{"title":"Solving SPDEs by a Least Squares Method","authors":"Hassan Manouzi","volume":85,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":71,"pagesEnd":75,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9997268","abstract":"

We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.<\/p>\r\n","references":" F. E. Benth and J. Gjerde, Convergence rates for finite element approximations of stochastic partial differential equations, Stochastics Stochastics Rep. 63 (1998) 313\u2013326.\r\n R. Glowinski, Numerical methods for nonlinear variational problems. Springer, 2008.\r\n H. Holden, T. Lindstr\u00f8m, B. \u00d8ksendal, J. Ub\u00f8e, and T.-S. Zhang, The pressure equation for fluid flow in a stochastic medium, Potential Analysis, 4 (1995) 655\u2013674.\r\n H. Holden, B. \u00d8ksendal, J. Ub\u00f8e, and T.-S. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Probability and its Applications. Birkh\u00a8auser, Boston, 1996.\r\n M. Kardar, Y.C. Zhang, Scaling of Directed Polymers in Random Media, Physical Review Letters, 58 (1987) 2087\u20132090.\r\n H. Manouzi, A finite element approximation of linear stochastic PDE\u2019s equations driven by a multiplicative white noise, International Journal of Computer Mathematics, 85 (2008) 527\u2013546.\r\n T. G. Theting, Solving Wick-stochastic boundary value problems using a finite element method, Stochastics Stochastics Rep. 70 (200) 241\u2013270.\r\n T.G. Theting, Solving Parabolic Wick-Stochastic Boundary Value Problems Using a Finite Element Method, Stochast. Stochast. Reports. 75 (2003) 57\u201392.\r\n G. V\u02daage, Hilbert space methods applied to elliptic stochastic partial differential equations, Stochastic analysis and related topics, Stochastic analysis and related topics, Progr. Probab. 38, Birkh\u00a8auser Boston, Boston, MA, 1996, pp. 281\u2013294.\r\n G.V\u02daage, Variational Methods for PDEs Applied to Stochastic Partial Differential Equations, Math. Scand. 82 (1998) 113\u2013137.\r\n Wuan Luo, Wiener chaos expansion and numerical solutions of stochastic partial differential equations. VDM Verlay Edition, 2010.\r\n T. Zhang, Characterization of white noise test functions and Hida distributions, Stochastics 41, pp 71\u201378, 1980.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 85, 2014"}