TY - JOUR

T1 - An equation-by-equation method for solving the multidimensional moment constrained maximum entropy problem

AU - Hao, Wenrui

AU - Harlim, John

N1 - Funding Information:
Hao’s research was partially supported by the American Heart Association (Grant 17SDG33660722), the National Science Foundation (Grant DMS-1818769) and an Institute for CyberScience Seed Grant. Harlim’s research was partially supported by the Office of Naval Research (Grant N00014-16-1-2888) and the National Science Foundation (Grant DMS-1619661). MSC2010: 65H10, 65H20, 94A17, 49M15. Keywords: homotopy continuation, moment constrained, maximum entropy, equation-by-equation method.
Publisher Copyright:
© 2018 Mathematical Sciences Publishers.

PY - 2018

Y1 - 2018

N2 - 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 sixmoment one-dimensional entropy problem with an explicit solution that contains components of order 100-103 in magnitude, (2) four-moment multidimensional entropy problems with explicit solutions where the resulting systems to be solved range from 70-310 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 fourmoment 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.

AB - 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 sixmoment one-dimensional entropy problem with an explicit solution that contains components of order 100-103 in magnitude, (2) four-moment multidimensional entropy problems with explicit solutions where the resulting systems to be solved range from 70-310 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 fourmoment 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.

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U2 - 10.2140/camcos.2018.13.189

DO - 10.2140/camcos.2018.13.189

M3 - Article

AN - SCOPUS:85049562654

SN - 1559-3940

VL - 13

SP - 189

EP - 214

JO - Communications in Applied Mathematics and Computational Science

JF - Communications in Applied Mathematics and Computational Science

IS - 2

ER -