Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations

Mathematical modeling and computer simulations are nowadays widely used tools to predict the behavior of problems in engineering and in the natural and social sciences. All such predictions are obtained by formulating mathematical models and then using computational methods to solve the corresponding problems. We use a probability theory approach for uncertainty quantification (UQ) since it is particularly well suited for SPDE models, and focus on the broad research areas of algorithmic development and numerical analysis for the discretization of systems of linear or nonlinear SPDEs, building upon and significantly extending our previous successful work. We conduct comprehensive theoretical and computational comparison of the efficiency, accuracy, and range of applicability of non-intrusive methods, such as stochastic collocation methods, and intrusive techniques, such as stochastic Galerkin methods, for solving SPDEs and for UQ applications. We extend the algorithmic and analysis advances wrought by these efforts to the even more challenging settings of optimal control and parameter identification problems for SPDEs. The parameter identification problem is especially important in the SPDE setting since it provides a very useful mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This effort builds on our previous work on adjoint and sensitivity-based methods for deterministic optimal control and parameter identification problems to develop similar methods for tracking statistical quantities of interest from the computational solutions of linear and nonlinear SPDEs driven by high-dimensional random inputs. Status/Progress 1. A generalized methodology for the solution of stochastic identification problems constrained by partial differential equations with random input data [5] We propose and analyze a scalable, parallel mechanism for stochastic identification/control for problems constrained by partial differential equations with random input data. Several identification objectives are discussed that either minimize the expectation of a tracking cost functional or minimize the difference of desired statistical quantities in the appropriate L norm, and the distributed parameters/control can both deterministic or stochastic. Given an objective we prove the existence of an optimal solution, establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations. The modeling process may describe the solution in terms of high dimensional spaces, particularly in the case when the input data (coefficients, forcing terms, boundary conditions, geometry, etc) are affected by a large amount of uncertainty. For higher accuracy, the computer simulation must increase the number of random variables (dimensions), and expend more effort approximating the quantity of interest in each individual dimension. Hence, we introduce a novel stochastic parameter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation FEM approach. This allows for decoupled, moderately high dimensional, parameterized computations of the stochastic optimality system, where at each collocation point, deterministic analysis and techniques can be utilized. The advantage of our approach is that it allows for the optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of the input random fields, given the probability distribution of some responses of the system (quantities of physical interest). Our rigorously derived error estimates, for the fully discrete problems, will be described and used to compare the efficiency of the method with several other techniques. Numerical examples illustrate the theoretical results and demonstrate the distinctions between the various stochastic identification objectives. The general framework of the problem is the following: we seek random parameters, coefficients κ(ω, x) and/or forcing terms f(ω, x), with x ∈ D ⊂ R, ω ∈ Ω, where (Ω,F , P ) a complete probability space, that minimize the mismatch between stochastic measured and simulated data. Here Ω is the set of outcomes, F ⊂ 2 is the σ-algebra of events and P : F → [0, 1] is a probability measure. There are two main ways of measuring this spatial-stochastic quantity: the expected value of spatial mismatch (̊see e.g. [8, 7] more ref’s) and the spatial mismatch of averages of the statistical quantities of interest. More precisely, we consider the minimization cost functionals of the type J (u, (κ, f)) (0.1) over all κ,f and random solutions u : Ω × D → R that satisfy P -almost everywhere in Ω, or in other words almost surely (a.s.), the following stochastic boundary value problem: L(κ)(u) = f in D (0.2) supplemented with appropriate boundary conditions. We consider the groundwater flow problem in a region D ⊂ R, d = 1, 2, 3, where the flux is related to the hydraulic head gradient by Darcy’s law. We model the uncertainties in the soil by describing the conductivity coefficient κ as a random field denoted κ(ω, x). Similarly, the stochastic forcing term f(ω, x) models the uncertainty in the sources and sinks.Therefore the hydraulic head u : Ω×D is also a random field satisfying the elliptic stochastic partial differential equation (SPDE): { −∇ · (κ(ω, x)∇u(ω, x)) = f(ω, x) in Ω×D, u = 0 on Ω× ∂D. (0.3) The linear elliptic SPDE (0.3) with κ(ω, ·) uniformly bounded and coercive, i.e. there exists κmin, κmax ∈ (0,+∞) such that P (ω ∈ Ω : κ(ω, x) ∈ [κmin, κmax]∀x ∈ D) = 1 (0.4) and f(ω, ·) square integrable with respect to P , satisfies assumptions A1 and A2 with W (D) = H 0 (D). We shall assume that D is a bounded and open subset of R, either with smooth boundary (of class C for instance) or convex. This implies that for every f ∈ LP (Ω;L(D)), problem (0.3) has a unique solution u ∈ LP (Ω;H 0 (D) ∩ H(D)). The solution to (0.3) must be understood in a variational sense, i.e., for given f ∈ LP (Ω, L(D)) we say that u ∈ LP (Ω, H 0 (D)) is a solution of E [∫