Preconditioning for linear systems arising in modeling and optimization

科学计算中的一个中心问题是大型线性方程组的求解。线性方程组作为一个子问题出现在计算科学和工程的几乎每一个分支中，包括固体力学、流体力学、电磁学、天体物理、化学、固体和高能物理，以及各种优化和统计问题。从设计汽车或飞机发动机，到大公司的工作人员调度，或者开发下一代成像设备，这些不同的问题都需要求解线性方程组。事实上，对于许多这样的问题，解线性方程组是计算最密集的步骤。自20世纪70年代以来，渐近性的考虑越来越倾向于求解大型线性方程组的迭代方法。迭代方法效率的关键是选择一个好的预条件。预条件是指一个矩阵，它逼近所考虑的系统，但又很容易求解。在我们以前与研究生和同事的工作中，我们发现了几个新的预条件器族，最近也是最令人兴奋的是基于潜在的图结构(即，关于科学问题背后的变量是如何相互联系的信息)。我建议为优化和科学建模中的许多应用开发新的预条件方法。我还计划将这些预条件条件应用于实际的科学问题。

A central problem in scientific computing is the solution of large systems of linear equations.  Linear equations arise as a subproblem in practically every branch of computational science and engineering, including solid mechanics, fluid mechanics, electromagnetics, astrophysics, chemistry, solid and high energy physics, as well as in all manner of optimization and statistical problems.  Problems as disparate as designing automobile or aircraft engines, scheduling work crews at a large company, or developing next-generation imaging devices all require solving linear equations.  Indeed, for many of these problems, solving linear equations is the most computationally intensive step.Since the 1970s, asymptotic considerations have increasingly favored iterative methods for solving very large linear systems.  The key to efficiency of iterative methods is the selection of a good preconditioner.  A preconditioner is defined as a matrix that approximates the system under consideration and yet is easy to solve.  In our previous work with graduate students and colleagues, we have discovered several new families of preconditioner, the most recent and perhaps most exciting being based on underlying graph structure (i.e., information about how the variables underlying the scientific problem are interconnected).  I propose to develop novel methods for preconditioning for many applications in optimization and scientific modeling.  I also plan to apply these preconditioners to practical scientific problems.