Reweighting for nonequilibrium Markov processes using sequential importance sampling methods

Capturing exponential variance using polynomial resources: applying tensor networks to nonequilibrium stochastic processes

Estimating the expected value of an observable appearing in a nonequilibrium stochastic process usually involves sampling. If the observable's variance is high, many samples are required. In contrast, we show that performing the same task without sampling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high-variance observable e^{-βW}, motivated by Jarzynski's equality, with W the work done quenching from equilibrium at inverse temperature β, is exactly and efficiently captured by tensor networks.

Multihistogram reweighting for nonequilibrium Markov processes using sequential importance sampling methods

We present a multihistogram reweighting technique for nonequilibrium Markov chains with discrete energies. The method generalizes the single-histogram method of Yin et al. [Phys. Rev. E 72, 036122 (2005)], making it possible to calculate the time evolution of observables at a posteriori chosen couplings based on a set of simulations performed at other couplings. In the same way as multihistogram reweighting in an equilibrium setting improves the practical reweighting range as well as use of available data compared to single-histogram reweighting, the method generalizes the multihistogram advantages to nonequilibrium simulations. We demonstrate the procedure for the Ising model with Metropolis dynamics, but stress that the method is generally applicable to a range of models and Monte Carlo update schemes.

Dynamic effect of overhangs and islands at the depinning transition in two-dimensional magnets

With the Monte Carlo methods, we systematically investigate the short-time dynamics of domain-wall motion in the two-dimensional random-field Ising model with a driving field (DRFIM). We accurately determine the depinning transition field and critical exponents. Through two different definitions of the domain interface, we examine the dynamics of overhangs and islands. At the depinning transition, the dynamic effect of overhangs and islands reaches maximum. We argue that this should be an important mechanism leading the DRFIM model to a different universality class from the Edwards-Wilkinson equation with quenched disorder.

Critical domain-wall dynamics of model B

With Monte Carlo methods, we simulate the critical domain-wall dynamics of model B, taking the two-dimensional Ising model as an example. In the macroscopic short-time regime, a dynamic scaling form is revealed. Due to the existence of the quasirandom walkers, the magnetization shows intrinsic dependence on the lattice size L . An exponent which governs the L dependence of the magnetization is measured to be sigma=0.243(8) .

Logarithmic correction to scaling in domain-wall dynamics at Kosterlitz-Thouless phase transitions

With Monte Carlo simulations, we investigate the relaxation dynamics of domain walls at the Kosterlitz-Thouless phase transition, taking the two-dimensional XY model as an example. The dynamic scaling behavior is carefully analyzed, and a domain-wall roughening process is observed. Two-time correlation functions are calculated and aging phenomena are investigated. Inside the domain interface, a strong logarithmic correction to scaling is detected.

A multiresolution stochastic level set method for Mumford-Shah image segmentation

The Mumford-Shah model is one of the most successful image segmentation models. However, existing algorithms for the model are often very sensitive to the choice of the initial guess. To make use of the model effectively, it is essential to develop an algorithm which can compute a global or near global optimal solution efficiently. While gradient descent based methods are well-known to find a local minimum only, even many stochastic methods do not provide a practical solution to this problem either. In this paper, we consider the computation of a global minimum of the multiphase piecewise constant Mumford-Shah model. We propose a hybrid approach which combines gradient based and stochastic optimization methods to resolve the problem of sensitivity to the initial guess. At the heart of our algorithm is a well-designed basin hopping scheme which uses global updates to escape from local traps in a way that is much more effective than standard stochastic methods. In our experiments, a very high-quality solution is obtained within a few stochastic hops whereas the solutions obtained with simulated annealing are incomparable even after thousands of steps. We also propose a multiresolution approach to reduce the computational cost and enhance the search for a global minimum. Furthermore, we derived a simple but useful theoretical result relating solutions at different spatial resolutions.

Short-time critical dynamics at perfect and imperfect surfaces

With Monte Carlo simulations, we study the dynamic relaxation at perfect and imperfect surfaces of the three-dimensional Ising model with an ordered initial state. The time evolution of the surface magnetization, the line magnetization of the defect line, and the corresponding susceptibilities and second cumulants is carefully examined. Universal dynamic scaling forms including a dynamic crossover scaling form are identified at the ordinary, special, and surface phase transitions. The critical exponents beta1 of the surface magnetization and beta2 of the line magnetization are extracted. The impact of the defect line on the universality classes is investigated.

A black-box re-weighting analysis can correct flawed simulation data

There is a great need for improved statistical sampling in a range of physical, chemical, and biological systems. Even simulations based on correct algorithms suffer from statistical error, which can be substantial or even dominant when slow processes are involved. Further, in key biomolecular applications, such as the determination of protein structures from NMR data, non-Boltzmann-distributed ensembles are generated. We therefore have developed the "black-box" strategy for re-weighting a set of configurations generated by arbitrary means to produce an ensemble distributed according to any target distribution. In contrast to previous algorithmic efforts, the black-box approach exploits the configuration-space density observed in a simulation, rather than assuming a desired distribution has been generated. Successful implementations of the strategy, which reduce both statistical error and bias, are developed for a one-dimensional system, and a 50-atom peptide, for which the correct 250-to-1 population ratio is recovered from a heavily biased ensemble.

Nonequilibrium critical dynamics with domain wall and surface

With Monte Carlo simulations, we investigate the relaxation dynamics with a domain wall for magnetic systems at the critical temperature. The dynamic scaling behavior is carefully analyzed, and a dynamic roughening process is observed. For comparison, similar analysis is applied to the relaxation dynamics with a free or disordered surface.

Short-time critical dynamics and aging phenomena in the two-dimensional XY model

With Monte Carlo methods, we simulate dynamic relaxation processes of the two-dimensional XY model at the Berezinskii-Kosterlitz-Thouless phase transition temperature and below, starting from both ordered and disordered initial states. The two-time correlation function A(t',t) is measured, and aging phenomena are investigated. The power-law correction in the spatial correlation length xi(t) for relaxation with an ordered initial state and the logarithmic correction for relaxation with a disordered initial state are carefully analyzed. The scaling functions of A(t',t) are then extracted.

Dynamic Monte Carlo simulations of the three-dimensional random-bond Potts model

The effect of random bonds on the phase transitions of the three-dimensional three-state Potts model is investigated with extensive dynamic Monte Carlo simulations. In the weakly disordered regime, the phase diagram is obtained with a recently suggested nonequilibrium reweighting method. The tricritical point separating the first- and second-order transitions is determined, and the critical exponents of the continuous phase transition induced by quenched randomness are estimated.