A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential

Wu, Zhizhang; Huang, Zhongyi

doi:10.1016/j.jcp.2016.04.051

Cited by 6 publications

(3 citation statements)

References 43 publications

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“…This model can be efficiently solved by a number of numerical schemes that make use of the periodic structure of the potential, e.g. the Bloch decomposition-based time-splitting pseudospectral method [19,20,42], the Gaussian beam method [24,25,38,43], and the frozen Gaussian approximation method [8]. With the recent development in nanotechnology, increasing interest has been shown in quantum heterostructures with tailored functionalities, such as heterojunctions, including the ferromagnet/metal/ferromagnet structure for giant megnetoresistance [44], the silicon-based heterojunction for solar cells [27], and quantum metamaterials [39].…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials

Wu,

Zhang

2021

Preprint

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In this paper, we analyze the convergence of the operator-compressed multiscale finite element method (OC MsFEM) for Schrödinger equations with general multiscale potentials in the semiclassical regime. In the OC MsFEM the multiscale basis functions are constructed by solving a constrained energy minimization. Under a mild assumption on the mesh size H, we prove the exponential decay of the multiscale basis functions so that localized multiscale basis functions can be constructed, which achieve the same accuracy as the global ones if the oversampling size m = O(log(1/H)). We prove the first-order convergence in the energy norm and second-order convergence in the L 2 norm for the OC MsFEM and super convergence rates can be obtained if the solution possesses sufficiently high regularity. By analysing the regularity of the solution, we also derive the dependence of the error estimates on the small parameters of the Schrödinger equation. We find that the OC MsFEM outperforms the finite element method (FEM) due to the super convergence behavior for high-regularity solutions and weaker dependence on the small parameters for low-regularity solutions in the presence of the multiscale potential. Finally, we present numerical results to demonstrate the accuracy and robustness of the OC MsFEM.

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Section: Introductionmentioning

confidence: 99%

Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials

Wu,

Zhang

2021

Preprint

Self Cite

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“…Convergence analyses on both methods for different kinds of differential equations have also been established [3,4,27,29,32,41]. For the linear Schrödinger equation, Wu and Huang applied the stochastic Galerkin method to the Schrödinger equation with a periodic potential and a random external potential [33]. Nevertheless, to the best of our knowledge, there is not any convergence result on both methods for the linear Schrödinger equation with a random potential.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs

Huang¹

2020

AAMM

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In this paper, we analyse the stochastic collocation method for a linear Schrödinger equation with random inputs, where the randomness appears in the potential and initial data and is assumed to be dependent on a random variable. We focus on the convergence rate with respect to the number of collocation points. Based on the interpolation theories, the convergence rate depends on the regularity of the solution with respect to the random variable. Hence, we investigate the dependence of the stochastic regularity of the solution on that of the random potential and initial data. We provide sufficient conditions on the random potential and initial data to ensure the smoothness of the solution and the spectral convergence. Finally, numerical results are presented to support our analysis.

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“…When the potential is deterministic, i.e., v ε (x, ω) = v ε (x), many numerical methods have been proposed; see [4,16,39,27,15,9,8] for example. When the potential is random, few works have been done; see [40,25]. As mentioned above, the major difficulty is that the wavefunction ψ ε develops high-frequency oscillations in both the physical space and the random space, which requires tremendous computational resources.…”

Section: Introductionmentioning

confidence: 99%

A Multiscale Finite Element Method for the Schrödinger Equation with Multiscale Potentials

Chen¹,

Ma²,

Zhang³

2019

SIAM J. Sci. Comput.

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The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial gridsize is only proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for Schrödinger equation with correlated random potentials in both 1D and 2D.

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A Bloch decomposition-based stochastic Galerkin method for quantum dynamics with a random external potential

Cited by 6 publications

References 43 publications

Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials

Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials

Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs

A Multiscale Finite Element Method for the Schrödinger Equation with Multiscale Potentials

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