Title: Lattice QCD Simulations towards Strong and Weak Coupling Limits
Description: Lattice gauge theory is a special regularization of continuum gauge theories and the numerical simulation of lattice quantum chromodynamics (QCD) remains as the only first principle method to study non-perturbative QCD at low energy. The lattice spacing a, which serves as the ultraviolet cut off, plays a significant role in determining error on any lattice simulation results. Physical results come from extrapolating a series of simulations with different values for a to a=0. Reducing the size of these errors for non-zero a improves the extrapolation and minimizes the error. In the strong coupling limit the coarse lattice spacing pushes the analysis of the finite lattice spacing error to its limit. Section 4 measures two renormalized physical observables, the neutral kaon mixing parameter BK and the Delta I=3/2 K pi pi decay amplitude A2 on a lattice with coarse lattice spacing of a ~ 1GeV and explores the a^2 scaling properties at this scale. In the weak coupling limit the lattice simulations suffer from critical slowing down where for the Monte Carlo Markov evolution the cost of generating decorrelated samples increases significantly as the lattice spacing decreases, which makes reliable error analysis on the results expensive.

Among the observables the topological charge of the configurations appears to have the longest integrated autocorrelation time. Based on a previous work where a diffusion model is proposed to describe the evolution of the topological charge, section 2 extends this model to lattices with dynamical fermions using a new numerical method that captures the behavior for different Fourier modes. Section 3 describes our effort to find a practical renormalization group transformation to transform lattice QCD between two different scales, whose knowledge could ultimately leads to a multi-scale evolution algorithm that solves the problem of critical slowing down. For a particular choice of action, we have found that doubling the lattice spacing of a fine lattice yields observables that agree at the few precent level with direct simulations on the coarser lattice. Section 5 aims at speeding up the lattice simulations in the weak coupling limit from the numerical method and hardware perspective. It proposes a preconditioner for solving the Dirac equation targeting the ensemble generation phase and details its implementation on currently the fastest supercomputer in the world.