Parallel optimized sampling for stochastic equations

Stochastic equations play an important role in computational science, due to their ability to treat a wide variety of complex statistical problems. However, current algorithms are strongly limited by their sampling variance, which scales proportionate to 1/N_S for N_S samples. In this paper, we obtain a new class of variance reduction methods for treating stochastic equations, called parallel optimized sampling. The objective of parallel optimized sampling is to reduce the sampling variance in the observables of an ensemble of stochastic trajectories. This is achieved through calculating a finite set of observables - typically statistical moments - in parallel, and minimizing the errors compared to known values. The algorithm is both numerically efficient and unbiased. Importantly, it does not increase the errors in higher order moments, and generally reduces such errors as well. The same procedure is applied both to initial ensembles and to changes in a finite time-step. Results of these methods show that errors in initially optimized moments can be reduced to the machine precision level, typically around 10^(-16) in current hardware. For nonlinear stochastic equations, sampled moment errors during time-evolution are larger than this, due to error propagation effects. Even so, we provide evidence for error reductions of up to two orders of magnitude in a nonlinear equation example, for low order moments, which is a large practical benefit. The sampling variance typically scales as 1/N_S, but with the advantage of a very much smaller prefactor than for standard, non-optimized methods.

Excitation spectrum of bosons in a finite one-dimensional circular waveguide via the Bethe ansatz

The exactly solvable Lieb-Liniger model of interacting bosons in one-dimension has attracted renewed interest as current experiments with ultra-cold atoms begin to probe this regime. Here we numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss the novel features of the solutions, and how they deviate from the well-known string solutions [H. B. Thacker, Rev. Mod. Phys. textbf{53}, 253 (1981)] at finite densities. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Finally we compare our results to those of exact diagonalization of the many-body Hamiltonian in a truncated basis. We also present excited state solutions and the excitation spectrum for the repulsive 1D Bose gas on a ring.