Computing with sparse and structured matrices: mathematical derivatives and beyond

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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