Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks

TL;DR

The paper tackles high-dimensional interpolation for piecewise smooth functions with kinks, where standard sparse grids struggle. It introduces two hp-adaptive sparse grid collocation algorithms that employ local h- and p-refinement via a hierarchical knot tree and hierarchical surplus, with either a greedy degree-selection strategy or a kink-detection procedure to identify non-smooth regions. Numerical experiments on four benchmark functions demonstrate that the hp-GSG methods outperform traditional linear, quadratic, and highest-degree approaches, especially in higher dimensions, by achieving higher accuracy with fewer points and reasonable computation times. The work advances uncertainty quantification and stochastic optimization by enabling efficient, non-smooth function approximation and has potential applications in gas-network stochastic control and other high-dimensional uncertainty problems.

Abstract

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.

Figures (12)

• Figure 1: The first $5$ levels of the equidistant knot tree on $[-1,1]$. For example, the ancestors of knot $x=-\frac{5}{8}$ are $S_A(-\frac{5}{8})=\{-\frac{3}{4},-\frac{1}{2},-1,0\}$.
• Figure 2: Interpolation across a kink for $f$ defined in \ref{['funcWithKink']}. The basis function $\varphi_{1,3}^{(3)}$ at knot $x_{1}^{(3)}=-0.75$ with highest possible polynomial degree $p=3$ uses a function value at $x=0$, which leads to a relatively bad approximation of $f(x)\equiv 0$ in its support $[-1,-0.5]$ away from the kink at $x=-0.45$. The greedy approach described in Section \ref{['sec:greedy']} restricts the used ancestors and yields a zero error there.
• Figure 3: Test problem with $f^{}_{\text{0}}$. Knot placement for locally linear polynomials (left) and $hp$-GSG-k (right) with thresholds $w_{\max}=10^{-3}$ and $w_{kink}=5$. Knots of basis functions for which a kink was detected within an interval of size smaller than $2^{-6}$ are colored in red.
• Figure 4: Test problem with $f^{}_{\text{0}}$. Comparison of the errors $\epsilon_{2}$ (left) and $\epsilon_{\infty}$ (right).
• Figure 5: Test problem with $f^{2}_{\text{1}}$. Comparison of the errors $\epsilon_{2}$ (left) and $\epsilon_{\infty}$ (right).

Theorems & Definitions (3)

• Remark 2.1
• Remark 2.2
• Remark 3.1