Computing with sparse and structured matrices: mathematical derivatives and beyond

我目前研究的中心主题之一是组合学，图论和数值线性代数的协同作用，以探索制定和解决重要科学问题的新的创新方法。随着高性能计算系统的技术进步，研究人员面临的一个主要挑战是设计有效的算法工具和框架，利用问题结构和计算系统架构来解决日益复杂和非常大的科学问题。精确的气候建模代码，如“社区气候系统模型”，例如，属于所谓的“千万亿次模拟”应用程序。这种模拟模型的一个必要的计算组成部分涉及大量输出参数相对于模型输入的一个小子集的“灵敏度”的有效计算。利用先验已知的信息，如稀疏性和结构的基本问题是设计有效的算法，这样的应用程序是至关重要的。另外，现代高性能计算系统的架构复杂性对计算系统的性能造成了相当大的影响。

One of the central themes of my current research is the synergy of combinatorics, graph theory, and numerical linear algebra to explore new and innovative ways of formulating and solving important scientific problems. With the technological advances in high performance computing systems a major challenge confronted by researchers is concerned with the design of effective algorithmic tools and frameworks that exploit the problem structure and the computing system architecture to solve increasingly complex and extremely large scientific problems. Accurate climate modeling code such as ``Community Climate System Model'', for example, belongs to the so called ``petascale simulation'' applications. One of the necessary computational components of such a simulation model is concerned with the efficient calculation of ``sensitivities'' of a large number of output parameters with respect to a small subset of model inputs. Exploiting a priori known information such as sparsity and structure of the underlying problem is crucial for designing effective algorithms for such applications. Additionally, the architectural complexity of modern high performance computing systems pose considerable