Derivatives of maximum-entropy basis functions on the boundary: Theory and computations

In this paper, we obtain explicit expressions to evaluate the derivatives of maximum-entropy (max-ent) basis function on the boundary of a convex domain. In the max-ent formulation, the basis functions are obtained by maximizing a concave functional subjected to linear constraints (reproducing conditions). In doing so, it is found that the Lagrange multipliers blow up when x, and the expressions for the derivatives of the max-ent basis functions in are of an indeterminate form for points on . We appeal to l'Hopital's rule to derive expressions to determine the derivatives of the basis functions. We consider the Shannon entropy functional and the relative entropy functional with different choices of the prior weight function. The first-order derivatives of all basis functions are bounded. In contrast, on an irregular grid with a certain nodal spacing, some of the second derivatives of the basis functions are unbounded on the boundary. Necessary and sufficient conditions on the priors to obtain bounded Lagrange multipliers are established. Optimal convergence rates for fourth-order problems are demonstrated for a Galerkin approach with a quadratically complete partition-of-unity enriched max-ent approximation.Copyright (c) 2013 John Wiley & Sons, Ltd.