Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices

We propose a Preconditioned Locally Harmonic Residual (PLHR) method for computing several interior eigenpairs of a generalized Hermitian eigenvalue problem, without traditional spectral transformations, matrix factorizations, or inversions. PLHR is based on a short-term recurrence, easily extended to a block form, computing eigenpairs simultaneously. PLHR can take advantage of Hermitian positive definite preconditioning, e.g., based on an approximate inverse of an absolute value of a shifted matrix, introduced in [SISC, 35 (2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is efficient and robust for certain classes of large-scale interior eigenvalue problems, involving Laplacian and Hamiltonian operators, especially if memory requirements are tight

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. 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Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver

In the numerical solution of large-scale eigenvalue problems, Davidson-type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson-type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi-Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non-symmetric problems. We present a parallel implementation that has been included in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with a Krylov-Schur eigensolver. © 2011 John Wiley and Sons, Ltd..The authors are indebted to Florian Merz for providing us with the test cases and for his useful suggestions. The authors acknowledge the computer resources provided by the Barcelona Supercomputing Center (BSC). This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519.Romero Alcalde, E.; Román Moltó, JE. (2011). Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi-Davidson eigensolver. Concurrency and Computation: Practice and Experience. 23:2179-2191. https://doi.org/10.1002/cpe.1740S2179219123Hochstenbach, M. E., & Notay, Y. (2009). Controlling Inner Iterations in the Jacobi–Davidson Method. SIAM Journal on Matrix Analysis and Applications, 31(2), 460-477. doi:10.1137/080732110Heuveline, V., Philippe, B., & Sadkane, M. (1997). Numerical Algorithms, 16(1), 55-75. doi:10.1023/a:1019126827697Arbenz, P., Bečka, M., Geus, R., Hetmaniuk, U., & Mengotti, T. (2006). On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Computing, 32(2), 157-165. doi:10.1016/j.parco.2005.06.005Genseberger, M. (2010). Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi–Davidson method for large scale eigenvalue problems. Applied Numerical Mathematics, 60(11), 1083-1099. doi:10.1016/j.apnum.2009.07.004Stathopoulos, A., & McCombs, J. R. (2010). PRIMME. ACM Transactions on Mathematical Software, 37(2), 1-30. doi:10.1145/1731022.1731031Baker, C. G., Hetmaniuk, U. L., Lehoucq, R. B., & Thornquist, H. K. (2009). Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Transactions on Mathematical Software, 36(3), 1-23. doi:10.1145/1527286.1527287Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPc. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Romero, E., Cruz, M. B., Roman, J. E., & Vasconcelos, P. B. (2011). A Parallel Implementation of the Jacobi-Davidson Eigensolver for Unsymmetric Matrices. High Performance Computing for Computational Science – VECPAR 2010, 380-393. doi:10.1007/978-3-642-19328-6_35Romero, E., & Roman, J. E. (2010). A Parallel Implementation of the Jacobi-Davidson Eigensolver and Its Application in a Plasma Turbulence Code. Lecture Notes in Computer Science, 101-112. doi:10.1007/978-3-642-15291-7_11Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen*). (1846). Journal für die reine und angewandte Mathematik (Crelles Journal), 1846(30), 51-94. doi:10.1515/crll.1846.30.51G. Sleijpen, G. L., & Van der Vorst, H. A. (1996). A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications, 17(2), 401-425. doi:10.1137/s0895479894270427Fokkema, D. R., Sleijpen, G. L. G., & Van der Vorst, H. A. (1998). Jacobi--Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils. SIAM Journal on Scientific Computing, 20(1), 94-125. doi:10.1137/s1064827596300073Morgan, R. B. (1991). Computing interior eigenvalues of large matrices. Linear Algebra and its Applications, 154-156, 289-309. doi:10.1016/0024-3795(91)90381-6Paige, C. C., Parlett, B. N., & van der Vorst, H. A. (1995). Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications, 2(2), 115-133. doi:10.1002/nla.1680020205Stathopoulos, A., Saad, Y., & Wu, K. (1998). Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods. SIAM Journal on Scientific Computing, 19(1), 227-245. doi:10.1137/s1064827596304162Sleijpen, G. L. G., Booten, A. G. L., Fokkema, D. R., & van der Vorst, H. A. (1996). Jacobi-davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT Numerical Mathematics, 36(3), 595-633. doi:10.1007/bf01731936Balay S Buschelman K Eijkhout V Gropp W Kaushik D Knepley M McInnes LC Smith B Zhang H PETSc users manual 2010Hernandez, V., Roman, J. E., & Tomas, A. (2007). Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Computing, 33(7-8), 521-540. doi:10.1016/j.parco.2007.04.004Dannert, T., & Jenko, F. (2005). Gyrokinetic simulation of collisionless trapped-electron mode turbulence. Physics of Plasmas, 12(7), 072309. doi:10.1063/1.1947447Roman, J. E., Kammerer, M., Merz, F., & Jenko, F. (2010). Fast eigenvalue calculations in a massively parallel plasma turbulence code. Parallel Computing, 36(5-6), 339-358. doi:10.1016/j.parco.2009.12.001Merz, F., & Jenko, F. (2010). Nonlinear interplay of TEM and ITG turbulence and its effect on transport. Nuclear Fusion, 50(5), 054005. doi:10.1088/0029-5515/50/5/054005Simoncini, V., & Szyld, D. B. (2002). Flexible Inner-Outer Krylov Subspace Methods. SIAM Journal on Numerical Analysis, 40(6), 2219-2239. doi:10.1137/s0036142902401074Morgan, R. B. (2002). GMRES with Deflated Restarting. SIAM Journal on Scientific Computing, 24(1), 20-37. doi:10.1137/s106482759936465

GENERALIZATIONS OF AN INVERSE FREE KRYLOV SUBSPACE METHOD FOR THE SYMMETRIC GENERALIZED EIGENVALUE PROBLEM

Symmetric generalized eigenvalue problems arise in many physical applications and frequently only a few of the eigenpairs are of interest. Typically, the problems are large and sparse, and therefore traditional methods such as the QZ algorithm may not be considered. Moreover, it may be impractical to apply shift-and-invert Lanczos, a favored method for problems of this type, due to difficulties in applying the inverse of the shifted matrix. With these difficulties in mind, Golub and Ye developed an inverse free Krylov subspace algorithm for the symmetric generalized eigenvalue problem. This method does not rely on shift-and-invert transformations for convergence acceleration, but rather a preconditioner is used. The algorithm suffers, however, in the presence of multiple or clustered eigenvalues. Also, it is only applicable to the location of extreme eigenvalues. In this work, we extend the method of Golub and Ye by developing a block generalization of their algorithm which enjoys considerably faster convergence than the usual method in the presence of multiplicities and clusters. Preconditioning techniques for the problems are discussed at length, and some insight is given into how these preconditioners accelerate the method. Finally we discuss a transformation which can be applied so that the algorithm extracts interior eigenvalues. A preconditioner based on a QR factorization with respect to the B-1 inner product is developed and applied in locating interior eigenvalues

Inexact and Nonlinear Extensions of the FEAST Eigenvalue Algorithm

Eigenvalue problems are a basic element of linear algebra that have a wide variety of applications. Common examples include determining the stability of dynamical systems, performing dimensionality reduction on large data sets, and predicting the physical properties of nanoscopic objects. Many applications require solving large dimensional eigenvalue problems, which can be very challenging when the required number of eigenvalues and eigenvectors is also large. The FEAST algorithm is a method of solving eigenvalue problems that allows one to calculate large numbers of eigenvalue/eigenvector pairs by using contour integration in the complex plane to divide the large number of desired pairs into many small groups; these small groups of eigenvalue/eigenvector pairs may then be simultaneously calculated independently of each other. This makes it possible to quickly solve eigenvalue problems that might otherwise be very difficult to solve efficiently. The standard FEAST algorithm can only be used to solve eigenvalue problems that are linear, and whose matrices are small enough to be factorized efficiently (thus allowing linear systems of equations to be solved exactly). This limits the size and the scope of the problems to which the FEAST algorithm may be applied. This dissertation describes extensions of the standard FEAST algorithm that allow it to efficiently solve nonlinear eigenvalue problems, and eigenvalue problems whose matrices are large enough that linear systems of equations can only be solved inexactly

Singular Value Computation and Subspace Clustering

In this dissertation we discuss two problems. In the first part, we consider the problem of computing a few extreme eigenvalues of a symmetric definite generalized eigenvalue problem or a few extreme singular values of a large and sparse matrix. The standard method of choice of computing a few extreme eigenvalues of a large symmetric matrix is the Lanczos or the implicitly restarted Lanczos method. These methods usually employ a shift-and-invert transformation to accelerate the speed of convergence, which is not practical for truly large problems. With this in mind, Golub and Ye proposes an inverse-free preconditioned Krylov subspace method, which uses preconditioning instead of shift-and-invert to accelerate the convergence. To compute several eigenvalues, Wielandt is used in a straightforward manner. However, the Wielandt deflation alters the structure of the problem and may cause some difficulties in certain applications such as the singular value computations. So we first propose to consider a deflation by restriction method for the inverse-free Krylov subspace method. We generalize the original convergence theory for the inverse-free preconditioned Krylov subspace method to justify this deflation scheme. We next extend the inverse-free Krylov subspace method with deflation by restriction to the singular value problem. We consider preconditioning based on robust incomplete factorization to accelerate the convergence. Numerical examples are provided to demonstrate efficiency and robustness of the new algorithm. In the second part of this thesis, we consider the so-called subspace clustering problem, which aims for extracting a multi-subspace structure from a collection of points lying in a high-dimensional space. Recently, methods based on self expressiveness property (SEP) such as Sparse Subspace Clustering and Low Rank Representations have been shown to enjoy superior performances than other methods. However, methods with SEP may result in representations that are not amenable to clustering through graph partitioning. We propose a method where the points are expressed in terms of an orthonormal basis. The orthonormal basis is optimally chosen in the sense that the representation of all points is sparsest. Numerical results are given to illustrate the effectiveness and efficiency of this method