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The Hierarchical subspace iteration method for Laplace-Beltrami Eigenproblems

ACM TRANSACTIONS ON GRAPHICS 2022年第2期41卷 p1-14页

作者： Nasikun, Ahmad Hildebrandt, Klaus Delft Univ Technol Delft Netherlands Univ Gadjah Mada Dept Tekn Elektro & Teknol Informasi Fak Tekn Jl Graf 2 Kampus UGM Yogyakarta 55281 Indonesia Delft Univ Technol Dept Intelligent Syst Van Mourik Broekmanweg 6 NL-2628 XE Delft Netherlands

Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace-Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The subspace iteration method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical subspace iteration method (HSIM), a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level. are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical subspace iteration method. Our experiments show that HSIM can solve Laplace Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.

关键词： Laplace-Beltrami operator Laplace matrix spectral methods multigrid eigensolver subspace iteration method

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The subspace iteration method for nonlinear eigenvalue problems occurring in the dynamics of structures with viscoelastic elements

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COMPUTERS & STRUCTURES 2021年 254卷 106571-106571页

作者： Lasecka-Plura, Magdalena Lewandowski, Roman Poznan Univ Tech Ul Piotrowo 5 PL-60965 Poznan Poland

The paper presents an extension of the subspace iteration method for application in systems with vis-coelastic damping elements, which are described by both classical and fractional models. The presented method enables determination of only a certain number of eigenvalues and associated eigenvectors. This is very useful, especially when the considered problem has many degrees of freedom, in which case it would be very time-consuming or even impossible to determine all the eigenvalues. At the same time, from the engineering point of view, it is not necessary to know all the eigenvalues. In this paper, the sub-space iteration method is used for the first time to solve the nonlinear eigenproblems which appear in the dynamic analysis of systems with viscoelastic damping elements. The proposed solution consists of assuming the number of eigenvalues to be determined, taking the initial point of iteration and then solv -ing the nonlinear reduced eigenproblem with Hermitian matrices in each iterative loop. The solution to the reduced eigenproblem is obtained using the continuation method, and it is an additional novelty in the paper. The correctness and effectiveness of the proposed approach are illustrated with numerical examples. (c) 2021 Elsevier Ltd. All rights reserved.

关键词： Nonlinear eigenvalue problem subspace iteration method Continuation method Fractional derivatives Viscoelastic material

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Calculation of coupled modes of fluid-structure systems by pseudo symmetric subspace iteration method

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SCIENTIA IRANICA 2019年第4期26卷 2100-2107页

作者： Lotfi, V Sani, A. Aftabi Amirkabir Univ Technol Dept Civil & Environm Engn Tehran Iran Ferdowsi Univ Mashhad Dept Civil Engn Mashhad Razavi Khorasan Iran

An efficient technique is proposed for calculation of coupled modes of fluid-structure interaction systems. The algorithm was developed with the symmetric matrix operation mentality such that apparently a symmetric eigen-problem was being solved. It was proven that each left eigen-vector was related to the corresponding right eigen-vector through a simple relation. Therefore, the subsequent transient analysis could readily be performed. Overall, the method seemed very efficient and was ideal to employ in general-purpose finite element programs for solving the above-mentioned eigen-problems. (C) 2019 Sharif University of Technology. All rights reserved.

关键词： Fluid-structure interaction Eigen-problem Pseudo symmetric technique subspace iteration method Coupled modes

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The subspace iteration method - Revisited

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COMPUTERS & STRUCTURES 2013年第1期126卷 177-183页

作者： Bathe, Klaus-Juergen MIT Cambridge MA 02139 USA

The objective in this paper is to present some recent developments regarding the subspace iteration method for the solution of frequencies and mode shapes. The developments pertain to speeding up the basic subspace iteration method by choosing an effective number of iteration vectors and by the use of parallel processing. The subspace iteration method lends itself particularly well to shared and distributed memory processing. We present the algorithms used and illustrative sample solutions. The present paper may be regarded as an addendum to the publications presented in the early 1970s, see Refs. [1,2], taking into account the changes in computers that have taken place. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Frequencies Mode shapes subspace iteration method Parallel processing Shared memory Distributed memory

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Adaptive Algorithms for Solving Eigenvalue Problems in the Variable Computer Environment of Supercomputers

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CYBERNETICS AND SYSTEMS ANALYSIS 2023年第3期59卷 480-492页

作者： Khimich, O. M. Popov, O. V. Chistyakov, O. V. Kokhanovskyi, V. O. Natl Acad Sci Ukraine VM Glushkov Inst Cybernet Kiev Ukraine Natl Tech Univ Ukraine Igor Sikorsky Kyiv Polytech Inst Kiev Ukraine

The authors propose software for the analysis and solution to the algebraic eigenvalue problem using an MIMD computer with GPUs, which includes parallel algorithms and programs with the functions of automatic adaptive configuration of the variable computer environment (multilevel parallelism, variable topology of interprocessor communications, mixed word length, caching, etc.) on the mathematical properties of the problem identified in the computer and the architectural features to ensure the reliability of the solution results and the efficient use of computing resources.

关键词： algebraic eigenvalue problems (AEVPs) changeable computer environment adaptive algorithms gradient conjugation method subspace iteration method multi-core computers of MIMD-architecture with graphic processors

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A rational-Chebyshev projection method for nonlinear eigenvalue problems

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2024年第6期31卷

作者： Tang, Ziyuan Saad, Yousef Univ Minnesota Dept Comp Sci & Engn Minneapolis MN 55455 USA

This article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function T(lambda)$$ T\left(\lambda \right) $$ under consideration is approximated by a matrix polynomial in lambda$$ \lambda $$. The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

关键词： Cauchy integral formula Krylov subspace method nonlinear eigenvalue problem polynomial approximation projection methods rational approximation subspace iteration method

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An Efficient Eigen-Solver and Some of Its Applications

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INTERNATIONAL JOURNAL FOR COMPUTATIONAL methodS IN ENGINEERING SCIENCE & MECHANICS 2019年第2期20卷 130-152页

作者： Rezaiee-Pajand, M. Sani, A. Aftabi Kazemiyan, M. S. Ferdowsi Univ Mashhad Dept Civil Engn Mashhad Razavi Khorasan Iran

This paper is mainly devoted to improve the efficiency of the subspace iteration strategy and propose practical new applications for the suggested scheme. In spite of utilizing subspace iteration method for many structures, it has not been mixed with the Sherman-Morrison-Woodbury process for problems with varying stiffness, yet. The authors' technique tries to achieve this goal. To corroborate the efficiency and capability of the present formulations, several numerical tests are performed. In order to assess the effects of various parameters on the eigenpairs of the problems, the required sensitivity analysis is also carried out.

关键词： subspace iteration method Sherman-Morrsion-Woodbury formulas natural frequency eigenvalue problem stiffness matrix sensitivity analysis

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Free vibration analysis of concrete arch dams by quadratic ideal-coupled method

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STRUCTURAL ENGINEERING AND MECHANICS 2018年第1期65卷 69-79页

作者： Rezaiee-Pajand, Mohammad Sani, Ahmad Aftabi Kazemiyan, Mohammad Sadegh Ferdowsi Univ Mashhad Dept Civil Engn Mashhad Iran

This paper is devoted to two new techniques for free vibration analysis of concrete arch dam-reservoir systems. The proposed schemes are quadratic ideal-coupled eigen-problems, which can solve the originally non-symmetric eigen-problem of the system. To find the natural frequencies and mode shapes, a new special-purpose eigen-value solution routine is developed. Moreover, the accuracy of the proposed approach is thoroughly assessed, and it is confirmed that the new scheme is very accurate under all practical conditions. It is also concluded that both decoupled and ideal-coupled strategy proposed in the previous works can be considered as special cases of the current more general procedure.

关键词： arch dam fluid-structure interaction decoupled method ideal-coupled method quadratic ideal-coupled method subspace iteration method

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An enhanced formulation of error bound in subspace iteration method

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN BIOMEDICAL ENGINEERING 2011年第1期27卷 113-127页

作者： Chen, Pu Gong, Yucai Chen, Yongqiang Kulasegaram, Sivakumar Peking Univ Dept Mech & Aerosp Engn Coll Engn Beijing 100871 Peoples R China Peking Univ LTCS Coll Engn Beijing 100871 Peoples R China Cardiff Univ Sch Engn Inst Theoret Appl & Computat Mech Cardiff Wales

In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.

关键词： subspace iteration method error bounds convergence criteria finite element

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Investigation on efficiency and applicability of subspace iteration method with accelerated starting vectors for calculating natural modes of structures

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STRUCTURAL ENGINEERING AND MECHANICS 2011年第5期37卷 561-573页

作者： Kim, B. W. Jung, H. J. Hong, S. Y. KORDI MOERI Marine Struct & Plant Res Dept Taejon South Korea Korea Adv Inst Sci & Technol Dept Civil & Environm Engn Taejon 305701 South Korea

For efficient calculation of natural modes of structures, a numerical scheme which accelerates convergence of the subspace iteration method by employing accelerated starting Lanczos vectors was proposed in 2005. This paper is an extension of the study. The previous study simply showed feasibility of the proposed method by analyzing structures with smaller degrees of freedom. While, the present study verifies efficiency of the proposed method more rigorously by comparing closeness of conventional and accelerated starting vectors to genuine eigenvectors. This study also analyzes an example structure with larger degrees of freedom and more complex constraints in order to investigate applicability of the proposed method.

关键词： natural modes of structures eigenvector subspace iteration method accelerated starting vector

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