Lattice Quantum Chromodynamics (QCD) is one of the world's top consumers of supercomputer cycles. As with many other applications, the computational bottlenecks are traced to numerical linear algebra problems. Lattice QCD gives rise to Hermitian matrix problems, where methods such as Conjugate Gradient (CG) or Lanczos converge optimally. Yet, the matrices are of enormous size, and preconditioners that speedup convergence by more than a factor of two or three have been elusive. Equally important, lattice QCD involves two of the still outstanding problems in numerical linear algebra: to find methods that solve optimally a linear system with multiple right hand sides and an eigenvalue problem for a large number of eigenpairs. This research aims at developing new methods, theory, and software that address the above numerical linear algebra problems, both in general and as guided by the physics of the particular QCD problems, thereby speeding considerably lattice QCD computations and enabling further understanding of the structure of matter.
The key contribution of this research is to provide a unified view of linear system and eigenvalue methods that leads to algorithms that solve both problems at once in a nearly optimal way. The investigators study new numerical techniques for efficient computation of quark propagators in lattice QCD and extend them for improving the efficiency of Hybrid Monte Carlo. They explore a new CG method that uses previously developed recurrence-like restarting to obtain eigenvectors and use the method to share information between successive correction equations in the Jacobi-Davidson eigensolvers. Finally the investigators develop well designed, tuned implementations of the above methods to the Chroma package for lattice QCD, and to the state-of-the-art eigenvalue package PRIMME.
