AN EQUATION-BY-EQUATION METHOD FOR SOLVING THE MULTIDIMENSIONAL MOMENT CONSTRAINED MAXIMUM ENTROPY PROBLEM
AN EQUATION-BY-EQUATION METHOD FOR SOLVING THE MULTIDIMENSIONAL MOMENT CONSTRAINED MAXIMUM ENTROPY PROBLEM
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DOI：
10.2140/camcos.2018.13.189
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发表时间：
2018-01-01
影响因子：
2.1
通讯作者：
Harlim, John
中科院分区：
文献类型：
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作者：
Hao, Wenrui;Harlim, John
提出了一种求解多维变量矩约束最大熵问题的非线性方程组的逐方程方法。EBE方法的设计结合了同伦延拓和牛顿迭代法的思想。理论上，我们建立在适当的条件下的局部收敛性，并表明，所提出的方法，几何上，通过搜索沿着表面对应的非线性问题的一个组件找到解决方案。我们将在各种数值例子上证明该方法的鲁棒性，包括（1）具有显式解的六阶矩一维熵问题，其包含数量级为100-103的分量，（2）具有显式解的四阶矩多维熵问题，其中所得到的待解系统的范围为70-310个方程，（3）二维熵问题的四到八阶矩，其解对应于风应力驱动的大尺度海洋模式的两个主导EOF的密度。在这种情况下，我们发现，EBE方法是更准确的相比，经典的牛顿的方法，MATLAB通用求解器，和以前开发的BFGS为基础的方法，这也是测试这个问题。第四个例子是fourmoment约束的五维熵的问题，其解决方案对应于多维密度的组件的解决方案的Kuramoto-Sivashinsky方程。对于这个例子的高维情况，EBE方法是上级的，因为它自动选择了一个子集的规定的时刻约束，最大熵的解决方案可以估计在所需的公差。这个选择特征是特别重要的，因为矩约束最大熵问题一般不一定有解。
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.