A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems from evolutionary PDEs

In this paper we propose to use model reduction techniques for speeding up the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for iteratively solving all-at-once systems from evolutionary PDEs. In particular, we use the reduced basis method to seek a low-dimensional approximation to the sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG preconditioning procedure. Different from the standard reduced order modeling that uses the separation of offline and online stages, we have to build the reduced order model (ROM) online for the considered systems at each iteration. Therefore, several heuristic acceleration techniques are introduced in the greedy basis generation algorithm, that is built upon a residual-based error indicator, to further boost up its computational efficiency. Several numerical experiments are conducted, which illustrate the favorable computational efficiency of our proposed ROM-accelerated ParaDIAG preconditioner, in comparison with the state of the art multigrid-based ParaDIAG preconditioner.

[1]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[2]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[3]  Martin J. Gander,et al.  A Direct Time Parallel Solver by Diagonalization for the Wave Equation , 2019, SIAM J. Sci. Comput..

[4]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[5]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[6]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[7]  Shuonan Wu,et al.  Parallel-in-time high-order BDF schemes for diffusion and subdiffusion equations , 2020, ArXiv.

[8]  K. Morton Numerical Solution of Convection-Diffusion Problems , 2019 .

[9]  Yanlai Chen,et al.  Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework , 2012, Journal of Scientific Computing.

[10]  Wolfgang Hackbusch,et al.  Elliptic Differential Equations: Theory and Numerical Treatment , 2017 .

[11]  A. Peirce Computer Methods in Applied Mechanics and Engineering , 2010 .

[12]  Yanlai Chen,et al.  Adaptive greedy algorithms based on parameter‐domain decomposition and reconstruction for the reduced basis method , 2019, International Journal for Numerical Methods in Engineering.

[13]  Volker John,et al.  Finite element methods for time-dependent convection – diffusion – reaction equations with small diffusion , 2008 .

[14]  Lili Ju,et al.  An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation , 2019, SIAM J. Sci. Comput..

[15]  Jun Liu,et al.  A Parallel-In-Time Block-Circulant Preconditioner for Optimal Control of Wave Equations , 2020, SIAM J. Sci. Comput..

[16]  Per-Gunnar Martinsson,et al.  Randomized Numerical Linear Algebra: Foundations & Algorithms , 2020, ArXiv.

[17]  M. Stynes,et al.  Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems , 1996 .

[18]  Tao Zhou,et al.  Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems , 2020, ESAIM: Control, Optimisation and Calculus of Variations.

[19]  Hans De Sterck,et al.  Convergence analysis for parallel‐in‐time solution of hyperbolic systems , 2019, Numer. Linear Algebra Appl..

[20]  Jan Dirk Jansen,et al.  Accelerating iterative solution methods using reduced‐order models as solution predictors , 2006 .

[21]  Dongwoo Sheen,et al.  A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature , 2000, Math. Comput..

[22]  N. Nguyen,et al.  Reduced-basis method for the iterative solution of parametrized symmetric positive-definite linear systems , 2018, 1804.06363.

[23]  Robert D. Falgout,et al.  Parallel time integration with multigrid , 2013, SIAM J. Sci. Comput..

[24]  Gene H. Golub,et al.  Matrix computations , 1983 .

[25]  Damiano Pasetto,et al.  A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations , 2017 .

[26]  S. Sen Reduced-Basis Approximation and A Posteriori Error Estimation for Many-Parameter Heat Conduction Problems , 2008 .

[27]  John W. Peterson,et al.  A high-performance parallel implementation of the certified reduced basis method , 2011 .

[28]  Martin J. Gander,et al.  A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods , 2016, SIAM Rev..

[29]  A. Wathen,et al.  A note on parallel preconditioning for all-at-once evolutionary PDEs , 2018, ETNA - Electronic Transactions on Numerical Analysis.

[30]  B. R. Noack Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 2013 .

[31]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[32]  Wolfgang Dahmen,et al.  Reduced Basis Greedy Selection Using Random Training Sets , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[33]  J. Varah A lower bound for the smallest singular value of a matrix , 1975 .

[34]  Martin J. Gander,et al.  Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems , 2014, SIAM J. Sci. Comput..

[35]  Valeria Simoncini,et al.  Flexible Inner-Outer Krylov Subspace Methods , 2002, SIAM J. Numer. Anal..

[36]  M. H. Baluch,et al.  An analytical solution of the diffusion- convection equation over a finite domain , 1983 .

[37]  B. Ong,et al.  Applications of time parallelization , 2020 .

[38]  Anthony Nouy,et al.  Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation , 2018, Advances in Computational Mathematics.

[39]  Jun Liu,et al.  A Fast Block α-Circulant Preconditoner for All-at-Once Systems From Wave Equations , 2020, SIAM J. Matrix Anal. Appl..

[40]  Randall J. LeVeque,et al.  Finite difference methods for ordinary and partial differential equations - steady-state and time-dependent problems , 2007 .

[41]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[42]  Sören Bartels,et al.  Numerical Approximation of Partial Differential Equations , 2016 .

[43]  Hai-Wei Sun,et al.  Fast Numerical Contour Integral Method for Fractional Diffusion Equations , 2016, J. Sci. Comput..

[44]  Nicola Guglielmi,et al.  Numerical inverse Laplace transform for convection-diffusion equations , 2020, Math. Comput..

[45]  Andrew J. Wathen,et al.  Preconditioning and Iterative Solution of All-at-Once Systems for Evolutionary Partial Differential Equations , 2018, SIAM J. Sci. Comput..

[46]  Martin J. Gander,et al.  ParaDIAG: Parallel-in-Time Algorithms Based on the Diagonalization Technique , 2020, ArXiv.

[47]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

[48]  Martin J. Gander,et al.  50 Years of Time Parallel Time Integration , 2015 .

[49]  K. J. in 't Hout,et al.  A Contour Integral Method for the Black-Scholes and Heston Equations , 2009, SIAM J. Sci. Comput..

[50]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[51]  A. Antoulas,et al.  Interpolatory Methods for Model Reduction , 2020 .

[52]  X.-L. Lin,et al.  An all-at-once preconditioner for evolutionary partial differential equations , 2020, ArXiv.

[53]  Steven L. Brunton,et al.  Data-Driven Science and Engineering , 2019 .

[54]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[55]  Valeria Simoncini,et al.  Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing , 2003, SIAM J. Sci. Comput..

[56]  Beatrice Meini,et al.  Numerical methods for structured Markov chains , 2005 .

[57]  Laurent Demanet,et al.  L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation , 2019, J. Comput. Phys..

[58]  Dongwoo Sheen,et al.  A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature , 2003 .

[59]  Julien Langou,et al.  Convergence in Backward Error of Relaxed GMRES , 2007, SIAM J. Sci. Comput..

[60]  A. Pinkus n-Widths in Approximation Theory , 1985 .