Stochastic Collocation with Kernel Density Estimation ∗ Howard

The stochastic collocation method has recently received much attention for solving partial differential equations posed with uncertainty, i.e., where coefficients in the differential operator, boundary terms or right-hand sides are random fields. Recent work has led to the formulation of an adaptive collocation method that is capable of accurately approximating functions with discontinuities and steep gradients. These methods, however, usually depend on an assumption that the random variables involved in expressing the uncertainty are independent with marginal probability distributions that are known explicitly. In this work we combine the adaptive collocation technique with kernel density estimation to approximate the statistics of the solution when the joint distribution of the random variables is unknown. 1. Problem Statement. Let (Ω,Σ, P ) be a complete probability space with sample space Ω, σ-algebra Σ ⊂ 2 and probability measure P : Σ → [0, 1]. Let D ⊂ R be a d-dimensional bounded domain with boundary ∂D. We investigate partial differential equations (PDEs) of the form L(x, ω;u) = f(x), ∀x ∈ D, ω ∈ Ω (1.1) B(x, ω;u) = g(x), ∀x ∈ ∂D, ω ∈ Ω. Here L is a partial differential operator with boundary operator B, both of which can depend on the random parameter ω. As a consequence of the Doob-Dynkin lemma, it follows that u is also a random field, dependent on both the spatial location x and the event ω. In order to work numerically with the expressions in (1.1), we must first represent the operators in terms of a finite number of random variables ξ = [ξ1, ξ2, ..., ξM ] . This is often accomplished using a truncated Karhunen-Loève (KL) expansion [13]. If we denote Γ = Image(ξ), then we can write (1.1) as L(x, ξ;u) = f(x), ∀x ∈ D, ξ ∈ Γ (1.2) B(x, ξ;u) = g(x), ∀x ∈ ∂D, ξ ∈ Γ. For a given realization of the random vector ξ, the system (1.2) is a deterministic partial differential equation that can be solved using a deterministic solver. Throughout this paper we assume that D, L, B, f , and g are defined so that the above problem (1.2) is well posed for all values of ξ ∈ Γ. In this paper we will explore several different sampling methods for solving the system (1.2). One is typically interested in methods that allow statistical properties of u to be computed. If ρ(ξ) denotes the joint probability density function of the random vector ξ, then the k moment of the solution u is defined as