Matrix equations and tensor problems

这是对应用数学中一些课题的研究建议。我的主要目标是推进非线性矩阵方程、张量特征值问题和张量方程的研究。非线性矩阵方程在许多应用中出现。它们可能出现在电信、计算机性能和库存控制中出现的排队问题的随机模型中。它们可能出现在纳米研究、经济模型研究等领域。在每一个实际应用中，都要找到一个满足矩阵方程的特殊矩阵。通过包括我在内的许多研究人员的努力，在各种应用中求解非线性矩阵方程的数值方法的设计和分析方面取得了很大进展。然而，许多有趣的（并且可能非常困难的）问题仍未得到解决。我的目标是解决其中的一些问题。它们可能涉及对已知在实践中工作良好的算法进行进一步的理论分析，或者设计更有效的算法来寻找特定方程的期望解，或者为一些值得进一步关注的矩阵方程建立一个工作理论。张量是矩阵的推广。张量特征值问题在谱超图理论和高阶马尔可夫链中都有应用。张量方程在数据挖掘和数值偏微分方程中都有应用。我对张量问题的研究将集中在非负张量和其他密切相关的张量上。我的目标是为解决张量特征值问题的一些现有数值方法建立一个可靠的理论，并设计新的和更有效的算法来解决张量方程。本课题的成功完成将对矩阵方程、张量特征值问题和张量方程的研究做出重要贡献。特别是张量分析是一个相对较新的和蓬勃发展的研究领域。到目前为止，加拿大对这一研究领域的贡献非常有限。我在这一领域的进一步工作将有助于提高加拿大研究在这一重要研究领域的知名度。

This is a proposal for research on some topics in applied mathematics. My main goal is to advance the study of nonlinear matrix equations, tensor eigenvalue problems, and tensor equations. Nonlinear matrix equations appear in many applications. They may appear in stochastic models for queueing problems arising in telecommunications, computer performance, and inventory control. They may appear in nano research, in the study of economical models, and so on. In each practical application, a special matrix satisfying the matrix equation is to be found. Great progress has been made on the design and analysis of numerical methods for solving nonlinear matrix equations from various applications, through the efforts of many researchers, myself included. However, a number of interesting (and potentially very difficult) problems remain unsolved. My goal is to solve some of these problems. They may involve further theoretical analysis of algorithms known to work well in practice, or the design of more efficient algorithms for finding the desired solution of a particular equation, or the establishment of a working theory for some matrix equations deserving further attention.  Tensors are generalizations of matrices. Tensor eigenvalue problems have applications in spectral hypergraph theory and higher order Markov chains. Tensor equations have applications in data mining and numerical partial differential equations. My research on tensor problems will be focused on nonnegative tensors and other closely related tensors. My goal is to establish a sound theory for some existing numerical methods that are useful in practice for solving tensor eigenvalue problems, and to design new and more efficient algorithms for solving tensor equations. The successful completion of my proposed research will be a significant contribution to the study of matrix equations, tensor eigenvalue problems and tensor equations. In particular, tensor analysis is a relatively new and flourishing research area. Canadian contributions to this research area is very limited so far. My further work in this area will help to improve the visibility of Canadian research in this important research area.