Construction, analysis and implementation of iterative methods for large systems of linear equations

建议在数值线性代数和科学计算领域进行研究。Krylov子空间方法是求解大型稀疏线性代数系统的一种重要且广泛使用的迭代方法。在过去的二十年里，这些方法的研究活动呈爆炸式增长，这是由于在许多科学和工程领域需要解决不断增加的维度的系统而引起的。这一发展主要致力于构建而不是分析方法。我们打算提高对Krylov子空间方法背后原理的理解，特别是对于非对称线性系统，从而为急需的全面分析基础做出贡献。虽然从数学的角度分析Krylov子空间方法是有趣的，但它对于方法在实践中的成功也是必不可少的。我们希望利用新的见解来解决一些现实世界和大规模应用中出现的线性系统，包括约束优化和辐射输运问题。同时，这些应用将指导我们的理论工作。

Proposed is research in the areas of numerical linearalgebra and scientific computing. The focus is on Krylov subspacemethods, an important and widely used class of iterative methodsfor solving large and sparse linear algebraic systems. The lasttwo decades have seen an explosion of research activity on thesemethods which was spurred by the need to solve systems of vastlyincreasing dimensions in many areas of science and engineering.This development was mainly devoted to theconstruction rather than the analysis of methods. We intend to improve the understanding of the principles behind Krylov subspace methods particularly for nonsymmetric linear systems and thereby tocontribute to a much needed thorough analyticfoundation. While the analysis of Krylov subspace methods isinteresting from a mathematical point of view, it is alsoessential for the success of the methods in practice. We wantto utilize new insights for solving linear systems arising inseveral real-world and large-scale applications, includingconstraint optimization and radiation transportproblems. Simultaneously, the applications will guide us in ourtheoretical work.