Computing with Sparse and Structured Matrices: Mathematical Derivatives and Beyond

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This research is mainly concerned with the design of efficient computational methods for the computation or estimation of mathematical derivatives and related scientific computing problems. Our approach is based on the synergy of combinatorics, graph theory, and numerical linear algebra to solve computational problems where the scale of the problem calls for innovative strategies for algorithm design and their computer implementation. An important component of our research methodology is the identification and exploitation of information such as structure, sparsity, and concurrency. ******Modelling and solving scientific problems arising in diverse application areas - from computational finance to meteorology to electricity grids, share a common theme: numerical calculations on matrices that are sparse or structured or both. The MIT general circulation model, MITgcm (a numerical model to study Earth's climate), is an example of the so called "exascale" application where, even one simulation run of the underlying model requires computational resources of an unprecedented scale. An essential calculation in such a numerical model is concerned with the evaluation of sensitivity of the model with respect to some model parameters that are unknown or poorly known. Excellent research in algorithmic differentiation (AD) techniques in the recent years enabled scientists to "automate" sensitivity calculation for MITgcm computer code. Exploiting information such as sparsity and structure of the underlying problem is crucial in designing effective algorithms for such applications. Moreover, the evolving architectural complexity of modern high-performance computing systems pose considerable challenge for effective software implementation of innovative algorithms. ******In broader terms, the results from this research  are expected to find applications in scientific and engineering calculations that involve solving system of nonlinear equations or optimization (minimization or maximization) of certain quantities. The training component (for graduate/undergraduate training) of this research will contribute  to the pool of highly qualified personnel (HQP) in Canada. All of the trainees supported by the past discovery grants went on (or currently considering) to undertake graduate studies (at M.Sc. or Ph.D. level) or are actively contributing to the Canadian Industry (e.g., Syncrude, Telus, University of Lethbridge).**