A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms

Mulin Cheng Thomas Y Hou Zhiwen Zhang

Numerical Analysis and Scientific Computing mathscidoc:1912.431036

Journal of Computational Physics, 242, 843-868, 2013.6
We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the KarhunenLoeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the
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@inproceedings{mulin2013a,
  title={A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms},
  author={Mulin Cheng, Thomas Y Hou, and Zhiwen Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211407998807600},
  booktitle={Journal of Computational Physics},
  volume={242},
  pages={843-868},
  year={2013},
}
Mulin Cheng, Thomas Y Hou, and Zhiwen Zhang. A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms. 2013. Vol. 242. In Journal of Computational Physics. pp.843-868. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224211407998807600.
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