Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU-time consumption. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two errors and the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to round-off errors, even when apparently good spectral convergence is achieved. The influence of round-off errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure

High Performance Computing for Stability Problems - Applications to Hydrodynamic Stability and Neutron Transport Criticality

In this work we examine two kinds of applications in terms of stability and perform numerical evaluations and benchmarks on parallel platforms. We consider the applicability of pseudospectra in the field of hydrodynamic stability to obtain more information than a traditional linear stability analysis can provide. Furthermore, we treat the neutron transport criticality problem and highlight the Davidson method as an attractive alternative to the so far widely used power method in that context

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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Ferronato, C. Janna, and G. Pini. 2012. Efficient parallel solution to large-size sparse eigenproblems with block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 5, 797--815.D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst. 1998. Jacobi--Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20, 1, 94--125.M. A. Freitag and A. Spence. 2007. Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem. Electron. Trans. Numer. Anal. 28, 40--64.M. Genseberger. 2010. Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. App. Numer. Math. 60, 11, 1083--1099.V. Hernandez, J. E. Roman, and A. Tomas. 2007. Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33, 7--8, 521--540.V. Hernandez, J. E. Roman, and V. Vidal. 2005. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 3, 351--362.V. Heuveline, B. Philippe, and M. Sadkane. 1997. Parallel computation of spectral portrait of large matrices by Davidson type methods. Numer. Algor. 16, 1, 55--75.M. E. Hochstenbach. 2005a. Generalizations of harmonic and refined Rayleigh-Ritz. Electron. Trans. Numer. Anal. 20, 235--252.M. E. Hochstenbach. 2005b. Variations on harmonic Rayleigh--Ritz for standard and generalized eigenproblems. Preprint, Department of Mathematics, Case Western Reserve University.M. E. Hochstenbach and Y. Notay. 2006. The Jacobi--Davidson method. GAMM Mitt. 29, 2, 368--382.F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang. 2010. A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation. J. Comput. Phys. 229, 8, 2932--2947.A. V. Knyazev. 2001. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 2, 517--541.A. V. Knyazev, M. E. Argentati, I. Lashuk, and E. E. Ovtchinnikov. 2007. Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput. 29, 5, 2224--2239.J. Kopal, M. Rozložník, M. Tuma, and A. Smoktunowicz. 2012. Rounding error analysis of orthogonalization with a non-standard inner product. Numer. Math. 52, 4, 1035--1058.D. Kressner. 2006. Block algorithms for reordering standard and generalized Schur forms. ACM Trans. Math. Softw. 32, 4, 521--532.R. B. Lehoucq, D. C. Sorensen, and C. Yang. 1998. ARPACK Users' Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA.Z. Li, Y. Saad, and M. Sosonkina. 2003. pARMS: a parallel version of the algebraic recursive multilevel solver. Numer. Linear Algebra Appl. 10, 5--6, 485--509.J. R. McCombs and A. Stathopoulos. 2006. Iterative validation of eigensolvers: a scheme for improving the reliability of Hermitian eigenvalue solvers. SIAM J. Sci. Comput. 28, 6, 2337--2358.F. Merz, C. Kowitz, E. Romero, J. E. Roman, and F. Jenko. 2012. Multi-dimensional gyrokinetic parameter studies based on eigenvalues computations. Comput. Phys. Commun. 183, 4, 922--930.R. B. Morgan. 1990. Davidson's method and preconditioning for generalized eigenvalue problems. J. Comput. Phys. 89, 241--245.R. B. Morgan. 1991. Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154--156, 289--309.R. B. Morgan and D. S. Scott. 1986. Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7, 3, 817--825.R. Natarajan and D. Vanderbilt. 1989. A new iterative scheme for obtaining eigenvectors of large, real-symmetric matrices. J. Comput. Phys. 82, 1, 218--228.M. Nool and A. van der Ploeg. 2000. 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Parametrically Forced Rotating and/or Stratified Confined Flows

abstract: The dynamics of a fluid flow inside 2D square and 3D cubic cavities under various configurations were simulated and analyzed using a spectral code I developed. This code was validated against known studies in the 3D lid-driven cavity. It was then used to explore the various dynamical behaviors close to the onset of instability of the steady-state flow, and explain in the process the mechanism underlying an intermittent bursting previously observed. A fairly complete bifurcation picture emerged, using a combination of computational tools such as selective frequency damping, edge-state tracking and subspace restriction. The code was then used to investigate the flow in a 2D square cavity under stable temperature stratification, an idealized version of a lake with warmer water at the surface compared to the bottom. The governing equations are the Navier-Stokes equations under the Boussinesq approximation. Simulations were done over a wide range of parameters of the problem quantifying the driving velocity at the top (e.g. wind) and the strength of the stratification. Particular attention was paid to the mechanisms associated with the onset of instability of the base steady state, and the complex nontrivial dynamics occurring beyond onset, where the presence of multiple states leads to a rich spectrum of states, including homoclinic and heteroclinic chaos. A third configuration investigates the flow dynamics of a fluid in a rapidly rotating cube subjected to small amplitude modulations. The responses were quantified by the global helicity and energy measures, and various peak responses associated to resonances with intrinsic eigenmodes of the cavity and/or internal retracing beams were clearly identified for the first time. A novel approach to compute the eigenmodes is also described, making accessible a whole catalog of these with various properties and dynamics. When the small amplitude modulation does not align with the rotation axis (precession) we show that a new set of eigenmodes are primarily excited as the angular velocity increases, while triadic resonances may occur once the nonlinear regime kicks in.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Low-dimensional models for fluid flow

Despite the temporal and spatial complexity of fluid flow, model dimensionality can often be greatly reduced while both capturing and illuminating the nonlinear dynamics of the flow. This dissertation follows the methodology of direct numerical simulation (DNS) followed by Proper Orthogonal Decomposition of temporally sampled DNS data to derive temporal and spatial eigenfunctions. The DNS calculations use Chorin's projection scheme; 2-d validation and results are presented for driven cavity and square cylinder wake flows. The flow velocity is expressed as a linear combination of the spatial eigenfunctions with time-dependent coefficients. Galerkin projection of these modes onto the Navier-Stokes equations obtains a dynamical system with quadratic nonlinearity and explicit Reynolds number (Re) dependence. Truncated to retain only the most energetic modes produces a low-dimensional model for the flow at the decomposition Re. This dissertation demonstrates that these low-dimensional models reproduce the flow dynamics, but with small errors in amplitude, phase, and particularly long term dynamics. A new stabilization algorithm is presented that projects the error onto the derived temporal eigenfunctions, then modifies the dynamical system coefficients to significantly reduce these errors. Its effectiveness is demonstrated with low-dimensional dynamical systems for driven cavity flow in the periodic regime, quasi-periodic flow at Re 10000, and the wake flow. This dissertation also addresses the task of obtaining more useful models that are valid over a range of Reynolds numbers. Straightforward Re-based parameter continuation applied to extrapolate the model proves inadequate for successful flow prediction. A new concept of parameterizing the dynamical system coefficients is introduced that utilizes the kinetic energy transfer between modes as a function of Re to predict the flow dynamics correctly. Results for the driven cavity flow include a minimal four-mode dynamical system that captures the flow dynamics for Re up to 10000. A four-mode dynamical system for the square cylinder wake flow demonstrates accurate amplitude predictions for Re up to 100. The most robust low-dimensional models are obtained by further including a model for the frequency variation with Re. Low-dimensional models that incorporate spatial mode changes with Re are developed and quantitatively assessed for both test flows

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

Estimation et réduction du coût d’algorithmes quantiques

Plusieurs algorithmes ont été marquants dans le développement de l’informatique quantique : on peut penser notamment à l’algorithme de Shor ou à l’algorithme de Grover. Bien que pour ce dernier, par exemple, nous ayons une accélération quadratique prouvée, il n’est pas clair que ce sera suffisant pour offrir un avantage pratique par rapport aux algorithmes classiques. D’abord, celui-ci est formulé en termes d’oracle, une boîte noire qui cache une sous-routine non comprise dans le calcul du coût. Ensuite, lorsque l’on prend en considération le surcoût engendré notamment par la correction d’erreur quantique, il est possible de perdre l’accélération promise. Mais également, la durée d’une porte logique quantique est considérablement plus long que son homologue classique. Quel est le coût réel d’un algorithme quantique lorsque l’on prend en considération toutes les sous-routines ? À quand un ordinateur quantique utile qui surpassera les performances d’un superordinateur ? Dans cette thèse, nous présenterons trois projets visant tous à estimer et réduire le coût d’algorithmes ou sous-routines quantiques. Les algorithmes abordés sont issus d’une discrétisation de l’évolution adiabatique. Le premier se concentre sur un algorithme de préparation d’état d’un système à N corps par une évolution adiabatique via l’effet Zénon. Le second porte sur une version quantique des algorithmes de marches aléatoires et de recuit simulé pouvant, par exemple, préparer un état stationnaire. Le dernier décrit un nouvel algorithme : une évolution adiabatique basée sur la réflexion. Celui-ci permet, entre autres, de résoudre des problèmes MAX-kSAT, une classe de problèmes NP-difficile. Avec ces projets, nous voulons, d’une part, proposer des algorithmes efficaces ainsi que leur implémentation de A à Z et, d’autre part, estimer les caractéristiques nécessaires à un ordinateur quantique utile (p. ex. taille, résistance au bruit, vitesse d’opération). Les résultats présentés démontrent le coût élevé associé aux algorithmes tolérants aux fautes. Bien qu’on s’attende à avoir une accélération par rapport au classique, lorsque l’on prend en considération le nombre de qubits physiques, le nombre d’opérations physiques et la durée de chacune de ces opérations, en incluant la correction d’erreur notamment, la taille des instances offrant un avantage réel est loin d’être atteignable pour les processeurs quantiques à court terme. Toutefois, en combinant des méthodes astucieuses et au moyen de différents procédés d’optimisation, il est possible de réduire considérablement le coût des algorithmes quantiques, et donc de réduire le délai pour atteindre la suprématie quantique