Efficient Basis Change for Sparse-Grid Interpolating Polynomials with Application to T-Cell Sensitivity Analysis

Sparse-grid interpolation provides good approximations to smooth functions in high dimensions based on relatively few function evaluations, but in standard form it is expressed in Lagrange polynomials. Here, we give a block-diagonal factorization of the basis-change matrix to give an efficient conversion of a sparse-grid interpolant to a tensored orthogonal polynomial (or gPC) representation. We describe how to use this representation to give an efficient method for estimating Sobol' sensitivity coefficients and apply this method to analyze and efficiently approximate a complex model of T-cell signaling events. 1. Introduction A common problem in many areas of computational mathematics is to approximate a given function based on a small number of functional evaluations or observations. This problem arises in numerical methods for PDE [1, 2], sensitivity analysis [3–5], uncertainty quantification [2], many areas of modeling [6, 7], and other settings. As a result, there are a large number of approaches to this problem, and the literature is large and growing quickly. In settings in which the points of evaluation may be chosen at will, two common approaches are generalized polynomial chaos (gPC) using cubature and sparse grid collocation [8]. In other settings in which the points of evaluation are given, common approaches include RS-HDMR, cut-HDMR, ANOVA decomposition, kriging, and moving least squares; see, for example, [9, Chapter 5] for a discussion of such methods and further references. Sparse grid collocation has been used widely in recent years as a means of providing a reasonable approximation to a smooth function, , defined on a hypercube in , based on relatively few function evaluations [2]. This method produces a polynomial interpolant using Lagrange interpolating polynomials based on function values at points in a union of product grids of small dimension [10, 11]. Using barycentric interpolation to evaluate the resulting polynomial [12], this method is a viable alternative to an expansion of in terms of a sum of products of one-dimensional orthogonal polynomials. This latter approach is known as generalized polynomial chaos (gPC) or spectral decomposition and is obtained via standard weighted techniques. However, the orthogonality implicit in the gPC representation often provides many advantages over the Lagrange representation, particularly in applications to differential equations, in which the gPC representation is closely related to spectral methods. Other advantages of the gPC representation include the ability to estimate convergence as