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

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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.