Vol. 13, No. 2, 2018

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An equation-by-equation method for solving the multidimensional moment constrained maximum entropy problem

Wenrui Hao and John Harlim

Vol. 13 (2018), No. 2, 189–214

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 100103 in magnitude, (2) four-moment multidimensional entropy problems with explicit solutions where the resulting systems to be solved range from 70310 equations, and (3) four- to eight-moment of a two-dimensional entropy problem, whose 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. The fourth example is four-moment constrained of up to five-dimensional entropy problems whose 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.

homotopy continuation, moment constrained, maximum entropy, equation-by-equation method
Mathematical Subject Classification 2010
Primary: 65H10, 65H20, 94A17, 49M15
Received: 4 July 2017
Revised: 17 January 2018
Accepted: 11 February 2018
Published: 5 June 2018
Wenrui Hao
Department of Mathematics
The Pennsylvania State University
University Park, PA
United States
John Harlim
Department of Mathematics
Department of Meteorology and Atmospheric Science
The Pennsylvania State University
University Park, PA
United States