Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

Lorin, Emmanuel; Yang, Xu; Antoine, Xavier

doi:10.1016/j.jcp.2016.02.035

Cited by 8 publications

(6 citation statements)

References 42 publications

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“…It is shown that these algorithms are fast and robust for complex linear problems. In [27], domain decomposition methods have been developed and combined with geometric optics and frozen gaussian approximation for computing the solution to the linear Schrödinger equations under and beyond the semi-classical regime. Finally, [9] is dedicated to the development of high-order transmission conditions for SWR methods applied to the Schrödinger equation, using only local operators.…”

Section: Introductionmentioning

confidence: 99%

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

Antoine

Lorin

2019

Journal of Computational and Applied Mathematics

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This paper is dedicated to the analysis of the rate of convergence of the classical and quasioptimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.

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Section: Introductionmentioning

confidence: 99%

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

Antoine

Lorin

2019

Journal of Computational and Applied Mathematics

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“…The behavior of the method shows that it can lead to fast and robust algorithms for complex linear problems. In [35], domain decomposition methods have been developed when using geometric optics and frozen gaussian approximations for computing the solution to linear Schrödinger equations under and beyond the semi-classical regime.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018

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This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for variable coefficients linear and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

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“…The behavior of the method shows that it can lead to fast and robust algorithms for complex linear problems. In [33], domain decomposition methods have been developed when using geometric optics and frozen gaussian approximation for computing the solution to linear Schrödinger equations beyond the semi-classical regime.…”

Section: Introductionmentioning

confidence: 99%

An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

Antoine

Lorin

2017

Numer. Math.

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The aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of Classical and Optimized Schwarz Waveform Relaxation (SWR) domain decomposition methods applied to the computation of the stationary states of the one-dimensional linear and nonlinear Schrödinger equations with a potential. Although SWR methods are currently used for efficiently solving high dimensional partial differential equations, their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension, except in simple cases. In this work, we tacke this problem for linear and nonlinear one-dimensional Schrödinger equations by developing techniques which can be extended to higher dimensional problems and other types of PDEs. The approach combines the method developed in [24] for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolical asymptotic expansions. For computing the stationary states, we consider the imaginary-time formulation of the Schrödinger equation based on the Continuous Normalized Gradient Flow (CNGF) method and use a semiimplicit Euler scheme for the discretization. Some numerical results in the one-dimensional case illustrate the analysis for both the linear Schrödinger and Gross-Pitaevskii equations.

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Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

Cited by 8 publications

References 42 publications

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

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