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

Accelerated graph-based spectral polynomial filters

Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructing accelerated polynomial filters by running flexible Krylov subspace based linear and eigenvalue solvers such as the Block Locally Optimal Preconditioned Conjugate Gradient (LOBPCG) method.Comment: 6 pages, 6 figures. Accepted to the 2015 IEEE International Workshop on Machine Learning for Signal Processin

Spectral acceleration of parallel iterative eigensolvers for large scale scientific computing

The computation of a number of the smallest eigenvalues of large and sparse matrices is crucial in various scientific applications, as the Finite Element solution of PDEs, electronic structure calculations or Laplacian of graphs, to mention a few. We propose in this contribution a parallel algorithm that is based on the spectral low-rank modification of a factorized sparse approximate inverse preconditioner (RFSAI) to accelerate the Newton-based iterative eigensolvers. Numerical results onto matrices arising from various realistic problems with size up to 5 million unknowns and 2.2 x 10^8 nonzero elements account for the efficiency and the scalability of the proposed RFSAI-updated preconditioner

A Jacobi-Davidson type method with a correction equation tailored for integral operators

The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-012-9656-9We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi–Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh–Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.This work was partially supported by European Regional Development Fund through COMPETE, FCT-Fundacao para a Ciencia e a Tecnologia through CMUP-Centro de Matematica da Universidade do Porto and Spanish Ministerio de Ciencia e Innovacion under projects TIN2009-07519 and AIC10-D-000600.Vasconcelos, PB.; D'almeida, FD.; Román Moltó, JE. (2013). A Jacobi-Davidson type method with a correction equation tailored for integral operators. Numerical Algorithms. 64(1):85-103. doi:10.1007/s11075-012-9656-9S85103641Absil, P.A., Mahony, R., Sepulchre, R., Dooren, P.V.: A Grassmann–Rayleigh quotient iteration for computing invariant subspaces. SIAM Rev. 44(1), 57–73 (2002)Ahues, M., Largillier, A., Limaye, B.V.: Spectral Computations with Bounded Operators. Chapman and Hall, Boca Raton (2001)Ahues, M., d’Almeida, F.D., Largillier, A., Titaud, O., Vasconcelos, P.: An L 1 refined projection approximate solution of the radiation transfer equation in stellar atmospheres. J. Comput. Appl. Math. 140(1–2), 13–26 (2002)Ahues, M., d’Almeida, F.D., Largillier, A., Vasconcelos, P.B.: Defect correction for spectral computations for a singular integral operator. Commun. Pure Appl. Anal. 5(2), 241–250 (2006)Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.): Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Society for Industrial and Applied Mathematics, Philadelphia (2000)Balay, S., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010)Chatelin, F.: Spectral Approximation of Linear Operators. SIAM, Philadelphia (2011)d’Almeida, F.D., Vasconcelos, P.B.: Convergence of multipower defect correction for spectral computations of integral operators. Appl. Math. Comput. 219(4), 1601–1606 (2012)Falgout, R.D., Yang, U.M.: Hypre: A library of high performance preconditioners. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) Computational Science - ICCS 2002, International Conference, Amsterdam, The Netherlands, April 21–24, 2002. Proceedings, Part III, Lecture Notes in Computer Science, vol. 2331, pp. 632–641. Springer (2002)Henson, V.E., Yang, U.M.: BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002)Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31(3), 351–362 (2005)Hernandez, V., Roman, J.E., Tomas, A., Vidal, V.: SLEPc Users Manual. Tech. Rep. DSIC-II/24/02 - Revision 3.1, D. Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2010)Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)Simoncini, V., Eldén, L.: Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT 42(1), 159–182 (2002)Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42(2), 267–293 (2000)Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001

Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

In this paper, we study a class of tuned preconditioners that will be designed to accelerate both the DACG-Newton method and the implicitly restarted Lanczos method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large-scale scientific computations. These tuning strategies are based on low-rank modifications of a given initial preconditioner. We present some theoretical properties of the preconditioned matrix. We experimentally show how the aforementioned methods benefit from the acceleration provided by these tuned/deflated preconditioners. Comparisons are carried out with the Jacobi-Davidson method onto matrices arising from various large realistic problems arising from finite element discretization of PDEs modeling either groundwater flow in porous media or geomechanical processes in reservoirs. The numerical results show that the Newton-based methods (which includes also the Jacobi-Davidson method) are to be preferred to the - yet efficiently implemented - implicitly restarted Lanczos method whenever a small to moderate number of eigenpairs is required. \ua9 2016 John Wiley & Sons, Ltd

Spectral preconditioners for the efficient numerical solution of a continuous branched transport model

We consider the efficient solution of sequences of linear systems arising in the numerical solution of a branched transport model whose long time solution for specific parameter settings is equivalent to the solution of the Monge\u2013Kantorovich equations of optimal transport. Galerkin Finite Element discretization combined with explicit Euler time stepping yield a linear system to be solved at each time step, characterized by a large sparse very ill conditioned symmetric positive definite (SPD) matrix . Extreme cases even prevent the convergence of Preconditioned Conjugate Gradient (PCG) with standard preconditioners such as an Incomplete Cholesky (IC) factorization of , which cannot always be computed. We investigate several preconditioning strategies that incorporate partial approximated spectral information. We present numerical evidence that the proposed techniques are efficient in reducing the condition number of the preconditioned systems, thus decreasing the number of PCG iterations and the overall CPU time

Multi-dimensional gyrokinetic parameter studies based on eigenvalue computations

Plasma microinstabilities, which can be described in the framework of the linear gyrokinetic equations, are routinely computed in the context of stability analyses and transport predictions for magnetic confinement fusion experiments. The GENE code, which solves the gyrokinetic equations, has been coupled to the SLEPc package for an efficient iterative, matrix-free, and parallel computation of rightmost eigenvalues. This setup is presented, including the preconditioner which is necessary for the newly implemented Jacobi-Davidson solver. The fast computation of instabilities at a single parameter set is exploited to make parameter scans viable, that is to compute the solution at many points in the parameter space. Several issues related to parameter scans are discussed, such as an efficient parallelization over parameter sets and subspace recycling. © 2011 Elsevier B.V. All rights reserved.E. Romero and J.E. Roman were supported by the Spanish Ministry of Science and Innovation (MICINN) under project number TIN2009-07519.Merz, F.; Kowitz, C.; Romero Alcalde, E.; Román Moltó, JE.; Jenko, F. (2012). Multi-dimensional gyrokinetic parameter studies based on eigenvalue computations. Computer Physics Communications. 183(4):922-930. https://doi.org/10.1016/j.cpc.2011.12.018S922930183