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- Title
High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables.
- Authors
Xie, Hongling
- Abstract
There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.
- Subjects
SIGNAL frequency estimation; STOCHASTIC differential equations; STOCHASTIC partial differential equations; DETERMINISTIC algorithms; RANDOM variables; GAUSSIAN measures; POLYNOMIAL chaos
- Publication
Advances in Mathematical Physics, 2023, p1
- ISSN
1687-9120
- Publication type
Article
- DOI
10.1155/2023/9974539