A Riemannian Fletcher-Reeves Conjugate Gradient Method for Doubly Stochastic Inverse Eigenvalue Problems
A Riemannian Fletcher-Reeves Conjugate Gradient Method for Doubly Stochastic Inverse Eigenvalue Problems
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DOI：
10.1137/15m1023051
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发表时间：
2016-02
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影响因子：
--
通讯作者：
Teng-Teng Yao-Teng;Zhengjian Bai;Zhi Zhao;W. Ching
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文献类型：
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作者：
Teng-Teng Yao-Teng;Zhengjian Bai;Zhi Zhao;W. Ching
我们考虑了从给定的谱数据重建双随机矩阵的逆特征值问题。我们将这个反问题转化为几个矩阵流形上的约束非线性最小二乘问题，以最小化等谱矩阵和双随机矩阵之间的距离。在此基础上，提出了求解约束非线性最小二乘问题的Riemannian Fletcher-Reeves共轭梯度法，并证明了其全局收敛性质。一个额外的收获是得到了一种新的黎曼等谱流动方法。我们的方法也被推广到规定条目的情况。最后，通过数值试验验证了该方法的有效性。
We consider the inverse eigenvalue problem of reconstructing a doubly stochastic matrix from the given spectrum data. We reformulate this inverse problem as a constrained nonlinear least squares problem over several matrix manifolds, which minimizes the distance between isospectral matrices and doubly stochastic matrices. Then a Riemannian Fletcher--Reeves conjugate gradient method is proposed for solving the constrained nonlinear least squares problem, and its global convergence is established. An extra gain is that a new Riemannian isospectral flow method is obtained. Our method is also extended to the case of prescribed entries. Finally, some numerical tests are reported to illustrate the efficiency of the proposed method.