arXiv subjects

Zhenli Xu

Publications and source records attributed to Zhenli Xu.

At least 55 records · Page 3Linked to original sources

arXiv2022

A Maxwell-Ampère Nernst-Planck Framework for Modeling Charge Dynamics

Understanding the properties of charge dynamics is crucial to many practical applications, such as electrochemical energy devices and transmembrane ion channels. This work proposes a Maxwell-Ampère Nernst-Planck (MANP) framework for the description of charge dynamics. The MANP model with a curl-free condition on the electric displacement is shown to be energy dissipative with respect to a convex free-energy functional, and demonstrated to be equivalent to the Poisson-Nernst-Planck model. By the energy dissipation law, the steady state of the MANP model reproduces the charge conserving Poisson--Boltzmann (PB) theory, providing an alternative energy stable approach to study the PB theory. In order to achieve the curl-free condition, a companion local curl-free relaxation algorithm, which is shown to naturally preserve the discrete Gauss's law and converge robustly with linear computational complexity, is developed for the MANP model. One of the main advantages of our development is that it can efficiently deal with space-dependent permittivity instead of solving the variable-coefficient Poisson's equation. Many-body effects such as ionic steric effects and Coulomb correlations can be incorporated within the MANP framework to derive modified MANP models for problems in which the mean-field approximation fails. Numerical results on the charge dynamics with such beyond mean-field effects in inhomogeneous dielectric environments are presented to demonstrate the performance of the MANP models in the description of charge dynamics, illustrating that the proposed MANP model provides a general framework for modeling charge dynamics.

physics.comp-ph↗

arXiv2022

An Energy-Conserving Fourier Particle-in-Cell Method with Asymptotic-Preserving Preconditioner for Vlasov-Ampère System with Exact Curl-Free Constraint

We present an efficient and accurate energy-conserving implicit particle-in-cell~(PIC) algorithm for the electrostatic Vlasov system, with particular emphasis on its high robustness for simulating complex plasma systems with multiple physical scales. This method consists of several indispensable elements: (\romannumeral1) the reformulation of the original Vlasov-Poisson system into an equivalent Vlasov-Ampère system with divergence-/curl-free constraints; (\romannumeral2) a novel structure-preserving Fourier spatial discretization, which exactly preserves these constraints at the discrete level; (\romannumeral3) a preconditioned Anderson-acceleration algorithm for the solution of the highly nonlinear system; and (\romannumeral4) a linearized and uniform approximation of the implicit Crank-Nicolson scheme for various Debye lengths, based on the generalized Ohm's law, which serves as an asymptotic-preserving preconditioner for the proposed method. Numerical experiments are conducted, and comparisons are made among the proposed energy-conserving scheme, the classical leapfrog scheme, and a Strang operator-splitting scheme to demonstrate the superiority of the proposed method, especially for plasma systems crossing physical scales.

physics.comp-ph↗

arXiv2022

An Asymptotic-Preserving and Energy-Conserving Particle-In-Cell Method for Vlasov-Maxwell Equations

In this paper, we develop an asymptotic-preserving and energy-conserving (APEC) Particle-In-Cell (PIC) algorithm for the Vlasov-Maxwell system. This algorithm not only guarantees that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model, but also preserves Gauss's law and energy conservation at the same time, thus it is promising to provide stable simulations of complex plasma systems even in the quasi-neutral regime. The key ingredients for achieving these properties include the generalized Ohm's law for electric field such that the asymptotic-preserving discretization can be achieved, and a proper decomposition of the effects of the electromagnetic fields such that a Lagrange multiplier method can be appropriately employed for correcting the kinetic energy. We investigate the performance of the APEC method with three benchmark tests in one dimension, including the linear Landau damping, the bump-on-tail problem and the two-stream instability. Detailed comparisons are conducted by including the results from the classical explicit leapfrog and the previously developed asymptotic-preserving PIC schemes. Our numerical experiments show that the proposed APEC scheme can give accurate and stable simulations both kinetic and quasi-neutral regimes, demonstrating the attractive properties of the method crossing scales.

physics.plasm-ph↗

arXiv2022

Improved random batch Ewald method in molecular dynamics simulations

The random batch Ewald (RBE) is an efficient and accurate method for molecular dynamics (MD) simulations of physical systems at the nano-/micro- scale. The method shows great potential to solve the computational bottleneck of long-range interactions, motivating a necessity to accelerating short-range components of the non-bonded interactions for a further speedup of MD simulations. In this work, we present an improved RBE method for the non-bonding interactions by introducing the random batch idea to constructing neighbor lists for the treatment of both the short-range part of the Ewald splitting and the Lennard-Jones potential. The efficiency of the novel neighbor list algorithm owes to the stochastic minibatch strategy which can significantly reduce the total number of neighbors. We obtan the error estimate and convergence by theoretical analysis and implement the improved RBE method in the LAMMPS package. Benchmark simulations are performed to demonstrate the accuracy and stability of the algorithm. Numerical tests on computer performance by conducting large-scaled MD simulations for systems including up to 0.1 billion water molecules, run on massive cluster with up to 50 thousand CPU cores, demonstrating the attractive features such as the high parallel scalability and memory-saving of the method in comparison to the existing methods.

physics.comp-ph↗

arXiv2022

Random batch sum-of-Gaussians method for molecular dynamics simulations of particle systems

We develop an accurate, highly efficient and scalable random batch sum-of-Gaussians (RBSOG) method for molecular dynamics simulations of systems with long-range interactions. The idea of the RBSOG method is based on a sum-of-Gaussians decomposition of the Coulomb kernel, and then a random batch importance sampling on the Fourier space is employed for approximating the summation of the Fourier expansion of the Gaussians with large bandwidths (the long-range components). The importance sampling significantly reduces the computational cost, resulting in a scalable algorithm by avoiding the use of communication-intensive fast Fourier transform. Theoretical analysis is present to demonstrate the unbiasedness of the approximate force, the controllability of variance and the weak convergence of the algorithm. The resulting method has $\mathcal{O}(N)$ complexity with low communication latency. Accurate simulation results on both dynamical and equilibrium properties of benchmark problems are reported to illustrate the attractive performance of the method. Simulations on parallel computing are also performed to show the high parallel efficiency. The RBSOG method can be straightforwardly extended to more general interactions with long ranged kernels, and thus is promising to construct fast algorithms of a series of molecular dynamics methods for various interacting kernels.

physics.comp-ph↗

arXiv2021

Linear-Scaling Selected Inversion based on Hierarchical Interpolative Factorization for Self Green's Function for Modified Poisson-Boltzmann Equation in Two Dimensions

This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete elliptic differential operator of the Debye-Hückel equation. Therefore, we develop a fast algorithm by a coupling of the selected inversion and hierarchical interpolative factorization. By the interpolative factorization, our new selected inverse algorithm achieves linear scaling to compute the diagonal of the inverse of this discrete operator. The accuracy and efficiency of the proposed algorithm will be demonstrated by extensive numerical results for solving MPB equations.

physics.comp-ph↗

arXiv2021

A random batch Ewald method for charged particles in the isothermal-isobaric ensemble

We develop an accurate, highly efficient and scalable random batch Ewald (RBE) method to conduct simulations in the isothermal-isobaric ensemble (the NPT ensemble) for charged particles in a periodic box. After discretizing the Langevin equations of motion derived using suitable Lagrangians, the RBE method builds the mini-batch strategy into the Fourier space in the Ewald summation for the pressure and forces so the computational cost is reduced from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ per time step. We implement the method in the LAMMPS package and report accurate simulation results for both dynamical quantities and statistics for equilibrium for typical systems including all-atom bulk water and a semi-isotropic membrane system. Numerical simulations on massive supercomputing cluster are also performed to show promising CPU efficiency of RBE.

physics.comp-ph↗

arXiv2021

A Kernel-Independent Sum-of-Exponentials Method

We propose an accurate algorithm for a novel sum-of-exponentials (SOE) approximation of kernel functions, and develop a fast algorithm for convolution quadrature based on the SOE, which allows an order $N$ calculation for $N$ time steps of approximating a continuous temporal convolution integral. The SOE method is constructed by a combination of the de la Vallée-Poussin sums for a semi-analytical exponential expansion of a general kernel, and a model reduction technique for the minimization of the number of exponentials under given error tolerance. We employ the SOE expansion for the finite part of the splitting convolution kernel such that the convolution integral can be solved as a system of ordinary differential equations due to the exponential kernels. The significant features of our algorithm are that the SOE method is efficient and accurate, and works for general kernels with controllable upperbound of positive exponents. We provide numerical analysis for both the new SOE method and the SOE-based convolution quadrature. Numerical results on different kernels, the convolution integral and integral equations demonstrate attractive performance of both accuracy and efficiency of the proposed method.

math.NA↗

arXiv2021

Superscalability of the random batch Ewald method

Coulomb interaction, following an inverse-square force-law, quantifies the amount of force between two stationary and electrically charged particles. The long-range nature of Coulomb interactions poses a major challenge to molecular dynamics simulations which are major tools for problems at the nano-/micro- scale. Various algorithms are developed to calculate the pairwise Coulomb interactions to a linear scaling but the poor scalability limits the size of simulated systems. Here, we conduct an efficient molecular dynamics algorithm with the random batch Ewald method on all-atom systems where the complete Fourier components in the Coulomb interaction are replaced by randomly selected mini-batches. By simulating the $N$-body systems up to 100 million particles using $10$ thousand CPU cores, we show that this algorithm furnishes $O(N)$ complexity, almost perfect scalability and an order of magnitude faster computational speed when compared to the existing state-of-the-art algorithms. Further examinations of our algorithm on distinct systems, including pure water, micro-phase-separated electrolyte and protein solution demonstrate that the spatiotemporal information on all time and length scales investigated and thermodynamic quantities derived from our algorithm are in perfect agreement with those obtained from the existing algorithms. Therefore, our algorithm provides a breakthrough solution on scalability of computing the Coulomb interaction. It is particularly useful and cost-effective to simulate ultra-large systems, which was either impossible or very costing to conduct using existing algorithms, thus would benefit the broad community of sciences.

physics.comp-ph↗

arXiv2021

Random-batch list algorithm for short-range molecular dynamics simulations

We propose a fast method for the calculation of short-range interactions in molecular dynamics simulations. The so-called random-batch list method is a stochastic version of the classical neighbor-list method to avoid the construction of a full Verlet list, which introduces two-level neighbor lists for each particle such that the neighboring particles are located in core and shell regions, respectively. Direct interactions are performed in the core region. For the shell zone, we employ a random batch of interacting particles to reduce the number of interaction pairs. The error estimate of the algorithm is provided. We investigate the Lennard-Jones fluid by molecular dynamics simulations, and show that this novel method can significantly accelerate the simulations with a factor of several fold without loss of the accuracy. This method is simple to implement, can be well combined with any linked cell methods to further speed up and scale up the simulation systems, and can be straightforwardly extended to other interactions such as Ewald short-range part, and thus it is promising for large-scale molecular dynamics simulations.

physics.comp-ph↗

arXiv2021

Modified Poisson-Nernst-Planck model with Coulomb and hard-sphere correlations

We develop a modified Poisson-Nernst-Planck model which includes both the long-range Coulomb and short-range hard-sphere correlations in its free energy functional such that the model can accurately describe the ion transport in complex environment and under a nanoscale confinement. The Coulomb correlation including the dielectric polarization is treated by solving a generalized Debye-Hückel equation which is a Green's function equation with the correlation energy of a test ion described by the self Green's function. The hard-sphere correlation is modeled through the modified fundamental measure theory. The resulting model is available for problems beyond the mean-field theory such as problems with variable dielectric media, multivalent ions, and strong surface charge density. We solve the generalized Debye-Hückel equation by the Wentzel-Kramers-Brillouin approximation, and study the electrolytes between two parallel dielectric surfaces. In comparison to other modified models, the new model is shown more accurate in agreement with particle-based simulations and capturing the physical properties of ionic structures near interfaces.

physics.comp-ph↗

arXiv2021

HSMA: An O(N) electrostatics package implemented in LAMMPS

We implement two recently developed fast Coulomb solvers, HSMA3D [J. Chem. Phys. 149 (8) (2018) 084111] and HSMA2D [J. Chem. Phys. 152 (13) (2020) 134109], into a new user package HSMA for molecular dynamics simulation engine LAMMPS. The HSMA package is designed for efficient and accurate modeling of electrostatic interactions in 3D and 2D periodic systems with dielectric effects at the O(N) cost. The implementation is hybrid MPI and OpenMP parallelized and compatible with existing LAMMPS functionalities. The vectorization technique following AVX512 instructions is adopted for acceleration. To establish the validity of our implementation, we have presented extensive comparisons to the widely used particle-particle particle-mesh (PPPM) algorithm in LAMMPS and other dielectric solvers. With the proper choice of algorithm parameters and parallelization setup, the package enables calculations of electrostatic interactions that outperform the standard PPPM in speed for a wide range of particle numbers.

physics.comp-ph↗

arXiv2021

A random batch Ewald method for particle systems with Coulomb interactions

We develop a random batch Ewald (RBE) method for molecular dynamics simulations of particle systems with long-range Coulomb interactions, which achieves an $O(N)$ complexity in each step of simulating the $N$-body systems. The RBE method is based on the Ewald splitting for the Coulomb kernel with a random "mini-batch" type technique introduced to speed up the summation of the Fourier series for the long-range part of the splitting. Importance sampling is employed to reduce the induced force variance by taking advantage of the fast decay property of the Fourier coefficients. The stochastic approximation is unbiased with controlled variance. Analysis for bounded force fields gives some theoretic support of the method. Simulations of two typical problems of charged systems are presented to illustrate the accuracy and efficiency of the RBE method in comparison to the results from the Debye-Hückel theory and the classical Ewald summation, demonstrating that the proposed method has the attractiveness of being easy to implement with the linear scaling and is promising for many practical applications.

physics.comp-ph↗

arXiv2021

A kernel-independent sum-of-Gaussians method by de la Vallée-Poussin sums

Approximation of interacting kernels by sum of Gaussians (SOG) is frequently required in many applications of scientific and engineering computing in order to construct efficient algorithms for kernel summation or convolution problems. In this paper, we propose a kernel-independent SOG method by introducing the de la Vallée-Poussin sum and Chebyshev polynomials. The SOG works for general interacting kernels and the lower bound of Gaussian bandwidths is tunable and thus the Gaussians can be easily summed by fast Gaussian algorithms. The number of Gaussians can be further reduced via the model reduction based on the balanced truncation based on the square root method. Numerical results on the accuracy and model reduction efficiency show attractive performance of the proposed method.

math.NA↗

arXiv2020

L1-based reduced over collocation and hyper reduction for steady state and time-dependent nonlinear equations

The task of repeatedly solving parametrized partial differential equations (pPDEs) in, e.g. optimization or interactive applications, makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents as such an option. Enabled by a mathematically rigorous error estimator, RBM constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold from which an approximate solution is computed. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrades online efficiency. In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation inherent to a traditional application of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced solution space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values to build the reduced solution space. Together, these two ingredients render the proposed L1-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in alternative RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of L1-ROC and its superior stability performance.

math.NA↗

arXiv2020

A random-batch Monte Carlo method for many-body systems with singular kernels

We propose a fast potential splitting Markov Chain Monte Carlo method which costs $O(1)$ time each step for sampling from equilibrium distributions (Gibbs measures) corresponding to particle systems with singular interacting kernels. We decompose the interacting potential into two parts, one is of long range but is smooth, and the other one is of short range but may be singular. To displace a particle, we first evolve a selected particle using the stochastic differential equation (SDE) under the smooth part with the idea of random batches, as commonly used in stochastic gradient Langevin dynamics. Then, we use the short range part to do a Metropolis rejection. Different from the classical Langevin dynamics, we only run the SDE dynamics with random batch for a short duration of time so that the cost in the first step is $O(p)$, where $p$ is the batch size. The cost of the rejection step is $O(1)$ since the interaction used is of short range. We justify the proposed random-batch Monte Carlo method, which combines the random batch and splitting strategies, both in theory and with numerical experiments. While giving comparable results for typical examples of the Dyson Brownian motion and Lennard-Jones fluids, our method can save more time when compared to the classical Metropolis-Hastings algorithm.

physics.comp-ph↗

arXiv2019

L1-ROC and R2-ROC: L1- and R2-based Reduced Over-Collocation methods for parametrized nonlinear partial differential equations

The onerous task of repeatedly resolving certain parametrized partial differential equations (pPDEs) in, e.g. the optimization context, makes it imperative to design vastly more efficient numerical solvers without sacrificing any accuracy. The reduced basis method (RBM) presents itself as such an option. With a mathematically rigorous error estimator, RBM seeks a surrogate solution in a carefully-built subspace of the parameter-induced high fidelity solution manifold. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrading to the online efficiency. In this paper, we augment and extend the EIM approach in the context of solving pPDEs in two different ways, resulting in the Reduced Over-Collocation methods (ROC). These are stable and capable of avoiding the efficiency degradation inherent to a direct application of EIM. There are two ingredients of these methods. First is a strategy to collocate at about twice as many locations as the number of bases for the surrogate space. The second is an efficient approach for the strategic selection of the parameter values to build the reduced solution space for which we study two choices, a recent empirical L1 approach and a new indicator based on the reduced residual. Together, these two ingredients render the schemes, L1-ROC and R2-ROC, online efficient and immune from the efficiency degradation of EIM for nonlinear and nonaffine problems offline and online. Numerical tests on three different families of nonlinear problems demonstrate the high efficiency and accuracy of these new algorithms and their superior stability performance.

math.NA↗

arXiv2018

A reduced basis method for the nonlinear Poisson-Boltzmann equation

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the {\em truth approximations} of the RBM upon which the fast algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.

math.NA↗