Efficiently transforming from values of a function on a sparse grid to basis coefficients

TL;DR

This paper introduces an efficient method for converting function values on sparse grids into basis coefficients, significantly reducing computational complexity by exploiting nesting properties and sequential evaluation.

Contribution

The paper presents a novel, efficient transformation technique from sparse grid function values to basis coefficients using nested basis functions and sequential sums.

Findings

Transformation cost scales as O(D[(b/(D+1))+1] N_sparse)

Method exploits nesting to improve efficiency

Compared with existing methods, offers reduced computational complexity

Abstract

In many contexts it is necessary to determine coefficients of a basis expansion of a function ${f}\left(x_1, \ldots, x_D\right) $ from values of the function at points on a sparse grid. Knowing the coefficients, one has an interpolant or a surrogate. For example, such coefficients are used in uncertainty quantification. In this chapter, we present an efficient method for computing the coefficients. It uses basis functions that, like the familiar piecewise linear hierarchical functions, are zero at points in previous levels. They are linear combinations of any, e.g. global, nested basis functions $\varphi_{i_k}^{\left(k\right)}\left(x_k\right)$. Most importantly, the transformation from function values to basis coefficients is done, exploiting the nesting, by evaluating sums sequentially. When the number of functions in level $\ell_k$ equals $\ell_k$ (i.e. when the level index is increased by one, only one point (function) is added) and the basis function indices satisfy ${\left\lVert\mathbf{i}-\mathbf{1}\right\lVert_1 \le b}$, the cost of the transformation scales as $\mathcal{O}\left(D \left[\frac{b}{D+1} + 1\right] N_\mathrm{sparse}\right)$, where $N_\mathrm{sparse}$ is the number of points on the sparse grid. We compare the cost of doing the transformation with sequential sums to the cost of other methods in the literature.