- ANOVA approximation with mixed tensor product basis on scattering points (arXiv)
Authors: Daniel Potts, Pascal Schroeder
Abstract: In this paper, we consider orthonormal bases generated by the tensor product of Fourier basis functions, semi-periodic cosine basis functions and Chebyshev basis functions. We address the high-dimensional approximation problem associated with this basis and design a fast algorithm to multiply the underlying matrix consisting of rows of non-uniform Fourier, non-uniform cosine, and non-uniform Chebyshev matrices. and transpose it. This leads to his ANOVA (analysis of variance) of functions with partially periodic boundary conditions using Fourier basis in some dimensions and semi-periodic cosine basis or Chebyshev basis in other dimensions. decomposition becomes possible. We consider sensitivity analysis in this setting to find a basis adapted to the underlying approximation problem. More precisely, we find the index set underlying the multidimensional series expansion. Additionally, we test this ANOVA approximation using mixed basis during numerical experiments to see the advantage of interpretable results.
2. Adaptive analysis of variance probabilistic Galerkin method (arXiv) for partial differential equations with random inputs
Authors: Wang Guangjie, Sumita Shirfu, Liao Qifeng
Abstract: The standard probabilistic Galerkin method is known to face challenges when solving PDEs with high-dimensional random inputs. This is usually due to the large number of probabilistic basis functions required. Good choice of effective basis functions is important so that the dimensionality of the probabilistic approximation space can be reduced. In this work, we focus on the probabilistic Galerkin approximation related to generalized polynomial chaos (gPC) and investigate an extension of gPC based on analysis of variance (ANOVA) decomposition. A concise form of the gPC extension is presented for each component function of his ANOVA extension, and an adaptive ANOVA procedure is proposed to construct the overall probabilistic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA probabilistic Galerkin method.
