(AREA: Numerical Computing and Optimization): Numerical Linear Algebra Problems and Quantum Chromodynamics

晶格量子色动力学（QCD）是世界上超级计算机周期的最大消耗者之一。与许多其他应用程序一样，计算瓶颈可以追溯到数值线性代数问题。晶格QCD产生了厄米矩阵问题，其中共轭梯度（CG）或Lanczos等方法是最优收敛的。然而，这些矩阵规模巨大，而将收敛速度提高两到三倍以上的前提条件一直是难以捉摸的。同样重要的是，点阵QCD涉及数值线性代数中两个仍然突出的问题：找到最优解具有多个右侧的线性系统的方法和大量特征对的特征值问题。本研究旨在开发新的方法、理论和软件，以解决上述数值线性代数问题，无论是在一般情况下还是在特定QCD问题的物理指导下，从而大大加快点阵QCD计算，并使人们能够进一步理解物质的结构。本研究的关键贡献在于提供了线性系统和特征值方法的统一视图，从而产生了以近乎最优的方式同时解决这两个问题的算法。研究了晶格QCD中夸克传播子高效计算的新数值技术，并对其进行了扩展，以提高混合蒙特卡罗算法的效率。他们探索了一种新的CG方法，该方法使用先前开发的递归式重新启动来获得特征向量，并使用该方法在Jacobi-Davidson特征解中的连续校正方程之间共享信息。最后，研究人员对上述方法进行了精心设计、优化的实现，以用于晶格QCD的chroroma包，以及最先进的特征值包prime。

Lattice Quantum Chromodynamics (QCD) is one of the world's topconsumers of supercomputer cycles. As with many other applications,the computational bottlenecks are traced to numerical linear algebraproblems. Lattice QCD gives rise to Hermitian matrix problems, wheremethods such as Conjugate Gradient (CG) or Lanczos convergeoptimally. Yet, the matrices are of enormous size, and preconditionersthat speedup convergence by more than a factor of two or three havebeen elusive. Equally important, lattice QCD involves two of the stilloutstanding problems in numerical linear algebra: to find methods thatsolve optimally a linear system with multiple right hand sides and aneigenvalue problem for a large number of eigenpairs. This research aims at developing new methods, theory, andsoftware that address the above numerical linear algebra problems,both in general and as guided by the physics of the particular QCDproblems, thereby speeding considerably lattice QCD computations andenabling further understanding of the structure of matter. The key contribution of this research is to provide a unified view oflinear system and eigenvalue methods that leads to algorithms thatsolve both problems at once in a nearly optimal way. The investigatorsstudy new numerical techniques for efficient computation of quarkpropagators in lattice QCD and extend them for improving theefficiency of Hybrid Monte Carlo. They explore a new CG method that usespreviously developed recurrence-like restarting to obtain eigenvectorsand use the method to share information between successive correctionequations in the Jacobi-Davidson eigensolvers. Finally the investigatorsdevelop well designed, tuned implementations of the above methods to theChroma package for lattice QCD, and to the state-of-the-art eigenvaluepackage PRIMME.