Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains [chapter]

Helmut Harbrecht, Michael Peters
2016 Lecture Notes in Computational Science and Engineering  
In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the influence of the random domain perturbation on the solution. We derive deterministic partial differential equations to approximate the random solution's mean and its covariance with leading order in the amplitude of the random domain perturbation. The partial differential equation
more » ... or the covariance is a tensor product Dirichlet problem which can efficiently be determined by Galerkin's method in the sparse tensor product space. We show that this Galerkin approximation coincides with the solution derived from the combination technique provided that the detail spaces in the related multiscale hierarchy are constructed with respect to Galerkin projections. This means that the combination technique does not impose an additional error in our construction. Numerical experiments quantify and qualify the proposed method. At first, we determine the boundary part u Γ J ∈ H 1 (D) such that Therefore, u Γ J | Γ ref is the L 2 -orthogonal projection of the Dirichlet datum g onto the discrete trace space V Γ J . Having u Γ J at hand, we can compute the domain part We use the conjugate gradient method to iteratively solve the systems of linear equations (20) and (21) . Using a nested iteration, combined with the BPX-preconditioner, cf. [3], in case of (21), results in a linear over-all complexity, see [15] . Moreover, from [2, Theorem 1], we obtain the following convergence result. Theorem 3. Let g ∈ H t (Γ ref ) for 0 t 3/2. Then, if g J ∈ V Γ J is given by (17) , the Galerkin solution u J to (15) satisfies Next, we deal with the tensor product boundary value problem (14) and discretize it in V J ⊗V J . We make the ansatz Combination technique based second moment analysis In complete analogy to the non-tensor product case, we obtain the solution scheme (1) Solve ( This lemma tells us that, given Cov[δ u] Γ ,Γ J , the computation of Cov[δ u] Λ ,Λ J for Λ ,Λ ∈ {D,Γ } decouples into J + 1 subproblems. It holds Cov[δ u]
doi:10.1007/978-3-319-28262-6_3 fatcat:qg4zrktrmfchxfqnzbm4efnm3y