Preconditioning for linear systems arising in modeling and optimization

科学计算中的一个中心问题是求解大型线性方程组。线性方程作为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、天体物理学、化学、固体和高能物理学，以及各种优化和统计问题。设计汽车或飞机发动机、在大公司安排工作人员或开发下一代成像设备等各种各样的问题都需要求解线性方程。事实上，对于许多这样的问题，求解线性方程是计算量最大的一步。自20世纪70年代以来，渐近考虑越来越倾向于求解超大线性系统的迭代方法。迭代方法效率的关键是选择好的预条件。预条件被定义为近似于所考虑的系统且易于求解的矩阵。在我们之前与研究生和同事的工作中，我们发现了几个新的预条件族，最近的，也许是最令人兴奋的是基于底层图结构（即，关于科学问题的变量如何相互关联的信息）。我建议为优化和科学建模中的许多应用开发新的预处理方法。我还打算将这些预处理应用到实际的科学问题中。

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.