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.

翻译：在很多情况下, 有必要从分散的网格点的函数值中确定 ${f} left (x_ 1,\ldots, x_D\right) 的基值扩张基值的系数 {f{f} left (x_ left (k\right) $。 了解系数, 一个人有一个内插或替代系数。 例如, 这种系数用于不确定性的量化。 在本章中, 我们提出了一个计算系数的有效方法。 它使用基础函数, 和熟悉的片段线性线性排序函数一样, 在前一级点零。 它们是任何( 例如) 全球、 嵌式基函数的线性组合 $\vphrei_i_ kleft (k\right)\\left (x_k\right) left (k\\k\right)\\\\\\\\\\\\\\\right$ blog。 基函数值通过对数值进行顺序评估。 当 $\ ell_ ell_ = $\\\\\\\\\\\\\\rexxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx