Advanced Numerical Methods for Computing Statistical Quantities of Interest from Solutions of Spdes

Computational simulation-based predictions are central to science and engineering and to risk assessment and decision making in economics, public policy, and military venues, including several of importance to Air Force missions. Unfortunately, predictions are often fraught with uncertainty so that effective means for quantifying that uncertainty are of paramount importance. The research effort investigates and resolves important algorithmic, mathematical, and practical issues related to the efficient, accurate, and robust computational determination of the quantities of interest used by engineers and decision makers that are determined from solutions of partial differential equations having random inputs. Notable accomplishments include the development of a novel approach to discretizing white noise and its application to the stochastic Navier-Stokes-Boussinesq system; development of efficient methods for solving high-dimensional backward stochastic differential equations; verifying the accuracy and effectiveness of approximate deconvolution models on the solution of the stochastic Navier-Stokes equations; error analyses of finite element approximations of the stochastic Stokes equations; error estimates for stochastic optimal Neumann boundary control problems; the development of ANOVA expansions and efficient sampling methods for parameter dependent nonlinear problems; and the analysis of nonlinear spectral eddy-viscosity models of turbulence. developing finite element methods for parameter identification and control of elliptic equations with random input data; developing stochastic reduced-order models that combine reducedorder models for spatial discretization and efficient stochastic parameter sampling techniques; using ANOVA expansions to study the impact of parameter dependent boundary conditions on solutions of nonlinear partial differential equations and related optimization problems; developing numerical methods for option pricing problems in the presence of random arbitrage return. 1. Novel algorithm for discretizing white noise [10] The goal of this project is to develop, analyze, and test a novel means for discretizing white noise in the context of partial differential equations. We describe our approach on the simple nonlinear parabolic equation; however, the approach is applicable and effective on more general systems, as is evidenced by the project described in Section 2. Consider the equation ut − κ∆u+ u = σdWt in (0, T ]×D (1) along with boundary and initial conditions, where the noise term is defined as