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

## TL;DR

The paper introduces an equation-by-equation (EBE) method combining homotopy continuation and Newton's methods to solve complex multidimensional maximum entropy problems, demonstrating robustness and improved accuracy over existing methods.

## Contribution

A novel EBE approach for solving nonlinear systems in maximum entropy problems, with proven local convergence and adaptive constraint selection for higher-dimensional cases.

## Key findings

- The EBE method successfully solves high-dimensional entropy problems with explicit solutions.
- It outperforms classical Newton and BFGS methods in accuracy.
- The method adapts to problems where solutions may not exist, selecting relevant constraints.

## Abstract

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.