Structured Eigenvalue Problems

Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude

A Jacobi-Davidson type method for the product eigenvalue problem

Abstract. We propose a Jacobi-Davidson type method to compute selected eigenpairs of the product eigenvalue problem Am · · · A1x = λx, where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi-Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers

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. 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Exponentially convergent data assimilation algorithm for Navier-Stokes equations

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data

Zerofinding of analytic functions by structured matrix methods

We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV--like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples