Computing with Sparse and Structured Matrices: Mathematical Derivatives and Beyond

本研究主要涉及数学导数计算或估计的有效计算方法的设计以及相关的科学计算问题。我们的方法是基于组合学、图论和数值线性代数的协同作用来解决计算问题，其中问题的规模需要算法设计和计算机实现的创新策略。我们研究方法的一个重要组成部分是识别和利用信息，如结构、稀疏性和并发性。******建模和解决不同应用领域中出现的科学问题-从计算金融到气象学到电网，都有一个共同的主题：对稀疏或结构化或两者兼而有之的矩阵进行数值计算。麻省理工学院的大气环流模型，MITgcm（一种研究地球气候的数值模型），是所谓的“百亿次”应用的一个例子，在这种应用中，即使是对底层模型进行一次模拟运行，也需要前所未有规模的计算资源。在这种数值模型中，一个重要的计算是关于模型对一些未知或不太清楚的模型参数的敏感性的评估。近年来在算法微分（AD）技术方面的出色研究使科学家能够“自动化”计算MITgcm计算机代码的灵敏度。利用潜在问题的稀疏性和结构等信息是为此类应用设计有效算法的关键。此外，现代高性能计算系统不断发展的架构复杂性对创新算法的有效软件实现提出了相当大的挑战。******从更广泛的角度来看，这项研究的结果有望在涉及解决非线性方程系统或某些数量的优化（最小化或最大化）的科学和工程计算中找到应用。本研究的培训部分（研究生/本科生培训）将有助于加拿大的高素质人才（HQP）。所有过去发现基金支持的学员都继续（或正在考虑）进行研究生学习（硕士或博士水平）或积极为加拿大工业做出贡献（例如，Syncrude, Telus，莱斯布里奇大学）

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).**