An adaptive high-order piecewise polynomial based sparse grid collocation method with applications

Abstract

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method. Error estimates are provided, and the numerical results in function interpolation, integration and some benchmark problems in uncertainty quantification are used to compare different collocation schemes.

An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

“…In this section, we first review the fundamentals of MRA of DG approximation spaces and the associated multiwavelets. Two classes of multiwavelets, namely the L 2 orthonormal Alpert's multiwavelets [2] and the interpolatory multiwavelets [44], are considered. We also introduce a set of key notations used throughout the paper.…”

“…Alpert's multiwavelets described in Section 2.1 are associated with the L 2 projection operator. The interpolatory multiwavelets introduced in [44] are constructed based on interpolation operators and also essential for efficient computation of integrals in the DG formulation, especially in high dimensions. In this work, only Lagrange interpolation is considered, while we note that Hermite interpolation can also be used but its implementation is more involved.…”

“…For the sparse grid space VM N or any adaptively chosen subspace of V M N , the interpolation operator, which is denoted by I M h in later sections, can be defined accordingly, by taking only multiwavelet basis functions that belong to that space. The fast algorithms which transform point values at interpolation points to hierarchical coefficients are given in [44]. The detailed formulas of the interpolation points and the associated interpolatory multiwavelets used in this work, we refer readers to [25,26].…”

“…The tensor-product Alpert's multiwavelets are used as the DG function space, following the approach developed in [45,20,21] for linear equations. Besides, the interpolatory multiwavelets for MRA quadrature [44] are used for computations of nonlinear terms. Numerical experiments for benchmark tests in up to four dimension verify the efficiency and efficacy of the method in capturing the viscosity solution of the HJ equations.…”

A sparse grid discrete ordinate discontinuous Galerkin method for the radiative transfer equation

“…This method is also applied to transport equations in [24,25], but the scattering effect is not considered. The adaptive analogues of their methods are also given in [25,41].…”