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和创新的方法。随着高性能计算系统技术的进步，如何设计有效的算法工具和框架来利用问题结构和计算系统体系结构来解决日益复杂和超大型的科学问题是研究人员面临的一个主要挑战。例如，“社区气候系统模型”等精确的气候建模代码属于所谓的“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是关于大量输出参数相对于一小部分模型输入的“灵敏度”的有效计算。利用先验的已知信息，如底层问题的稀疏性和结构，对于为此类应用设计有效的算法至关重要。此外，现代高性能计算系统的架构复杂性也不容忽视

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