocity\u0000is very very small in comparison to the sound speed in the medium, the solution\u0000of the compressible system converges to the solution of its incompressible\u0000counter part. Standard numerical schemes fail to respect this transition\u0000property and hence are plagued with inaccuracies as well as instabilities. In\u0000this paper we introduce an extra flux term to the momentum flux. This extra\u0000term is brought to fore by looking at the incompressibility constraints of the\u0000asymptotic limit system. This extra flux term enables us to get a suitable flux\u0000splitting, so that an additive IMEX-RK scheme could be applied. Using an\u0000elliptic reformulation the scheme boils down to just solving a linear elliptic\u0000problem for the density and then explicit updates for the momentum. The IMEX\u0000schemes developed are shown to be formally asymptotically consistent with the\u0000low Mach number limit of the Euler equations. A second order space time fully\u0000discrete scheme is obtained in the finite volume framework using a combination\u0000of Rusanov flux for the explicit part and simple central differences for the\u0000implicit part. Numerical results are reported which elucidate the theoretical\u0000assertions regarding the scheme and its robustness.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"389 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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