Computing with Sparse and Structured Matrices: Mathematical Derivatives and Beyond

本研究主要是设计有效的计算方法来计算或估计数学导数及相关的科学计算问题。我们的方法是基于组合数学，图论和数值线性代数的协同作用，以解决计算问题的规模的问题，要求创新的策略，算法设计和计算机实现。我们的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。** 在不同的应用领域（从计算金融到气象学再到电网）中，建模和解决科学问题有一个共同的主题：稀疏或结构化或两者兼而有之的矩阵上的数值计算。麻省理工学院的大气环流模型，MITgcm（一个研究地球气候的数值模型），是所谓的“艾级”应用程序的一个例子，即使是一个模拟运行的基础模型需要前所未有的规模的计算资源。在这种数值模型中，一个重要的计算是评估模型对某些未知或知之甚少的模型参数的敏感性。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”MITgcm计算机代码的灵敏度计算。利用信息，如稀疏性和结构的基本问题是至关重要的，在设计有效的算法，这样的应用程序。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求解非线性方程组或优化（最小化或最大化）某些数量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）将有助于加拿大高素质人才库（HQP）。所有由过去的发现赠款支持的受训人员都继续（或正在考虑）进行研究生学习（硕士或博士）。或积极为加拿大工业做出贡献（例如，Syncrude，Telus，University of Lethbridge）.**

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).**