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首页> 美国政府科技报告 >Numerical Methods for Parabolic Equations and Inequalities in Very High Dimensions.
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Numerical Methods for Parabolic Equations and Inequalities in Very High Dimensions.

机译：极高维抛物方程和不等式的数值方法。

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The main goal of this projects was to investigate the numerical solution of parabolic equations in very high dimension, the principal motivation being the solution of the so-called Zakai equation from nonlinear filtering. As it will be seen in Section 2, below, this equation is of the advection-diffusion type and therefore is closely related to other equations of this type such as the Navier-Stokes equations from Fluid Dynamics; since these equations are also of interest for the Department of Defense some aspects of their numerical solution, byproducts of the present investigation will be considered in the present document. The content of this report is the following: In section 2, we describe the Zakai and its origin. In Section 3, we consider the solution of time dependent problems by operator splitting methods, and show in Sections 4, 5, 6, how these methods apply to the Zakai equation, to parabolic inequalities of obstacle type and to Navier-Stokes equations. In Section 7 we comment on the solution of advection-diffusion problems by up winding, particle and characteristics methods. In Section 8 we go back to the Zakai equation and its stochastic dynamical system origin and comment about the feasibility of the Zakai equation approach for the analysis of such systems. (KR)

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