An improved algorithm for the multidimensional moment-constrained maximum entropy problem

The maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints. In the recent paper we introduced new computational framework for the moment-constrained maximum entropy problem in a multidimensional domain, and developed a simple numerical algorithm capable of computing maximum entropy problem in a two-dimensional domain with moment constraints of order up to 4. Here we design an improved numerical algorithm for computing the maximum entropy problem in a two- and higher-dimensional domain with higher order moment constraints. The algorithm features multidimensional orthogonal polynomial basis in the dual space of Lagrange multipliers to achieve numerical stability and rapid convergence of Newton iterations. The new algorithm is found to be capable of solving the maximum entropy problem in the two-dimensional domain with moment constraints of order up to 8, in the three-dimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. The two- and higher-dimensional maximum entropy test problems in the current work are based upon long-term statistics of numerical simulation of the real-world geophysical model for wind stress driven oceanic currents such as the Gulf Stream and the Kuroshio.