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

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.