A stable parareal-like method for the second order wave equation

Nguyen, Hieu; Tsai, Richard

doi:10.1016/j.jcp.2019.109156

Cited by 15 publications

(12 citation statements)

References 37 publications

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“…In that spirit, interesting ideas have emerged recently in the scientific community for the design of coarse level propagators more suited for PinT integration of hyperbolic problems, see e.g. [5,31,36], and testing these new methods under the stress of a random initial guess with precise accuracy requirements of the computed solution is an important task. The difficulty of good coarse corrections is reminiscent of the tremendous difficulties faced when trying to solve Helmholtz problems using multilevel techniques, see for example [7,8].…”

Section: Application To Pint Methodsmentioning

confidence: 99%

Toward error estimates for general space-time discretizations of the advection equation

Gander

Lunet

2020

Comput. Visual Sci.

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We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.

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Section: Application To Pint Methodsmentioning

confidence: 99%

Toward error estimates for general space-time discretizations of the advection equation

Gander

Lunet

2020

Comput. Visual Sci.

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“…In [37], the numerical solution in a coarse mesh is used in a neural network to predict the solution in a fine mesh. In [40], the solution of a second order wave equation and its gradient computed by a first order scheme is used to predict the solution computed by a higher order scheme. In this study, we introduce a novel neural network, the 2-Coarse-Grid neural network (2CGNN) to accurately compute solutions of systems of conservation laws which may contain shocks and contacts.…”

Section: -Coarse-grid Neural Networkmentioning

confidence: 99%

“…Meanwhile, to explore the possibility of learning temporal stages in the future while data may not be available, Psaros et al [39] extended PINN to the meta-learning framework. In Nguyen and Tsai et al [40], a neural network is developed for solving second order wave equations by using the solution and its gradient computed from a low order scheme as input to predict the higher order solution.…”

Section: Introductionmentioning

confidence: 99%

Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems

Huang¹,

Liu²,

Yang³

2021

Preprint

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Recent research works for solving partial differential equations (PDEs) with deep neural networks (DNNs) have demonstrated that spatiotemporal function approximators defined by auto-differentiation are effective for approximating nonlinear problems, e.g. the Burger's equation, heat conduction equations, Allen-Cahn and other reaction-diffusion equations, and Navier-Stokes equation. Meanwhile, researchers apply automatic differentiation in physics-informed neural network (PINN) to solve nonlinear hyperbolic systems based on conservation laws with highly discontinuous transition, such as Riemann problem, by inverse problem formulation in data-driven approach. However, it remains a challenge for forward methods using DNNs without knowing part of the solution to resolve discontinuities in nonlinear conservation laws. In this study, we incorporate 1st order numerical schemes into DNNs to set up the loss functional approximator instead of auto-differentiation from traditional deep learning framework, e.g. TensorFlow package, which improves the effectiveness of capturing discontinuities in Riemann problems. In particular, the 2-Coarse-Grid neural network (2CGNN) and 2-Diffusion-Coefficient neural network (2DCNN) are introduced in this work. We use 2 solutions of a conservation law from a converging sequence, computed from a low-cost numerical scheme, and in a domain of dependence of a space-time grid point as the input for a neural network to predict its high-fidelity solution at the grid point. Despite smeared input solutions, they output sharp approximations to solutions containing shocks and contacts and are efficient to use once trained.

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“…However, for wave propagation problems the performance of these mainstream algorithms is somewhat disappointing, because the convergence rate heavily depends on the dissipativity (see [38,39] for discussions). There are also many efforts toward ameliorating the convergence behavior of the iterative PinT algorithms via improving the the coarse grid correction [8,10,15,33,37], but as pointed out in [36] these modified algorithms either need significant additional computation burden (leading to further degradation of efficiency) or have very limited applicability.…”

Section: Introductionmentioning

confidence: 99%

A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

Li¹,

Wang²,

Wu³

et al. 2021

Preprint

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In this paper, we propose a direct parallel-in-time (PinT) algorithm for timedependent problems with first-or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix V grows as n is increased, where n is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing V and V −1 , by which we prove that Cond 2 (V ) = O(n 2 ). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

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A stable parareal-like method for the second order wave equation

Cited by 15 publications

References 37 publications

Toward error estimates for general space-time discretizations of the advection equation

Toward error estimates for general space-time discretizations of the advection equation

Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems

A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

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