Volume 12, Issue 1
Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs

Adv. Appl. Math. Mech., 12 (2020), pp. 30-56.

Published online: 2019-12

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• Abstract

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.

• Keywords

Schrodinger equation, stochastic collocation methods, convergence analysis, uncertainty quantification.

65M12, 81Q05, 60H25

wzz14@mails.tsinghua.edu.cn (Zhizhang Wu)

zhuang@math.tsinghua.edu.cn (Zhongyi Huang)

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@Article{AAMM-12-30, author = {Wu , Zhizhang and Huang , Zhongyi }, title = {Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {12}, number = {1}, pages = {30--56}, abstract = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0008}, url = {http://global-sci.org/intro/article_detail/aamm/13418.html} }
TY - JOUR T1 - Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs AU - Wu , Zhizhang AU - Huang , Zhongyi JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 30 EP - 56 PY - 2019 DA - 2019/12 SN - 12 DO - http://dor.org/10.4208/aamm.OA-2019-0008 UR - https://global-sci.org/intro/article_detail/aamm/13418.html KW - Schrodinger equation, stochastic collocation methods, convergence analysis, uncertainty quantification. AB -

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.

Zhizhang Wu & Zhongyi Huang. (2019). Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs. Advances in Applied Mathematics and Mechanics. 12 (1). 30-56. doi:10.4208/aamm.OA-2019-0008
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