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A Robust Numerical Method for the Random Interface Grating Problem via Shape Calculus, Weak Galerkin Method, and Low-Rank Approximation

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Abstract

We present an efficient numerical algorithm to solve random interface grating problems based on a combination of shape derivatives, the weak Galerkin method, and a low-rank approximation technique. By using the asymptotic perturbation approach via shape derivative, we estimate the expectation and the variance of the random solution in terms of the magnitude of the perturbation. To effectively capture the severe oscillations of the random solution with high resolution near the interface, we use weak Galerkin method to solve the Helmholtz equation related to the grating interface problem at each realization. To effectively compute the variance operator, we use an efficient low-rank approximation method based on a pivoted Cholesky decomposition to compute the two-point correlation function. Two numerical experiments are conducted to demonstrate the efficiency of our algorithm.

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Acknowledgements

The work of G. Bao is supported in part by a NSFC Innovative Group Fun (No. 11621101), an Integrated Project of the Major Research Plan of NSFC (No. 91630309), and an NSFC A3 Project (No. 11421110002), and the Fundamental Research Funds for the Central Universities. The work of Y.Z. Cao is supported in part by the National Science Foundation under the Grant Numbers DMS1620027 and DMS1620150. The work of K. Zhang is supported in part by China Natural National Science Foundation (91630201, U1530116, 11471141, 11771179, 11726102), and by the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University. They also wish to thank the high performance computing center of Jilin university and computing center of Jilin province for essential computing support.

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Authors and Affiliations

  1. Department of Mathematics, Jilin University, Changchun, Jilin, People’s Republic of China

    Gang Bao, Yongle Hao & Kai Zhang

  2. Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China

    Gang Bao

  3. Department of Mathematics and Statistics, Auburn University, Auburn, USA

    Yanzhao Cao

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  1. Gang Bao
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Correspondence to Kai Zhang.

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Bao, G., Cao, Y., Hao, Y. et al. A Robust Numerical Method for the Random Interface Grating Problem via Shape Calculus, Weak Galerkin Method, and Low-Rank Approximation. J Sci Comput 77, 419–442 (2018). https://doi.org/10.1007/s10915-018-0712-z

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