A block Krylov subspace time‐exact solution method for linear ordinary differential equation systems

Botchev, Mike A.

doi:10.1002/nla.1865

Cited by 28 publications

(58 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Details of the method are given in [2]. We follow the method of lines approach [55], i.e., the PDE is discretized in space first.…”

Section: Exponential Block Krylov Methodmentioning

confidence: 99%

“…There are several possible choices for p(t), among which, cubic piecewise polynomials. Then, the approximation error in the source term, g(t) − U p(t) , can be easily controlled within a desired tolerance, depending on the number of samples in [0, ∆T ], and the number of singular values truncated (see [2]). …”

Section: Exponential Block Krylov Methodmentioning

confidence: 99%

“…The truncated SVD approximation of the source term (2.4) facilitates the solution of the initial value problem (IVP) in (2.1) by the block Krylov subspace method [2]. Here, the block Krylov subspace at iteration l is defined as…”

Section: Exponential Block Krylov Methodmentioning

confidence: 99%

“…Our method adopts the Paraexp method within a waveform relaxation framework, which enables the extension of parallel time integration to nonlinear PDEs. The method is inspired by and relies on recent techniques in Krylov subspace matrix exponential methods such as shiftand-invert acceleration [22,40,51], restarting [1,10,16,45,50] and using block Krylov subspaces [2] to handle nonautonomous PDEs efficiently. The method also relies on a singular value decomposition (SVD) of source terms in the PDE, which is used to construct the block Krylov subspace.…”

mentioning

confidence: 99%

See 3 more Smart Citations

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Kooij

Botchev

Geurts

2017

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.Key words. parallel computing, exponential integrators, partial differential equations, parallel in time, block Krylov subspace.AMS subject classifications. 65F60, 65L05, 65Y051. Introduction. Recent developments in hardware architecture, bringing to practice computers with hundreds of thousands of cores, urge the creation of new, as well as a major revision of existing numerical algorithms [14]. To be efficient on such massively parallel platforms, the algorithms need to employ all possible means to parallelize the computations. When solving partial differential equations (PDEs) with a time-dependent solution, an important way to parallelize the computations is, next to the parallelization across space, parallelization across time. This adds a new dimension of parallelism with which the simulations can be implemented. In this paper we present a new time-parallel integration method extending the Paraexp method [21] to nonlinear partial differential equations using Krylov methods and waveform relaxation.Several approaches to parallelize the simulation of time-dependent solutions in time can be distinguished. The first important class of the methods are the waveform relaxation methods [6,38,44,56], including the space-time multigrid methods for parabolic PDEs [8,24,30,37]. The key idea is to start with an approximation to the numerical solution for the whole time interval of interest and update the solution, solving an easier-to-solve approximate system in time. The Parareal method [36], which attracted significant attention recently, is a prime example related to the class of waveform relaxation methods [19].Parallel Runge-Kutta methods and general linear methods, where the parallelism is determined and restricted by the number of stages or steps, form another class of the time-parallel methods [8,12,52]. Time-parallel schemes can also be obtained

show abstract

“…Details of the method are given in [2]. We follow the method of lines approach [55], i.e., the PDE is discretized in space first.…”

Section: Exponential Block Krylov Methodmentioning

confidence: 99%

Section: Exponential Block Krylov Methodmentioning

confidence: 99%

Section: Exponential Block Krylov Methodmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Kooij

Botchev

Geurts

2017

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…This approach is very similar to waveform relaxation [108]. Our method relies on the exponential block Krylov (EBK) method [11] which we use to solve both the homogeneous and nonhomogeneous subproblems in the Paraexp framework. The EBK method is a highly efficient exponential integrator that is based on a singular value decomposition (SVD) of the source term, e.g., arising from the linearized convection term at the previous iteration level.…”

Section: Parallelism In Timementioning

confidence: 99%

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

Kooij¹

View full text Add to dashboard Cite

Fast solvers for simulation, inversion, and control of wave propagation problems

Borzì

Oosterlee

2013

Numerical Linear Algebra App

View full text Add to dashboard Cite

From 26 to 28 of September 2011, a European Science Foundation (ESF) workshop 'Fast solvers for simulation, inversion, and control of wave propagation problems', OPTPDE 2011, was organized, in Würzburg, Germany, with approximately 50 participants from many different countries.The workshop aimed at fostering development and application of fast computational techniques to direct and inverse wave problems, which are of primal importance in strategic engineering areas. Regarding applications, there were several researchers dealing with propagating and standing wave phenomena, and also inverse problems concerning design, control, and parameter estimation were often discussed. Within this general framework, the following topics were addressed: modeling, data assimilation, fast computational strategies for direct and inverse wave problems, Helmholtz problems, model order reduction, uncertainty quantification, and related applications.This special issue in Numerical Linear Algebra with Applications gives an overview of some of the highly interesting achievements that were presented during the workshop in Würzburg.In the contribution [1], 'Convergence Analysis for Hyperbolic Evolution Problems in Mixed Norm', D. Boffi, A. Buffa, and L. Gastaldi present a convergence analysis for the space discretization of hyperbolic evolution problems in mixed form. In particular, this paper discusses the relation between the approximation of an underlying eigenvalue problem and the space discretization of the evolution problem.In the paper [2], 'A Block Krylov Subspace Time-Exact Solution Method for Linear ODE Systems' by M.A. Botchev, a residual-based block Krylov subspace method that solves certain classes of linear systems of ordinary differential equations is discussed. This method consists of two steps that involve an accurate piecewise polynomial approximation of the source term, constructed with

show abstract

A block Krylov subspace time‐exact solution method for linear ordinary differential equation systems

Cited by 28 publications

References 38 publications

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

Fast solvers for simulation, inversion, and control of wave propagation problems

Contact Info

Product

Resources

About