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The maximum volume principle is investigated as a means to solve thefollowing problem: Given a set of arbitrary interpolation nodes, how to choosea set of polynomial basis functions for which the Lagrange interpolationproblem is well-defined with reasonable interpolation error? The interpolationerror is controlled by the Lebesgue constant of multivariate polynomialinterpolation and it is proven that the Lebesgue constant can effectively bebounded by the reciprocals of the volume (i.e., determinant in modulus) and theminimal singular value of the multidimensional Vandermonde matrix associatedwith the interpolation problem. This suggests that a large volume of theVandermonde system can be used as an indicator of accuracy and stability of theresulting interpolating polynomial. Numerical examples demonstrate that theapproach outlined in this paper works remarkably well in practicalcomputations.

On applying the maximum volume principle to a basis selection problem in multivariate polynomial interpolation