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Cristian Predescu

Publications and source records attributed to Cristian Predescu.

At least 19 recordsLinked to original sources
arXiv2021

Times Square sampling: an adaptive algorithm for free energy estimation

Estimating free energy differences, an important problem in computational drug discovery and in a wide range of other application areas, commonly involves a computationally intensive process of sampling a family of high-dimensional probability distributions and a procedure for computing estimates based on those samples. The variance of the free energy estimate of interest typically depends strongly on how the total computational resources available for sampling are divided among the distributions, but determining an efficient allocation is difficult without sampling the distributions. Here we introduce the Times Square sampling algorithm, a novel on-the-fly estimation method that dynamically allocates resources in such a way as to significantly accelerate the estimation of free energies and other observables, while providing rigorous convergence guarantees for the estimators. We also show that it is possible, surprisingly, for on-the-fly free energy estimation to achieve lower asymptotic variance than the maximum-likelihood estimator MBAR, raising the prospect that on-the-fly estimation could reduce variance in a variety of other statistical applications.

math.ST↗
arXiv2019

The $\textit{u}$-series: A separable decomposition for electrostatics computation with improved accuracy

The evaluation of electrostatic energy for a set of point charges in a periodic lattice is a computationally expensive part of molecular dynamics simulations (and other applications) because of the long-range nature of the Coulomb interaction. A standard approach is to decompose the Coulomb potential into a near part, typically evaluated by direct summation up to a cutoff radius, and a far part, typically evaluated in Fourier space. In practice, all decomposition approaches involve approximations---such as cutting off the near-part direct sum---but it may be possible to find new decompositions with improved tradeoffs between accuracy and performance. Here we present the $\textit{u-series}$, a new decomposition of the Coulomb potential that is more accurate than the standard (Ewald) decomposition for a given amount of computational effort, and achieves the same accuracy as the Ewald decomposition with approximately half the computational effort. These improvements, which we demonstrate numerically using a lipid membrane system, arise because the $\textit{u}$-series is smooth on the entire real axis and exact up to the cutoff radius. Additional performance improvements over the Ewald decomposition may be possible in certain situations because the far part of the $\textit{u}$-series is a sum of Gaussians, and can thus be evaluated using algorithms that require a separable convolution kernel; we describe one such algorithm that reduces communication latency at the expense of communication bandwidth and computation, a tradeoff that may be advantageous on modern massively parallel supercomputers.

physics.comp-ph↗
arXiv2007

Entropic effects in large-scale Monte Carlo simulations

The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Renyi divergence of order 1/2. We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for non-vanishing acceptance probabilities in the limit of large dimensions. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart Monte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.

physics.comp-ph↗
arXiv2007

Sampling diffusive transition paths

We address the problem of sampling double-ended diffusive paths. The ensemble of paths is expressed using a symmetric version of the Onsager-Machlup formula, which only requires evaluation of the force field and which, upon direct time discretization, gives rise to a symmetric integrator that is accurate to second order. Efficiently sampling this ensemble requires avoiding the well-known stiffness problem associated with sampling infinitesimal Brownian increments of the path, as well as a different type of stiffness associated with sampling the coarse features of long paths. The fine-feature sampling stiffness is eliminated with the use of the fast sampling algorithm (FSA), and the coarse-feature sampling stiffness is avoided by introducing the sliding and sampling (S&S) algorithm. A key feature of the S&S algorithm is that it enables massively parallel computers to sample diffusive trajectories that are long in time. We use the algorithm to sample the transition path ensemble for the structural interconversion of the 38-atom Lennard-Jones cluster at low temperature.

cond-mat.stat-mech↗
arXiv2006

Generalized moments of spectral functions from short-time correlation functions

We present an integral transformation capable of extracting moments of arbitrary Paley-Wiener entire functions against a given spectral distribution based solely on short-time values of the correlation function in a small open disk about the origin. The integral is proven to converge absolutely to the expected result for those correlation functions that can be extended analytically to the entire complex plane, with the possible exception of two branch cuts on the imaginary axis. It is only the existence of an analytic continuation that is required and not the actual values away from the small disk about the origin. If the analytic continuation exists only for a strip |Im(z)| < τ_0, then the integral transformation remains valid for all Paley-Wiener functions obtained by Fourier-Laplace transforming a compactly supported distribution, with the support included in the interval (-2τ_0, 2τ_0). Finally, if the support of the distribution is contained in the interval $(-τ_0, τ_0)$, then the generalized moment can be evaluated from the short-time values of the correlation function exponentially fast

math-ph↗
arXiv2005

Design of high-order short-time approximations as a problem of matching the covariance of a Brownian motion

One of the outstanding problems in the numerical discretization of the Feynman-Kac formula calls for the design of arbitrary-order short-time approximations that are constructed in a stable way, yet only require knowledge of the potential function. In essence, the problem asks for the development of a functional analogue to the Gauss quadrature technique for one-dimensional functions. In PRE 69, 056701 (2004), it has been argued that the problem of designing an approximation of order νis equivalent to the problem of constructing discrete-time Gaussian processes that are supported on finite-dimensional probability spaces and match certain generalized moments of the Brownian motion. Since Gaussian processes are uniquely determined by their covariance matrix, it is tempting to reformulate the moment-matching problem in terms of the covariance matrix alone. Here, we show how this can be accomplished.

math-ph↗
arXiv2005

Thermodynamics and equilibrium structure of Ne_38 cluster: Quantum Mechanics versus Classical

The equilibrium properties of classical LJ_38 versus quantum Ne_38 Lennard-Jones clusters are investigated. The quantum simulations use both the Path-Integral Monte-Carlo (PIMC) and the recently developed Variational-Gaussian-Wavepacket Monte-Carlo (VGW-MC) methods. The PIMC and the classical MC simulations are implemented in the parallel tempering framework. The VGW method is used to locate and characterize the low energy states of Ne_38, which are then further refined by PIMC calculations. Unlike the classical case, the ground state of Ne_38 is a liquid-like structure. Among the several liquid-like states with energies below the two symmetric states (O_h and C_5v), the lowest two exhibit strong delocalization over basins associated with at least two classical local minima. Because the symmetric structures do not play an essential role in the thermodynamics of Ne_38, the quantum heat capacity is a featureless curve indicative of the absence of any structural transformations. Good agreement between the two methods, VGW and PIMC, is obtained.

physics.chem-ph↗
arXiv2005

Moments of spectral functions: Monte Carlo evaluation and verification

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely-differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval $(-β\hbar / 2, β\hbar/2)$. The algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is non-linear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.

cond-mat.stat-mech↗
arXiv2004

On the efficient Monte Carlo implementation of path integrals

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.

cond-mat.stat-mech↗
arXiv2004

The fast sampling algorithm for Lie-Trotter products

A fast algorithm for path sampling in path integral Monte Carlo simulations is proposed. The algorithm utilizes the Levy-Ciesielski implementation of Lie-Trotter products to achieve a mathematically proven computational cost of n*log_2(n) with the number of time slices n, despite the fact that each path variable is updated separately, for reasons of optimality. In this respect, we demonstrate that updating a group of random variables simultaneously results in loss of efficiency.

cond-mat.stat-mech↗
arXiv2004

Reconstruction of thermally-symmetrized quantum autocorrelation functions from imaginary-time data

In this paper, I propose a technique for recovering quantum dynamical information from imaginary-time data via the resolution of a one-dimensional Hamburger moment problem. It is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin. A general class of reconstruction algorithms is then identified, according to Theorem 3. The technique is advocated as especially effective for a certain class of quantum problems in continuum space, for which only a few moments are necessary. For such problems, it is argued that the derivatives at origin can be evaluated by Monte Carlo simulations via estimators of finite variances in the limit of an infinite number of path variables. Finally, a maximum entropy inversion algorithm for the Hamburger moment problem is utilized to compute the quantum rate of reaction for a one-dimensional symmetric Eckart barrier.

physics.chem-ph↗
arXiv2004

Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation

We perform a thorough analysis of the relationship between discrete and series representation path integral methods, which are the main numerical techniques used in connection with the Feynman-Kac formula. First, a new interpretation of the so-called standard discrete path integral methods is derived by direct discretization of the Feynman-Kac formula. Second, we consider a particular random series technique based upon the Levy-Ciesielski representation of the Brownian bridge and analyze its main implementations, namely the primitive, the partial averaging, and the reweighted versions. It is shown that the n=2^k-1 subsequence of each of these methods can also be interpreted as a discrete path integral method with appropriate short-time approximations. We therefore establish a direct connection between the discrete and the random series approaches. In the end, we give sharp estimates on the rates of convergence of the partial averaging and the reweighted Levy-Ciesielski random series approach for sufficiently smooth potentials. The asymptotic rates of convergence are found to be O(1/n^2), in agreement with the rates of convergence of the best standard discrete path integral techniques.

cond-mat.stat-mech↗
arXiv2004

Phase changes in selected Lennard-Jones X_{13-n}Y_n clusters

Detailed studies of the thermodynamic properties of selected binary Lennard-Jones clusters of the type X_{13-n}Y_n (where n=1,2,3) are presented. The total energy, heat capacity and first derivative of the heat capacity as a function of temperature are calculated by using the classical and path integral Monte Carlo methods combined with the parallel tempering technique. A modification in the phase change phenomena from the presence of impurity atoms and quantum effects is investigated.

cond-mat.stat-mech↗
arXiv2004

Reconstruction of silicon surfaces: a stochastic optimization problem

Over the last two decades, scanning tunnelling microscopy (STM) has become one of the most important ways to investigate the structure of crystal surfaces. STM has helped achieve remarkable successes in surface science such as finding the atomic structure of Si(111) and Si(001). For high-index Si surfaces the information about the local density of states obtained by scanning does not translate directly into knowledge about the positions of atoms at the surface. A commonly accepted strategy for identifying the atomic structure is to propose several possible models and analyze their corresponding {\em simulated} STM images for a match with the experimental ones. However, the number of good candidates for the lowest-energy structure is very large for high-index surfaces, and heuristic approaches are not likely to cover all the relevant structural models. In this article, we take the view that finding the atomic structure of a surface is a problem of stochastic optimization, and we address it as such. We design a general technique for predicting the reconstruction of silicon surfaces with arbitrary orientation, which is based on parallel-tempering Monte Carlo simulations combined with an exponential cooling. The advantages of the method are illustrated using the Si(105) surface as example, with two main results: (a) the correct single-step rebonded structure [e.g., Fujikawa {\em et al.}, Phys. Rev. Lett. 88, 176101 (2002)] is obtained even when starting from the paired-dimer model [Mo {\em et al.}, Phys. Rev. Lett. 65, 1020 (1990)] that was assumed to be correct for many years, and (b) we have found several double-step reconstructions that have lower surface energies than any previously proposed double-step models.

cond-mat.mtrl-sci↗
arXiv2003

Upon the existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods

In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is ``experimental'', I only provide a ``physicist's'' proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in the Appendix, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present article is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.

math-ph↗