arXiv subjects

Xingjie Helen Li

Publications and source records attributed to Xingjie Helen Li.

17 recordsLinked to original sources

arXiv2026

Numerical stability revisited: A family of benchmark problems for the analysis of explicit stochastic differential equation integrators

We revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of the (1) Euler-Maruyama, (2) Milstein, (3) Stochastic Heun, and (4) three-stage Runge-Kutta schemes. Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs.

math.NA↗

arXiv2024

A Comprehensive Study of Covid 19 in Florida

Within the likes of any highly contagious and unpredictable disease, lies a predictable and attainable growth rate that researchers can find in order to make logistical conclusions about that particular disease and its affected regions' counterparts. The foundation that researchers pull from when studying a particular disease and looking for its growth rate is the Susceptible-Infected-Removed (SIR) model, presented by a series of differential equations. The issue with the SIR model lies not in its complexity, but actually its simplicity and lack of a potentially high-finite amount of factors; the limit being bounded by the amount of data available for that particular factor. Our research involves the application of multiple regressions to pinpoint and identify our Covid lockdown periods, followed by the modification of the SIR model. This involved creating new model approximations such as the time-delayed SIR model and the reinfected SIR model in order to take into account factors such as incubation and reinfection, and get the lowest error discrepancy as possible for our infection rate. We were able to conclude that the more factors that we took into account, our error rate became lower and our results became more accurate. We could also identify outlier Metros and draw certain conclusions on performance level and the reasons behind them. We then moved on to find correlations, if any, between the infection rates and outside factors. We looked at demographic and weather data to demonstrate whether correlations appeared. We found that there are a few factors with high correlations, including graduate education and low temperatures.

stat.AP↗

arXiv2024

Automated construction of effective potential via algorithmic implicit bias

We introduce a novel approach for decomposing and learning every scale of a given multiscale objective function in $\mathbb{R}^d$, where $d\ge 1$. This approach leverages a recently demonstrated implicit bias of the optimization method of gradient descent by Kong and Tao, which enables the automatic generation of data that nearly follow Gibbs distribution with an effective potential at any desired scale. One application of this automated effective potential modeling is to construct reduced-order models. For instance, a deterministic surrogate Hamiltonian model can be developed to substantially soften the stiffness that bottlenecks the simulation, while maintaining the accuracy of phase portraits at the scale of interest. Similarly, a stochastic surrogate model can be constructed at a desired scale, such that both its equilibrium and out-of-equilibrium behaviors (characterized by auto-correlation function and mean path) align with those of a damped mechanical system with the original multiscale function being its potential. The robustness and efficiency of our proposed approach in multi-dimensional scenarios have been demonstrated through a series of numerical experiments. A by-product of our development is a method for anisotropic noise estimation and calibration. More precisely, Langevin model of stochastic mechanical systems may not have isotropic noise in practice, and we provide a systematic algorithm to quantify its covariance matrix without directly measuring the noise. In this case, the system may not admit closed form expression of its invariant distribution either, but with this tool, we can design friction matrix appropriately to calibrate the system so that its invariant distribution has a closed form expression of Gibbs.

math.NA↗

arXiv2023

Dynamical properties of coarse-grained linear SDEs

Coarse-graining or model reduction is a term describing a range of approaches used to extend the time-scale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analysed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics such as the mean-squared displacement display systematic error, even when a system exhibits large time-scale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.

math.DS↗

arXiv2023

A One-Dimensional Symmetric Force-Based Blending Method for Atomistic-to-Continuum Coupling

Inspired by the blending method developed by [P. Seleson, S. Beneddine, and S. Prudhome, \emph{A Force-Based Coupling Scheme for Peridynamics and Classical Elasticity}, (2013)] for the nonlocal-to-local coupling, we create a symmetric and consistent blended force-based Atomistic-to-Continuum (a/c) scheme for the atomistic chain in one-dimensional space. The conditions for the well-posedness of the underlying model are established by analyzing an optimal blending size and blending type to ensure the $H^1$ semi-norm stability for the blended force-based operator. We present several numerical experiments to test and confirm the theoretical findings.

math.NA↗

arXiv2021

Maximum Principle Preserving Finite Difference Scheme for 1-D Nonlocal-to-Local Diffusion Problems

In a recent paper (see [7]), a quasi-nonlocal coupling method was introduced to seamlessly bridge a nonlocal diffusion model with the classical local diffusion counterpart in a one-dimensional space. The proposed coupling framework removes inconsistency on the interface, preserves the balance of fluxes, and satisfies the maximum principle of diffusion problem. However, the numerical scheme proposed in that paper does not maintain all of these properties on a discrete level. In this paper we resolve these issues by proposing a new finite difference scheme that ensures the balance of fluxes and the discrete maximum principle. We rigorously prove these results and provide the stability and convergence analyses accordingly. In addition, we provide the Courant-Friedrichs-Lewy (CFL) condition for the new scheme and test a series of benchmark examples which confirm the theoretical findings.

math.NA↗

arXiv2018

Coarse-graining of overdamped Langevin dynamics via the Mori-Zwanzig formalism

The Mori-Zwanzig formalism is applied to derive an equation for the evolution of linear observables of the overdamped Langevin equation. To illustrate the resulting equation and its use in deriving approximate models, a particular benchmark example is studied both numerically and via a formal asymptotic expansion. The example considered demonstrates the important of memory effects in determining the correct temporal behaviour of such systems.

math.DS↗

arXiv2017

A quasinonlocal coupling method for nonlocal and local diffusion models

In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new coupling framework removes interfacial inconsistency, ensures the flux balance, and satisfies energy conservation as well as the maximum principle, whereas none of existing coupling methods for nonlocal-to-local coupling satisfies all of these properties. We establish the well-posedness and provide the stability analysis of the coupling method. We investigate the difference to the local limiting problem in terms of the nonlocal interaction range. Furthermore, we propose a first order finite difference numerical discretization and perform several numerical tests to confirm the theoretical findings. In particular, we show that the resulting numerical result is free of artifacts near the boundary of the domain where a classical local boundary condition is used, together with a coupled fully nonlocal model in the interior of the domain.

math.NA↗

arXiv2017

Quasinonlocal coupling of nonlocal diffusions

We developed a new self-adjoint, consistent, and stable coupling strategy for nonlocal diffusion models, inspired by the quasinonlocal atomistic-to-continuum method for crystalline solids. The proposed coupling model is coercive with respect to the energy norms induced by the nonlocal diffusion kernels as well as the $L^2$ norm, and it satisfies the maximum principle. A finite difference approximation is used to discretize the coupled system, which inherits the property from the continuous formulation. Furthermore, we design a numerical example which shows the discrepancy between the fully nonlocal and fully local diffusions, whereas the result of the coupled diffusion agrees with that of the fully nonlocal diffusion.

math.NA↗

arXiv2014

Analysis of Blended Atomistic/Continuum Hybrid Methods

We present a comprehensive error analysis of two prototypical atomistic-to-continuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods. Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (i) new force and energy consistency error estimates; and (ii) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.

math.NA↗

arXiv2013

Theory-based Benchmarking of the Blended Force-Based Quasicontinuum Method

We formulate an atomistic-to-continuum coupling method based on blending atomistic and continuum forces. Our precise choice of blending mechanism is informed by theoretical predictions. We present a range of numerical experiments studying the accuracy of the scheme, focusing in particular on its stability. These experiments confirm and extend the theoretical predictions, and demonstrate a superior accuracy of B-QCF over energy-based blending schemes.

math.NA↗

arXiv2013

Numerical solution of Dyson Brownian motion and a sampling scheme for invariant matrix ensembles

The Dyson Brownian Motion (DBM) describes the stochastic evolution of $N$ points on the line driven by an applied potential, a Coulombic repulsion and identical, independent Brownian forcing at each point. We use an explicit tamed Euler scheme to numerically solve the Dyson Brownian motion and sample the equilibrium measure for non-quadratic potentials. The Coulomb repulsion is too singular for the SDE to satisfy the hypotheses of rigorous convergence proofs for tamed Euler schemes. Nevertheless, in practice the scheme is observed to be stable for time steps of O(1/N^2) and to relax exponentially fast to the equilibrium measure with a rate constant of O(1) independent of $N$. Further, this convergence rate appears to improve with $N$ in accordance with O(1/N) relaxation of local statistics of the Dyson Brownian motion. This allows us to use the Dyson Brownian motion to sample $N\times N$ Hermitian matrices from the invariant ensembles. The computational cost of generating $M$ independent samples is O(MN^4) with a naive scheme, and O(MN^3\log N) when a fast multipole method is used to evaluate the Coulomb interaction.

math.NA↗

arXiv2011

Positive-Definiteness of the Blended Force-Based Quasicontinuum Method

The development of consistent and stable quasicontinuum models for multi-dimensional crystalline solids remains a challenge. For example, proving stability of the force-based quasicontinuum (QCF) model remains an open problem. In 1D and 2D, we show that by blending atomistic and Cauchy--Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width.

math.NA↗

arXiv2011

Lattice Stability for Atomistic Chains Modeled by Local Approximations of the Embedded Atom Method

The accurate approximation of critical strains for lattice instability is a key criterion for predictive computational modeling of materials. In this paper, we present a comparison of the lattice stability for atomistic chains modeled by the embedded atom method (EAM) with their approximation by local Cauchy-Born models. We find that both the volume-based local model and the reconstruction-based local model can give O(1) errors for the critical strain since the embedding energy density is generally strictly convex. The critical strain predicted by the volume-based model is always larger than that predicted by the atomistic model, but the critical strain for reconstruction-based models can be either larger or smaller than that predicted by the atomistic model.

math.NA↗

arXiv2011

A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods

We give computational results to study the accuracy of several quasicontinuum methods for two benchmark problems - the stability of a Lomer dislocation pair under shear and the stability of a lattice to plastic slip under tensile loading. We find that our theoretical analysis of the accuracy near instabilities for one-dimensional model problems can successfully explain most of the computational results for these multi-dimensional benchmark problems. However, we also observe some clear discrepancies, which suggest the need for additional theoretical analysis and benchmark problems to more thoroughly understand the accuracy of quasicontinuum methods.

math.NA↗

arXiv2010

A Generalized Quasi-Nonlocal Atomistic-to-Continuum Coupling Method with Finite Range Interaction

The accurate and efficient computation of the deformation of crystalline solids requires the coupling of atomistic models near lattice defects such as cracks and dislocations with coarse-grained models away from the defects. Quasicontinuum methods utilize a strain energy density derived from the Cauchy-Born rule for the coarse-grained model. Several quasicontinuum methods have been proposed to couple the atomistic model with the Cauchy-Born strain energy density. The quasi-nonlocal coupling method is easy to implement and achieves a reasonably accurate coupling for short range interactions. In this paper, we give a new formulation of the quasi-nonlocal method in one space dimension that allows its extension to arbitrary finite range interactions. We also give an analysis of the stability and accuracy of a linearization of our generalized quasi-nonlocal method that holds for strains up to lattice instabilities.

math.NA↗

arXiv2010

An Analysis of the Quasi-Nonlocal Quasicontinuum Approximation of the Embedded Atom Model

The quasi-nonlocal quasicontinuum method (QNL) is a consistent hybrid coupling method for atomistic and continuum models. Embedded atom models are empirical many-body potentials that are widely used for FCC metals such as copper and aluminum. In this paper, we consider the QNL method for EAM potentials, and we give a stability and error analysis for a chain with next-nearest neighbor interactions. We identify conditions for the pair potential, electron density function, and embedding function so that the lattice stability of the atomistic and the EAM-QNL models are asymptotically equal.

math.NA↗