2024/04/03 by Hendrik Wilka, Wilka, Hendrik, Jens Lang +1
Computer Science · Engineering · Mathematics · #65D05 #65D15 #65D40 #Advanced Numerical Analysis Techniques #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2404.02556
openalex publication_date 2024/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Four numerical benchmark examples with different dimensions are discussed and comparison with locally linear, quadratic and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods.