Experiments on a Parallel Nonlinear Jacobi–Davidson Algorithm

AbstractThe Jacobi–Davidson (JD) algorithm is very well suited for the computation of a few eigen-pairs of large sparse complex symmetric nonlinear eigenvalue problems. The performance of JD crucially depends on the treatment of the so-called correction equation, in particular the preconditioner, and the initial vector. Depending on the choice of the spectral shift and the accuracy of the solution, the convergence of JD can vary from linear to cubic. We investigate parallel preconditioners for the Krylov space method used to solve the correction equation.We apply our nonlinear Jacobi–Davidson (NLJD) method to quadratic eigenvalue problems that originate from the time-harmonic Maxwell equation for the modeling and simulation of resonating electromagnetic structures

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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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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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