27 research outputs found
Accurate and efficient splitting methods for dissipative particle dynamics
We study numerical methods for dissipative particle dynamics (DPD), which is
a system of stochastic differential equations and a popular stochastic
momentum-conserving thermostat for simulating complex hydrodynamic behavior at
mesoscales. We propose a new splitting method that is able to substantially
improve the accuracy and efficiency of DPD simulations in a wide range of the
friction coefficients, particularly in the extremely large friction limit that
corresponds to a fluid-like Schmidt number, a key issue in DPD. Various
numerical experiments on both equilibrium and transport properties are
performed to demonstrate the superiority of the newly proposed method over
popular alternative schemes in the literature
Time correlation functions of equilibrium and nonequilibrium Langevin dynamics: Derivations and numerics using random numbers
We study the time correlation functions of coupled linear Langevin dynamics
without and with inertia effects, both analytically and numerically. The model
equation represents the physical behavior of a harmonic oscillator in two or
three dimensions in the presence of friction, noise, and an external field with
both rotational and deformational components. This simple model plays pivotal
roles in understanding more complicated processes. The presented analytical
solution serves as a test of numerical integration schemes, its derivation is
presented in a fashion that allows to be repeated directly in a classroom.
While the results in the absence of fields (equilibrium) or confinement (free
particle) are omnipresent in the literature, we write down, apparently for the
first time, the full nonequilibrium results that may correspond, e.g., to a
Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow
field. We demonstrate how the inertia results reduce to their noninertia
counterparts in the nontrivial limit of vanishing mass. While the results are
derived using basic integrations over Dirac delta distributions, we mention its
relationship with alternative approaches involving (i) Fourier transforms, that
seems advantageous only if the measured quantities also reside in Fourier
space, and (ii) a Fokker--Planck equation and the moments of the probability
distribution. The results, verified by numerical experiments, provide
additional means of measuring the performance of numerical methods for such
systems. It should be emphasized that this manuscript provides specific details
regarding the derivations of the time correlation functions as well as the
implementations of various numerical methods, so that it can serve as a
standalone piece as part of education in the framework of stochastic
differential equations and calculus.Comment: 35 pages, 5 figure
Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting
We study the optimal design of numerical integrators for dissipative systems,
for which there exists an underlying thermodynamic structure known as GENERIC
(general equation for the nonequilibrium reversible-irreversible coupling). We
present a frame-work to construct structure-preserving integrators by splitting
the system into reversible and irreversible dynamics. The reversible part,
which is often degenerate and reduces to a Hamiltonian form on its symplectic
leaves, is solved by using a symplectic method (e.g., Verlet) with degenerate
variables being left unchanged, for which an associated modified Hamiltonian
(and subsequently a modified energy) in the form of a series expansion can be
obtained by using backward error analysis. The modified energy is then used to
construct a modified friction matrix associated with the irreversible part in
such a way that a modified degeneracy condition is satisfied. The modified
irreversible dynamics can be further solved by an explicit midpoint method if
not exactly solvable. Our findings are verified by various numerical
experiments, demonstrating the superiority of structure-preserving integrators
over alternative schemes in terms of not only the accuracy control of both
energy conservation and entropy production but also the preservation of the
conformal symplectic structure in the case of linearly damped systems
Extended stochastic dynamics: theory, algorithms, and applications in multiscale modelling and data science
This thesis addresses the sampling problem in a high-dimensional space, i.e., the
computation of averages with respect to a defined probability density that is a
function of many variables. Such sampling problems arise in many application
areas, including molecular dynamics, multiscale models, and Bayesian sampling
techniques used in emerging machine learning applications. Of particular interest
are thermostat techniques, in the setting of a stochastic-dynamical system,
that preserve the canonical Gibbs ensemble defined by an exponentiated energy
function. In this thesis we explore theory, algorithms, and numerous applications
in this setting.
We begin by comparing numerical methods for particle-based models. The
class of methods considered includes dissipative particle dynamics (DPD) as well
as a newly proposed stochastic pairwise Nosé-Hoover-Langevin (PNHL) method.
Splitting methods are developed and studied in terms of their thermodynamic
accuracy, two-point correlation functions, and convergence. When computational
efficiency is measured by the ratio of thermodynamic accuracy to CPU time, we
report significant advantages in simulation for the PNHL method compared to
popular alternative schemes in the low-friction regime, without degradation of
convergence rate.
We propose a pairwise adaptive Langevin (PAdL) thermostat that fully captures
the dynamics of DPD and thus can be directly applied in the setting of
momentum-conserving simulation. These methods are potentially valuable for
nonequilibrium simulation of physical systems. We again report substantial improvements
in both equilibrium and nonequilibrium simulations compared to popular
schemes in the literature. We also discuss the proper treatment of the Lees-Edwards boundary conditions, an essential part of modelling shear flow.
We also study numerical methods for sampling probability measures in high
dimension where the underlying model is only approximately identified with a
gradient system. These methods are important in multiscale modelling and in
the design of new machine learning algorithms for inference and parameterization
for large datasets, challenges which are increasingly important in "big data"
applications. In addition to providing a more comprehensive discussion of
the foundations of these methods, we propose a new numerical method for the
adaptive Langevin/stochastic gradient Nosé-Hoover thermostat that achieves a
dramatic improvement in numerical efficiency over the most popular stochastic
gradient methods reported in the literature. We demonstrate that the newly established
method inherits a superconvergence property (fourth order convergence
to the invariant measure for configurational quantities) recently demonstrated in
the setting of Langevin dynamics.
Furthermore, we propose a covariance-controlled adaptive Langevin (CCAdL)
thermostat that can effectively dissipate parameter-dependent noise while maintaining
a desired target distribution. The proposed method achieves a substantial
speedup over popular alternative schemes for large-scale machine learning applications
Distribution-Free Model-Agnostic Regression Calibration via Nonparametric Methods
In this paper, we consider the uncertainty quantification problem for
regression models. Specifically, we consider an individual calibration
objective for characterizing the quantiles of the prediction model. While such
an objective is well-motivated from downstream tasks such as newsvendor cost,
the existing methods have been largely heuristic and lack of statistical
guarantee in terms of individual calibration. We show via simple examples that
the existing methods focusing on population-level calibration guarantees such
as average calibration or sharpness can lead to harmful and unexpected results.
We propose simple nonparametric calibration methods that are agnostic of the
underlying prediction model and enjoy both computational efficiency and
statistical consistency. Our approach enables a better understanding of the
possibility of individual calibration, and we establish matching upper and
lower bounds for the calibration error of our proposed methods. Technically,
our analysis combines the nonparametric analysis with a covering number
argument for parametric analysis, which advances the existing theoretical
analyses in the literature of nonparametric density estimation and quantile
bandit problems. Importantly, the nonparametric perspective sheds new
theoretical insights into regression calibration in terms of the curse of
dimensionality and reconciles the existing results on the impossibility of
individual calibration. To our knowledge, we make the first effort to reach
both individual calibration and finite-sample guarantee with minimal
assumptions in terms of conformal prediction. Numerical experiments show the
advantage of such a simple approach under various metrics, and also under
covariates shift. We hope our work provides a simple benchmark and a starting
point of theoretical ground for future research on regression calibration.Comment: Accepted at NeurIPS 2023 and update a camera-ready version; Add some
experiments and literature review
Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming
In this paper, we study the predict-then-optimize problem where the output of
a machine learning prediction task is used as the input of some downstream
optimization problem, say, the objective coefficient vector of a linear
program. The problem is also known as predictive analytics or contextual linear
programming. The existing approaches largely suffer from either (i)
optimization intractability (a non-convex objective function)/statistical
inefficiency (a suboptimal generalization bound) or (ii) requiring strong
condition(s) such as no constraint or loss calibration. We develop a new
approach to the problem called \textit{maximum optimality margin} which designs
the machine learning loss function by the optimality condition of the
downstream optimization. The max-margin formulation enjoys both computational
efficiency and good theoretical properties for the learning procedure. More
importantly, our new approach only needs the observations of the optimal
solution in the training data rather than the objective function, which makes
it a new and natural approach to the inverse linear programming problem under
both contextual and context-free settings; we also analyze the proposed method
under both offline and online settings, and demonstrate its performance using
numerical experiments.Comment: to be published in ICML 202
Assessing numerical methods for molecular and particle simulation
We discuss the design of state-of-the-art numerical methods for molecular
dynamics, focusing on the demands of soft matter simulation, where the purposes
include sampling and dynamics calculations both in and out of equilibrium. We
discuss the characteristics of different algorithms, including their essential
conservation properties, the convergence of averages, and the accuracy of
numerical discretizations. Formulations of the equations of motion which are
suited to both equilibrium and nonequilibrium simulation include Langevin
dynamics, dissipative particle dynamics (DPD), and the more recently proposed
"pairwise adaptive Langevin" (PAdL) method, which, like DPD but unlike Langevin
dynamics, conserves momentum and better matches the relaxation rate of
orientational degrees of freedom. PAdL is easy to code and suitable for a
variety of problems in nonequilibrium soft matter modeling, our simulations of
polymer melts indicate that this method can also provide dramatic improvements
in computational efficiency. Moreover we show that PAdL gives excellent control
of the relaxation rate to equilibrium. In the nonequilibrium setting, we
further demonstrate that while PAdL allows the recovery of accurate shear
viscosities at higher shear rates than are possible using the DPD method at
identical timestep, it also outperforms Langevin dynamics in terms of stability
and accuracy at higher shear rates
Adaptive Thermostats for Noisy Gradient Systems
We study numerical methods for sampling probability measures in high
dimension where the underlying model is only approximately identified with a
gradient system. Extended stochastic dynamical methods are discussed which have
application to multiscale models, nonequilibrium molecular dynamics, and
Bayesian sampling techniques arising in emerging machine learning applications.
In addition to providing a more comprehensive discussion of the foundations of
these methods, we propose a new numerical method for the adaptive
Langevin/stochastic gradient Nos\'{e}--Hoover thermostat that achieves a
dramatic improvement in numerical efficiency over the most popular stochastic
gradient methods reported in the literature. We also demonstrate that the newly
established method inherits a superconvergence property (fourth order
convergence to the invariant measure for configurational quantities) recently
demonstrated in the setting of Langevin dynamics. Our findings are verified by
numerical experiments