项目名称: 张量特征值的算法研究
项目编号: No.11201092
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 陈震
作者单位: 贵州师范大学
项目金额: 23万元
中文摘要: 张量特征值的概念从2005年提出至今只有七年的时间,它在盲信号分离、核磁共振成像及量子物理等领域有着广泛的应用前景。因此,张量特征值问题是一个新兴的、极富生命力的研究方向。本项目拟做如下的研究工作:讨论张量特征值的谱理论,建立张量特征值的估计式,为算法提供必备的理论依据;结合广义Rayleigh商讨论张量奇异值与特征值的联系,设计求非负张量最大奇异值的迭代算法,并分析算法的收敛性、稳定性等问题;讨论张量特征值在医学图像中的应用,并针对高阶马尔可夫链过程中的转移概率张量,讨论其最大特征值所对应的特征向量的计算及收敛性问题,从而进一步研究随机张量所特有的诸多性质。在本项目中涉及的所有新设计、改进的算法都将用数值试验来检验其有效性及理论分析的正确性。
中文关键词: 张量;特征值;谱理论;张量方程;迭代算法
英文摘要: It just has been seven years from the concept of eigenvalue of tensor was put forward in 2005. It is widely applied in the fields of blind source separation, magnetic resonance imaging, molecular conformation and so on. Thus, the tensor eigenvalue problem is a new research field of vitality. This project attempts to study the eigenvalue of tensor from three aspects: (1) To study the spectral theory of tensors and establish the eigenvalue inclusion theorem for tensors, which provides the theory basis for numerical algorithm; (2) To explore the connection between the tensor eigenvalue and singular value by combining with the general Rayleigh quotient, and design an iterative algorithm for finding the largest singular value of a nonnegative tensor, and further analyze the convergence and stability of the algorithm; (3) To discuss the application of the tensor eigenvalues in medical image, and study a transition probability tensor which arises in a high-order Markov chain, and discuss the computation and convergence of eigenvector corresponding to the largest eigenvalue of a transition probability tensor, and further study the specific propositions of random tensor. Numerical experiments will be done to demonstrate the efficiency of all new designed or improved algorithms involved in this project and verify the
英文关键词: tensor;eigenvalue;spectral theory;tensor equation;iterative algorithms