Efficient Basis Change and Regularization for Sparse-grid Interpolating Polynomials

Sparse-grid interpolation provides good approximations to smooth functions in high dimensions based on relatively few function evaluations, but in standard form is expressed in Lagrange polynomials and requires function values at all points of a sparse grid. Here we give a block-diagonal factorization of the matrix for changing basis from a Lagrange polynomial formulation of a sparse-grid interpolant to a tensored orthogonal polynomial (or gPC) representation. For fixed maximum degree of interpolation, the resulting change of basis algorithm is linear in the number of points of evaluation as dimension increases. Additionally, we use this factorization with `1 and minimum Sobolev norm (MSN) regularization to provide good interpolants even when function values are not available at a significant fraction of points of the sparse grid or are subject to measurment error.