High-order spectral method of density estimation for stochastic differential equation driven by multivariate Gaussian random variables. (English) Zbl 1531.65204
The scientific problem addressed in the paper involves the development of efficient and high-order numerical methods for density estimation in Stochastic Partial Differential Equations (SPDEs) driven by multivariate Gaussian random variables. Previous research in this area has primarily focused on SPDEs with independent random variables, with less attention given to those driven by multivariate Gaussian random variables.
To solve this problem, the author proposes a high-order algorithm based on the generalized Polynomial Chaos (gPC) approach. The methodology involves several key steps: firstly, constructing a new multivariate orthogonal basis through the Gauss-Schmidt orthogonalization process; secondly, assuming that the unknown function in the SPDE can be expanded in terms of this stochastic gPC basis; thirdly, implementing the stochastic gPC expansion for the SPDE in multivariate Gaussian measure space; and finally, performing numerical calculations to derive deterministic differential equations for the coefficients of the expansion. Additionally, the paper utilizes a high-order gPC-based algorithm for both density and moment estimation.
The main findings of the research are significant. The newly proposed numerical method is applied to well-known random function stochastic equations, specifically a 1D wave equation and a 2D Schnakenberg model, both driven by bivariate Gaussian random variables. The efficiency of the proposed method is compared with the traditional Monte-Carlo method. The results demonstrate that the proposed high-order spectral method provides a more efficient and accurate approach for density estimation in stochastic differential equations driven by multivariate Gaussian random variables. The paper significantly contributes to the field by providing a novel method that enhances the precision and efficiency of density estimation in complex SPDEs.
To solve this problem, the author proposes a high-order algorithm based on the generalized Polynomial Chaos (gPC) approach. The methodology involves several key steps: firstly, constructing a new multivariate orthogonal basis through the Gauss-Schmidt orthogonalization process; secondly, assuming that the unknown function in the SPDE can be expanded in terms of this stochastic gPC basis; thirdly, implementing the stochastic gPC expansion for the SPDE in multivariate Gaussian measure space; and finally, performing numerical calculations to derive deterministic differential equations for the coefficients of the expansion. Additionally, the paper utilizes a high-order gPC-based algorithm for both density and moment estimation.
The main findings of the research are significant. The newly proposed numerical method is applied to well-known random function stochastic equations, specifically a 1D wave equation and a 2D Schnakenberg model, both driven by bivariate Gaussian random variables. The efficiency of the proposed method is compared with the traditional Monte-Carlo method. The results demonstrate that the proposed high-order spectral method provides a more efficient and accurate approach for density estimation in stochastic differential equations driven by multivariate Gaussian random variables. The paper significantly contributes to the field by providing a novel method that enhances the precision and efficiency of density estimation in complex SPDEs.
Reviewer: Denys Dutykh (Le Bourget-du-Lac)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65C05 | Monte Carlo methods |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 62G07 | Density estimation |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
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