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An equation-by-equation (EBE) method is proposed to solve a system ofnonlinear equations arising from the moment constrained maximum entropy problemof multidimensional variables. The design of the EBE method combines ideas fromhomotopy continuation and Newton's iterative methods. Theoretically, weestablish the local convergence under appropriate conditions and show that theproposed method, geometrically, finds the solution by searching along thesurface corresponding to one component of the nonlinear problem. We willdemonstrate the robustness of the method on various numerical examples,including: (1) A six-moment one-dimensional entropy problem with an explicitsolution that contains components of order $10^0-10^3$ in magnitude; (2)Four-moment multidimensional entropy problems with explicit solutions where theresulting systems to be solved ranging from $70-310$ equations; (3) Four- toeight-moment of a two-dimensional entropy problem, which solutions correspondto the densities of the two leading EOFs of the wind stress-driven large-scaleoceanic model. In this case, we find that the EBE method is more accuratecompared to the classical Newton's method, the MATLAB generic solver, and thepreviously developed BFGS-based method, which was also tested on this problem.(4) Four-moment constrained of up to five-dimensional entropy problems whichsolutions correspond to multidimensional densities of the components of thesolutions of the Kuramoto-Sivashinsky equation. For the higher dimensionalcases of this example, the EBE method is superior because it automaticallyselects a subset of the prescribed moment constraints from which the maximumentropy solution can be estimated within the desired tolerance. This selectionfeature is particularly important since the moment constrained maximum entropyproblems do not necessarily have solutions in general.

An Equation-By-Equation Method for Solving the Multidimensional Moment Constrained Maximum Entropy Problem