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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Correlated volumes for extended wavefunctions on a random-regular graph

We analyze the ergodic properties of a metallic wavefunction for the Anderson model in a disordered random-regular graph with branching number $k=2.$ A few q-moments $I_q$ associated with the zero energy eigenvector are numerically computed up to sizes $N=4\times 10^6.$ We extract their corresponding fractal dimensions $D_q$ in the thermodynamic limit together with correlated volumes $N_q$ that control finite-size effects. At intermediate values of disorder $W,$ we obtain ergodicity $D_q=1$ for $q=1,2$ and correlation volumes that increase fast upon approaching the Anderson transition $\log(\log(N_q))\sim W.$ We then focus on the extraction of the volume $N_0$ associated with the typical value of the wavefunction $e^{},$ which follows a similar tendency as the ones for $N_1$ or $N_2.$ Its value at intermediate disorders is close, but smaller, to the so-called ergodic volume previously found via the super-symmetric formalism and belief propagator algorithms. None of the computed correlated volumes shows a tendency to diverge up to disorders $W\approx 15$, specifically none with exponent $\nu=1/2$. Deeper in the metal, we characterize the crossover to system sizes much smaller than the first correlated volume $N_1\gg N.$ Once this crossover has taken place, we obtain evidence of a scaling in which the derivative of the first fractal dimension $D_1$ behaves critically with an exponent $\nu=1.

NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc

[EN] SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.This work was partially funded by the Spanish Agencia Estatal de Investigacion AEI http://ciencia.gob.es under grants TIN2016-75985-P AEI and PID2019-107379RB-I00 AEI (including European Commission FEDER funds).Campos, C.; Roman, JE. (2021). NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc. ACM Transactions on Mathematical Software. 47(3):1-29. https://doi.org/10.1145/3447544S12947

Acoustic modal analysis with heat release fluctuations using nonlinear eigensolvers

Closed combustion devices like gas turbines and rockets are prone to thermoacoustic instabilities. Design engineers in the industry need tools to accurately identify and remove instabilities early in the design cycle. Many different approaches have been developed by the researchers over the years. In this work we focus on the Helmholtz wave equation based solver which is found to be relatively fast and accurate for most applications. This solver has been a subject of study in many previous works. The Helmholtz wave equation in frequency space reduces to a nonlinear eigenvalue problem which needs to be solved to compute the acoustic modes. Most previous implementations of this solver have relied on linearized solvers and iterative methods which as shown in this work are not very efficient and sometimes inaccurate. In this work we make use of specialized algorithms implemented in SLEPc that are accurate and efficient for computing eigenvalues of nonlinear eigenvalue problems. We make use of the n-tau model to compute the reacting source terms in the Helmholtz equation and describe the steps involved in deriving the Helmholtz eigenvalue equation and obtaining its solution using the SLEPc library

MPI-CUDA parallel linear solvers for block-tridiagonal matrices in the context of SLEPc's eigensolvers

[EN] We consider the computation of a few eigenpairs of a generalized eigenvalue problem Ax = lambda Bx with block-tridiagonal matrices, not necessarily symmetric, in the context of Krylov methods. In this kind of computation, it is often necessary to solve a linear system of equations in each iteration of the eigensolver, for instance when B is not the identity matrix or when computing interior eigenvalues with the shift-and-invert spectral transformation. In this work, we aim to compare different direct linear solvers that can exploit the block-tridiagonal structure. Block cyclic reduction and the Spike algorithm are considered. A parallel implementation based on MPI is developed in the context of the SLEPc library. The use of GPU devices to accelerate local computations shows to be competitive for large block sizes.This work was supported by Agencia Estatal de Investigacion (AEI) under grant TIN2016-75985-P, which includes European Commission ERDF funds. Alejandro Lamas Davina was supported by the Spanish Ministry of Education, Culture and Sport through a grant with reference FPU13-06655.Lamas Daviña, A.; Roman, JE. (2018). MPI-CUDA parallel linear solvers for block-tridiagonal matrices in the context of SLEPc's eigensolvers. Parallel Computing. 74:118-135. https://doi.org/10.1016/j.parco.2017.11.006S1181357

KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners

[EN] Contemporary applications in computational science and engineering often require the solution of linear systems which may be of different sizes, shapes, and structures. The goal of this paper is to explain how two libraries, PETSc and HPDDM, have been interfaced in order to offer end-users robust overlapping Schwarz preconditioners and advanced Krylov methods featuring recycling and the ability to deal with multiple right-hand sides. The flexibility of the implementation is showcased and explained with minimalist, easy-to-run, and reproducible examples, to ease the integration of these algorithms into more advanced frameworks. The examples provided cover applications from eigenanalysis, elasticity, combustion, and electromagnetism.Jose E. Roman was supported by the Spanish Agencia Estatal de Investigacion (AEI) under project SLEPc-DA (PID2019-107379RB-I00)Jolivet, P.; Roman, JE.; Zampini, S. (2021). KSPHPDDM and PCHPDDM: Extending PETSc with advanced Krylov methods and robust multilevel overlapping Schwarz preconditioners. Computers & Mathematics with Applications. 84:277-295. https://doi.org/10.1016/j.camwa.2021.01.0032772958

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library

Parallel Direct Solution of the Covariance-Localized Ensemble Square Root Kalman Filter Equations with Matrix Functions

[EN] Recently, the serial approach to solving the square root ensemble Kalman filter (ESRF) equations in the presence of covariance localization was found to depend on the order of observations. As shown previously, correctly updating the localized posterior covariance in serial requires additional effort and computational expense. A recent work by Steward et al. details an all-at-once direct method to solve the ESRF equations in parallel. This method uses the eigenvectors and eigenvalues of the forward observation covariance matrix to solve the difficult portion of the ESRF equations. The remaining assimilation is easily parallelized, and the analysis does not depend on the order of observations. While this allows for long localization lengths that would render local analysis methods inefficient, in theory, an eigenpair-based method scales as the cube number of observations, making it infeasible for large numbers of observations. In this work, we extend this method to use the theory of matrix functions to avoid eigenpair computations. The Arnoldi process is used to evaluate the covariance-localized ESRF equations on the reduced-order Krylov subspace basis. This method is shown to converge quickly and apparently regains a linear scaling with the number of observations. The method scales similarly to the widely used serial approach of Anderson and Collins in wall time but not in memory usage. To improve the memory usage issue, this method potentially can be used without an explicit matrix. In addition, hybrid ensemble and climatological covariances can be incorporated.This research was partially funded by the NOAA Hurricane Forecast Improvement Project Award NA14NWS4680022. This work was partially supported by Agencia Estatal de Investigacion (AEI) under Grant TIN2016-75985-P, which includes European Commission ERDF funds. Alejandro Lamas Davina was supported by the Spanish Ministry of Education, Culture and Sport through a grant with reference FPU13-06655. The fourth author's work was in part carried out under the auspices of CIMAS, a joint institute of the University of Miami and NOAA, Cooperative Agreement NA15OAR4320064. The authors acknowledge the NOAA Research and Development High Performance Computing Program for providing computing and storage resources that have contributed to the research results reported within this paper (http://rdhpcs.noaa.gov). We thank Jeff Anderson, Shu-Chih Yang, and three anonymous reviewers for their helpful comments and contributions. We also thank Hui Christophersen for providing technical assistance.Steward, JL.; Roman, JE.; Lamas Daviña, A.; Aksoy, A. (2018). Parallel Direct Solution of the Covariance-Localized Ensemble Square Root Kalman Filter Equations with Matrix Functions. Monthly Weather Review. 146(9):2819-2836. https://doi.org/10.1175/MWR-D-18-0022.1S28192836146

Calculation of multiple eigenvalues of the neutron diffusion equation discretized with a parallelized finite volume method

[EN] The spatial distribution of the neutron flux within the core of nuclear reactors is a key factor in nuclear safety. The easiest and fastest way to determine it is by solving the eigenvalue problem of the neutron diffusion equation, which only contains spatial derivatives. The approximation of these derivatives is performed by discretizing the geometry and using numerical methods. In this work, the authors used a finite volume method based on a polynomial expansion of the neutron flux. Once these terms are discretized, a set of matrix equations is obtained, which constitutes the eigenvalue problem. A very effective class of methods for the solution of eigenvalue problems are those based on projection onto a low-dimensional subspace, such as Krylov subspaces. Thus, the SLEPc library was used for solving the eigenvalue problem by means of the Krylov-Schur method, which also uses projection methods of PETSc for solving linear systems. This work includes a complete sensitivity analysis of different issues: mesh, polynomial terms, linear systems solvers and parallelization.This work has been partially supported by the Spanish Ministerio de Eduacion Cultura y Deporte under the grant FPU13/01009, the Spanish Ministerio de Ciencia e Innovacion under the project ENE2014-59442-P, the Spanish Ministerio de Economia y Competitividad and the European Fondo Europeo de Desarrollo Regional (FEDER) under the project ENE2015-68353-P (MINECO/FEDER), the Generalitat Valenciana under the project PROMETEOII/2014/008, and the Spanish Ministerio de Economia y Competitividad and the European Fondo Europeo de Desarrollo Regional (FEDER) under the project TIN2016-075985-P.Bernal-Garcia, A.; Roman, JE.; Miró Herrero, R.; Verdú Martín, GJ. (2018). Calculation of multiple eigenvalues of the neutron diffusion equation discretized with a parallelized finite volume method. Progress in Nuclear Energy. 105:271-278. https://doi.org/10.1016/j.pnucene.2018.02.006S27127810