Ubiquitous Doubling Algorithms for Nonlinear Matrix Equations and Applications

首席研究员（PI）的研究旨在解决最优控制理论的实际问题，从高速列车的振动分析到纳米研究中的量子输运。这些应用的基础是需要有效和准确地求解重要非线性矩阵方程的公式，这些方程对整个模拟的整体性能和保真度有重要影响。过去的经验表明，需要求解非线性矩阵方程的实际相关模拟非常具有挑战性，以至于计算出的解可能完全错误。该项目旨在通过发展一个完整的理论，设计创新的算法来更好地反映问题结构，并设计更健壮的实现来改变现状。除了推进这些非线性矩阵方程的研究外，PI还将招收和培养计算数学和跨学科研究方面的研究生。本项目将在数学理论、计算方法和软件的背景下促进对高影响非线性矩阵方程的理解和解决技术。几十年来，计算科学家和工程师一直在努力计算这些非线性矩阵方程的可靠数值解，但收效甚微。例如，马尔可夫调制流体流动理论中的m矩阵代数Riccati方程的解矩阵的项表示事件的概率，而微小的项表示罕见但仍然重要的事件，因此需要精确计算。高速列车振动分析的特征值如果处理不当，变化幅度很大，超出了IEEE双精度所能处理的范围。在纳米量子输运研究中，倍增算法在非摄动非线性矩阵方程上可能完全不收敛。这些困难通常不是由问题的规模造成的，而是由于它们固有的数学性质。随着本课题的顺利完成，一个完整的、连贯的、统一的加倍算法框架，以及相关的理论将得到发展。我们将对加倍算法有更深入的了解，并有可能确定重要的应用，其中加倍算法可以被利用，并且比目前最先进的方法更快。更重要的是，所开发的理论将确保新的算法将产生比目前可能的更可靠、更准确的数值结果。

The principal investigator's (PI's) research is aimed at the solution of practical problems of optimal control theory from areas as diverse as vibration analysis for high speed trains and quantum transport in nano research. Underlying these applications are formulations that require the efficient and accurate solution of important nonlinear matrix equations which critically influence the overall performance and fidelity of the entire simulations. Past experience has shown that practically relevant simulations requiring solutions of nonlinear matrix equations are notoriously challenging to the point that computed solutions may be completely erroneous. This project aims at changing the status quo by developing a complete theory, devising innovative algorithms to better reflect problem structures, and designing more robust implementations. In addition to advancing research in these nonlinear matrix equations, the PI will recruit and train graduate students in computational mathematics and interdisciplinary studies.This project will advance the understanding and solution techniques for high impact nonlinear matrix equations in the context of mathematical theory, computational methods, and software. For decades, computational scientists and engineers have been struggling to compute trustworthy numerical solutions to some of these nonlinear matrix equations with limited success. For example, the entries of the solution matrices to M-matrix algebraic Riccati equations from Markov-modulated fluid flow theory represent probabilities of events, and tiny entries indicate events that are rare but still important, and thus need to be computed accurately. The eigenvalues from vibration analysis of high speed trains vary widely in magnitude beyond the ranges that the IEEE double precision can handle if not dealt with correctly. The doubling algorithm may not converge at all on the unperturbed nonlinear matrix equations for quantum transport in nano research. These difficulties are often not caused by the large sizes of the problems but rather are more due to their inherent mathematical properties. With the successful completion of this project, a complete and coherent unifying framework of doubling algorithms, along with the relevant theory, will be developed. A much deeper understanding of doubling algorithms will be gained, and it is possible that important applications will be identified where doubling algorithms can be utilized and are faster than the current state-of-the-art methods. More significantly, the theory developed will assure that the new algorithms will produce more trustworthy accurate numerical results than is currently possible.