CONSTRUCTING NESTED NODAL SETS FOR MULTIVARIATE POLYNOMIAL INTERPOLATION

We present a robust method for choosing multivariate polynomial interpolation nodes. Our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity, a variant of a weighted Lebesgue function. Nodes are therefore chosen that tend to control oscillations in the resulting interpolant. This method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space. Our method is therefore flexible: nested nodal sets are produced in spaces of arbitrary dimensions, and the number of nodes added at each stage can be arbitrary. The algorithm produces a nodal set given a probability measure on the input space, thus parameterizing interpolants with respect to finite measures. We present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics: the Lebesgue constant, the Vandermonde determinant, and the Vandermonde condition number. We also show that a nongreedy version of the nodal array has a strong connection with equilibrium measures from weighted pluripotential theory.