On the Closest Vector Problem with a Distance Guarantee

We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any $\alpha < 1/2$, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most $\alpha$ times the length of the shortest nonzero lattice vector, requires as preprocessing advice only $N \approx \widetilde{O}(n \exp(\alpha^2 n /(1-2\alpha)^2))$ vectors, and runs in time $\widetilde{O}(nN)$. As our second main contribution, we present reductions showing that it suffices to solve CVP, both in its plain and preprocessing versions, when the input target point is within some bounded distance of the lattice. The reductions are based on ideas due to Kannan and a recent sparsification technique due to Dadush and Kun. Combining our reductions with the LLM algorithm gives an approximation factor of $O(n/\sqrt{\log n})$ for search CVPP, improving on the previous best of $O(n^{1.5})$ due to Lagarias, Lenstra, and Schnorr. When combined with our improved algorithm we obtain, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance Decoding and the Closest Vector Problem with Preprocessing". Conference on Computational Complexity (2014

Simulation of Field Theories in Wavelet Representation

The field is expanded in a wavelet series and the wavelet coefficients are varied in a simulation of the 2D $\phi^4$ field theory. The drastically reduced autocorrelations result in a substantial decrease of computing requirements, compared to those in local Metropolis simulations. A large part of the improvement is shown to be the result of an additional freedom in the choice of the allowed range of change at the Metropolis update of wavelet components, namely the range can be optimized independently for all wavelet sizes.Comment: 10 pages, LaTeX with 8 figures, Swansea preprint SWAT/3

Problems in Lattice Gauge Fixing

We review many topics and results about numeric gauge fixing in lattice QCD.Comment: 47 pages, 16 eps figures. Review article sent to IJMP

Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator

In lattice QCD computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a precondtioner to solve the "clover improved" Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used seperately in lattice QCD before. We show in extensive numerical test conducted with a parallel production code implementation that considerable speed-up over conventional Krylov subspace methods, domain decomposition methods and other hierarchical approaches for realistic system sizes can be achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to mixed-precision odd-even preconditioned BiCGStab. Results of numerical experiments changed slightly due to more systematic use of odd-even preconditionin

Bogoliubov Excitations of Disordered Bose-Einstein Condensates

We describe repulsively interacting Bose-Einstein condensates in spatially correlated disorder potentials of arbitrary dimension. The first effect of disorder is to deform the mean-field condensate. Secondly, the quantum excitation spectrum and condensate population are affected. By a saddle-point expansion of the many-body Hamiltonian around the deformed mean-field ground state, we derive the fundamental quadratic Hamiltonian of quantum fluctuations. Importantly, a basis is used such that excitations are orthogonal to the deformed condensate. Via Bogoliubov-Nambu perturbation theory, we compute the effective excitation dispersion, including mean free paths and localization lengths. Corrections to the speed of sound and average density of states are calculated, due to correlated disorder in arbitrary dimensions, extending to the case of weak lattice potentials.Comment: 23 pages, 11 figure