  An equation-by-equation (EBE) method is proposed to solve a system of
nonlinear equations arising from the moment constrained maximum entropy problem
of multidimensional variables. The design of the EBE method combines ideas from
homotopy continuation and Newton's iterative methods. Theoretically, we
establish the local convergence under appropriate conditions and show that the
proposed method, geometrically, finds the solution by searching along the
surface corresponding to one component of the nonlinear problem. We will
demonstrate the robustness of the method on various numerical examples,
including: (1) A six-moment one-dimensional entropy problem with an explicit
solution that contains components of order $10^0-10^3$ in magnitude; (2)
Four-moment multidimensional entropy problems with explicit solutions where the
resulting systems to be solved ranging from $70-310$ equations; (3) Four- to
eight-moment of a two-dimensional entropy problem, which solutions correspond
to the densities of the two leading EOFs of the wind stress-driven large-scale
oceanic model. In this case, we find that the EBE method is more accurate
compared to the classical Newton's method, the MATLAB generic solver, and the
previously developed BFGS-based method, which was also tested on this problem.
(4) Four-moment constrained of up to five-dimensional entropy problems which
solutions correspond to multidimensional densities of the components of the
solutions of the Kuramoto-Sivashinsky equation. For the higher dimensional
cases of this example, the EBE method is superior because it automatically
selects a subset of the prescribed moment constraints from which the maximum
entropy solution can be estimated within the desired tolerance. This selection
feature is particularly important since the moment constrained maximum entropy
problems do not necessarily have solutions in general.
