A Geometric Nonlinear Conjugate Gradient Method for Stochastic Inverse Eigenvalue Problems
A Geometric Nonlinear Conjugate Gradient Method for Stochastic Inverse Eigenvalue Problems
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DOI：
10.1137/140992576
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发表时间：
2016-07
期刊：
影响因子：
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通讯作者：
Zhi Zhao;X. Jin;Zhengjian Bai
中科院分区：
文献类型：
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作者：
Zhi Zhao;X. Jin;Zhengjian Bai
在这篇文章中，我们主要研究从给定的谱重构随机矩阵的随机逆特征值问题。我们直接将随机逆特征值问题转化为多个矩阵流形上的约束优化问题，以最小化等谱矩阵和随机矩阵之间的距离。然后，我们提出了一种基于几何Polak-Ribiere-Polyak的非线性共轭梯度法来求解约束优化问题。证明了该方法的全局收敛性质。我们的方法也可以推广到具有指定项的随机逆特征值问题。一个额外的优势是我们的模型产生了新的等谱流动方法。最后，我们给出了一些数值测试，以说明所提出的方法在求解随机逆特征值问题和给定条目的情况下的有效性。
In this paper, we focus on the stochastic inverse eigenvalue problem of reconstructing a stochastic matrix from the prescribed spectrum. We directly reformulate the stochastic inverse eigenvalue problem as a constrained optimization problem over several matrix manifolds to minimize the distance between isospectral matrices and stochastic matrices. Then we propose a geometric Polak--Ribiere--Polyak-based nonlinear conjugate gradient method for solving the constrained optimization problem. The global convergence of the proposed method is established. Our method can also be extended to the stochastic inverse eigenvalue problem with prescribed entries. An extra advantage is that our models yield new isospectral flow methods. Finally, we report some numerical tests to illustrate the efficiency of the proposed method for solving the stochastic inverse eigenvalue problem and the case of prescribed entries.