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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An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.Comment: 23 Pages, 6 figures. First revision of an invited submission to special issue of Concurrency and Computation: Practice and Experienc

Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems

In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.Comment: 12 Pages, 5 figures. Accepted for publication on Computer Physics Communication

A polynomial Jacobi-Davidson solver with support for non-monomial bases and deflation

[EN] Large-scale polynomial eigenvalue problems can be solved by Krylov methods operating on an equivalent linear eigenproblem (linearization) of size d center dot n where d is the polynomial degree and n is the problem size, or by projection methods that keep the computation in the n-dimensional space. Jacobi-Davidson belongs to the latter class of methods, and, since it is a preconditioned eigensolver, it may be competitive in cases where explicitly computing a matrix factorization is exceedingly expensive. However, a fully fledged implementation of polynomial Jacobi-Davidson has to consider several issues, including deflation to compute more than one eigenpair, use of non-monomial bases for the case of large degree polynomials, and handling of complex eigenvalues when computing in real arithmetic. We discuss these aspects and present computational results of a parallel implementation in the SLEPc library.This work was supported by Agencia Estatal de Investigación (AEI) under Grant TIN2016-75985-P, which includes European Commission ERDF funds.Campos, C.; Jose E. Roman (2020). A polynomial Jacobi-Davidson solver with support for non-monomial bases and deflation. BIT Numerical Mathematics. 60(2):295-318. https://doi.org/10.1007/s10543-019-00778-zS295318602Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26(3), 640–659 (2005)Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W., Karpeyev, D., Kaushik, D., Knepley, M., May, D., McInnes, L.C., Mills, R., Munson, T., Rupp, K., Sanan, P., Smith, B., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. 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The parallel computation of the smallest eigenpair of an acoustic problem with damping

Acoustic problems with damping may give rise to large quadratic eigenproblems. Efficient and parallelizable algorithms are required for solving these problems. The recently proposed Jacobi-Davidson method is well suited for parallel computing: no matrix decomposition and no back or forward substitutions are needed. This paper describes the parallel solution of the smallest eigenpair of a realistic and very large quadratic eigenproblem with the Jacobi-Davidson method

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). 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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. 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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. 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