arXiv subjects

Gideon Simpson

Publications and source records attributed to Gideon Simpson.

At least 19 recordsLinked to original sources

arXiv2026

Non-asymptotic uniform in time error bounds for new and old numerical schemes for SPDEs

We study numerical schemes for Stochastic Partial Differential Equations (SPDEs). We introduce a general method of proof of non-asymptotic uniform in time error bounds on numerical integrators for SPDEs, ensuring the schemes capture both the transient and the long term dynamics faithfully. We then consider SPDEs with non-globally Lipshitz nonlinearities, which include for example the stochastic Allen-Cahn equation and some stochastic advection-diffusion equations. For the case of Allen-Cahn type SPDEs we show that the classic semi-implicit Euler time-discretization can exhibit finite time blow up. This motivates analysing other schemes which do not suffer from this blow-up problem. We consider three numerical schemes for SPDEs with non globally Lipshitz nonlinearity: a fully implicit scheme and two tamed schemes. For these schemes we prove non-asymptotic uniform in time error bounds by leveraging our general criterion, and provide numerical comparisons. While the main emphasis in this paper is on the properties of the time-discretization, the schemes we consider are full space-time discretization of the SPDE.

math.NA↗

arXiv2025

Reducing Weighted Ensemble Variance With Optimal Trajectory Management

Weighted ensemble (WE) is an enhanced path-sampling method that is conceptually simple, widely applicable, and statistically exact. In a WE simulation, an ensemble of trajectories is periodically pruned or replicated to enhance sampling of rare transitions and improve estimation of mean first passage times (MFPTs). However, poor choices of the parameters governing pruning and replication can lead to high-variance MFPT estimates. Our previous work [J. Chem. Phys. 158, 014108 (2023)] presented an optimal WE parameterization strategy and applied it in low-dimensional example systems. The strategy harnesses estimated local MFPTs from different initial configurations to a single target state. In the present work, we apply the optimal parameterization strategy to more challenging, high-dimensional molecular models, namely, synthetic molecular dynamics (MD) models of Trp-cage folding and unfolding, as well as atomistic MD models of NTL9 folding in high-friction and low-friction continuum solvents. In each system we use WE to estimate the MFPT for folding or unfolding events. We show that the optimal parameterization reduces the variance of MFPT estimates in three of four systems, with dramatic improvement in the most challenging atomistic system. Overall, the parameterization strategy improves the accuracy and reliability of WE estimates for the kinetics of biophysical processes.

physics.chem-ph↗

arXiv2024

Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators

The Kuramoto model (KM) of $n$ coupled phase-oscillators is analyzed in this work. The KM on a Cayley graph possesses a family of steady state solutions called twisted states. Topologically distinct twisted states are distinguished by the winding number $q\in\mathbb{Z}$. These states are known to be stable for small enough $q$. In the presence of small noise, the KM exhibits metastable transitions between $q$-twisted states: A typical trajectory remains in the basin of attraction of a given $q$-twisted state for an exponentially long time, but eventually transitions to the vicinity of another such state. In the course of this transition, it passes in close proximity of a saddle of Morse index $1$, called a relevant saddle. In this work, we provide an exhaustive analysis of metastable transitions in the stochastic KM with nearest-neighbor coupling. We start by analyzing the equilibria and their stability. First, we identify all equilibria in this model. Using the discrete Fourier transform and eigenvalue estimates for rank-1 perturbations of symmetric matrices, we classify the equilibria by their Morse indices. In particular, we identify all stable equilibria and all relevant saddles involved in the metastable transitions. Further, we use Freidlin-Wentzell theory and the potential-theoretic approach to metastability to establish the metastable hierarchy and sharp estimates of Eyring-Kramers type for the transition times. The former determines the precise order, in which the metastable transitions occur, while the latter characterizes the times between successive transitions. The theoretical estimates are complemented by numerical simulations and a careful numerical verification of the transition times. Finally, we discuss the implications of this work for the KM with other coupling types including nonlocal coupling and the continuum limit as $n$ tends to infinity.

math.PR↗

arXiv2024

Counting the number of stationary solutions of Partial Differential Equations via infinite dimensional sampling

This paper is concerned with the problem of counting solutions of stationary nonlinear Partial Differential Equations (PDEs) when the PDE is known to admit more than one solution. We suggest tackling the problem via a sampling-based approach. We test our proposed methodology on the McKean-Vlasov PDE, more precisely on the problem of determining the number of stationary solutions of the McKean-Vlasov (or porous medium) equation.

math.AP↗

arXiv2024

RiteWeight: Randomized Iterative Trajectory Reweighting for Steady-State Distributions Without Discretization Error

A significant challenge in molecular dynamics (MD) simulations is ensuring that sampled configurations converge to the equilibrium or nonequilibrium stationary distribution of interest. Lack of convergence constrains the estimation of free energies, rates, and mechanisms of complex molecular events. Here, we introduce the "Randomized ITErative trajectory reWeighting" (RiteWeight) algorithm to estimate a stationary distribution from unconverged simulation data. This method iteratively reweights trajectory segments in a self-consistent way by solving for the stationary distribution of a Markov state model (MSM), updating segment weights, and employing a new random clustering in each iteration. The iterative random clustering mitigates the phase-space discretization error inherent in existing trajectory reweighting techniques and yields quasi-continuous configuration-space distributions. We present mathematical analysis of the algorithm's fixed points as well as empirical validation using both synthetic MD Trp-cage trajectories, for which the stationary solution is exactly calculable, and standard atomistic MD Trp-cage trajectories extracted from a long reference simulation. In both test systems, we find that RiteWeight corrects flawed distributions and generates accurate observables for equilibrium and nonequilibrium steady states. The results highlight the value of correcting the underlying trajectory distribution rather than using a standard MSM

physics.comp-ph↗

arXiv2021

Iterate Averaging, the Kalman Filter, and 3DVAR for Linear Inverse Problem

It has been proposed that classical filtering methods, like the Kalman filter and 3DVAR, can be used to solve linear statistical inverse problems. In the work of Iglesias, Lin, Lu, & Stuart (2017), error estimates were obtained for this approach. By optimally tuning a regularization parameter in the filters, the authors were able to show that the mean squared error could be systematically reduced. Building on the aforementioned work of Iglesias, Lin, Lu, & Stuart, we prove that by (i) considering the problem in a weaker norm and (ii) applying simple iterate averaging of the filter output, 3DVAR will converge in mean square, unconditionally on the choice of parameter. Without iterate averaging, 3DVAR cannot converge by running additional iterations with a fixed choice of parameter. We also establish that the Kalman filter's performance in this setting cannot be improved through iterate averaging. We illustrate our results with numerical experiments that suggest our convergence rates are sharp.

math.NA↗

arXiv2021

Longitudinal Distance: Towards Accountable Instance Attribution

Previous research in interpretable machine learning (IML) and explainable artificial intelligence (XAI) can be broadly categorized as either focusing on seeking interpretability in the agent's model (i.e., IML) or focusing on the context of the user in addition to the model (i.e., XAI). The former can be categorized as feature or instance attribution. Example- or sample-based methods such as those using or inspired by case-based reasoning (CBR) rely on various approaches to select instances that are not necessarily attributing instances responsible for an agent's decision. Furthermore, existing approaches have focused on interpretability and explainability but fall short when it comes to accountability. Inspired in case-based reasoning principles, this paper introduces a pseudo-metric we call Longitudinal distance and its use to attribute instances to a neural network agent's decision that can be potentially used to build accountable CBR agents.

cs.AI↗

arXiv2021

A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and Q-Wiener process describing the interactions among the particles in the network and stochastic forcing respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct a semidiscrete (discrete in space and continuous in time) and a fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler-Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence estimate is expressed in terms of the regularity of the graphon, Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity including small-world and power-law are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler-Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.

math.NA↗

arXiv2021

Unbiased estimation of equilibrium, rates, and committors from Markov state model analysis

Markov state models (MSMs) have been broadly adopted for analyzing molecular dynamics trajectories, but the approximate nature of the models that results from coarse-graining into discrete states is a long-known limitation. We show theoretically that, despite the coarse graining, in principle MSM-like analysis can yield unbiased estimation of key observables. We describe unbiased estimators for equilibrium state populations, for the mean first-passage time (MFPT) of an arbitrary process, and for state committors - i.e., splitting probabilities. Generically, the estimators are only asymptotically unbiased but we describe how extension of a recently proposed reweighting scheme can accelerate relaxation to unbiased values. Exactly accounting for 'sliding window' averaging over finite-length trajectories is a key, novel element of our analysis. In general, our analysis indicates that coarse-grained MSMs are asymptotically unbiased for steady-state properties only when appropriate boundary conditions (e.g., source-sink for MFPT estimation) are applied directly to trajectories, prior to calculation of the appropriate transition matrix.

physics.comp-ph↗

arXiv2020

A splitting method to reduce MCMC variance

We explore whether splitting and killing methods can improve the accuracy of Markov chain Monte Carlo (MCMC) estimates of rare event probabilities, and we make three contributions. First, we prove that "weighted ensemble" is the only splitting and killing method that provides asymptotically consistent estimates when combined with MCMC. Second, we prove a lower bound on the asymptotic variance of weighted ensemble's estimates. Third, we give a constructive proof and numerical examples to show that weighted ensemble can approach this optimal variance bound, in many cases reducing the variance of MCMC estimates by multiple orders of magnitude.

math.NA↗

arXiv2019

Sampling from Rough Energy Landscapes

We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features, or "roughness," corresponds to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and analysis we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of fine scale roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open questions and numerical challenges are also highlighted.

math.NA↗

arXiv2018

Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture

Probability currents are fundamental in characterizing the kinetics of non-equilibrium processes. Notably, the steady-state current $J_{ss}$ for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from source to sink. Because transient non-equilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on $J_{ss}$ -- and hence on the MFPT -- as the system evolves in time. Here we show that $J_{ss}$ is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time $t$ for one-dimensional systems undergoing over-damped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating $J_{ss}$ and the long-timescale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.

cond-mat.stat-mech↗

arXiv2017

Existence theory for magma equations in dimension two and higher

We examine a degenerate, dispersive, nonlinear wave equation related to the evolution of partially molten rock in dimensions two and higher. This simplified model, for a scalar field capturing the melt fraction by volume, has been studied by direct numerical simulation where it has been observed to develop stable solitary waves. In this work, we prove local in time well-posedness results for the time dependent equation, on both the whole space and the torus, for dimensions two and higher. We also prove the existence of the solitary wave solutions in dimensions two and higher.

math.AP↗

arXiv2017

Spin-Diffusions and Diffusive Molecular Dynamics

Metastable condensed matter typically fluctuates about local energy minima at the femtosecond time scale before transitioning between local minima after nanoseconds or microseconds. This vast scale separation limits the applicability of classical molecular dynamics methods and has spurned the development of a host of approximate algorithms. One recently proposed method is diffusive molecular dynamics which aims to integrate a system of ordinary differential equations describing the likelihood of occupancy by one of two species, in the case of a binary alloy, while quasistatically evolving the locations of the atoms. While diffusive molecular dynamics has shown to be efficient and provide agreement with observations, it is fundamentally a model, with unclear connections to classical molecular dynamics. In this work, we formulate a spin-diffusion stochastic process and show how it can be connected to diffusive molecular dynamics. The spin-diffusion model couples a classical overdamped Langevin equation to a kinetic Monte Carlo model for exchange amongst the species of a binary alloy. Under suitable assumptions and approximations, spin-diffusion can be shown to lead to diffusive molecular molecular dynamics type models. The key assumptions and approximations include a well defined time scale separation, a choice of spin exchange rates, a low temperature approximation, and a mean field type approximation. We derive several models from different assumptions and show their relationship to diffusive molecular dynamics. Differences and similarities amongst the models are explored in a simple test problem.

physics.comp-ph↗

arXiv2016

Conservative Integrators for a Toy Model of Weak Turbulence

Weak turbulence is a phenomenon by which a system generically transfers energy from low to high wave numbers, while persisting for all finite time. It has been conjectured by Bourgain that the 2D defocusing nonlinear Schr\"odinger equation (NLS) on the torus has this dynamic, and several analytical and numerical studies have worked towards addressing this point. In the process of studying the conjecture, Colliander, Keel, Staffilani, Takaoka, and Tao introduced a "toy model" dynamical system as an approximation of NLS, which has been subsequently studied numerically. In this work, we formulate and examine several numerical schemes for integrating this model equation. The model has two invariants, and our schemes aim to conserve at least one of them. We prove convergence in some cases, and our numerical studies show that the schemes compare favorably to others, such as Trapezoidal Rule and fixed step fourth order Runge-Kutta. The preservation of the invariants is particularly important in the study of weak turbulence as the energy transfer tends to occur on long time scales.

math.NA↗

arXiv2016

Local structure of singular profiles for a Derivative Nonlinear Schr\"odinger Equation

The Derivative Nonlinear Schr\"odinger equation is an $L^2$-critical nonlinear dispersive equation model for Alfv\'en waves in space plasmas. Recent numerical studies on an $L^2$-supercritical extension of this equation provide evidence of finite time singularities. Near the singular point, the solution is described by a universal profile that solves a nonlinear elliptic eigenvalue problem depending only on the strength of the nonlinearity. In the present work, we describe the deformation of the profile and its parameters near criticality, combining asymptotic analysis and numerical simulations.

math.AP↗

arXiv2015

Existence and Stability Properties of Radial Bound States for Schr\"odinger-Poisson with an External Coulomb Potential in Three Space Dimensions

We consider radial solutions to the Schr\"odinger-Poisson system in three dimensions with an external smooth potential with Coulomb-like decay. Such a system can be viewed as a model for the interaction of dark matter with a bright matter background in the non-relativistic limit. We find that there are infinitely many critical points of the Hamiltonian, subject to fixed mass, and that these bifurcate from solutions to the associated linear problem at zero mass. As a result, each branch has a different topological character defined by the number of zeros of the radial states. We construct numerical approximations to these nonlinear states along the first several branches. The solution branches can be continued, numerically, to large mass values, where they become asymptotic, under a rescaling, to those of the Schr\"odinger-Poisson problem with no external potential. Our numerical computations indicate that the ground state is orbitally stable, while the excited states are linearly unstable for sufficiently large mass.

math.AP↗

arXiv2015

Relative Entropy Minimization over Hilbert Spaces via Robbins-Monro

One way of getting insight into non-Gaussian measures, posed on infinite dimensional Hilbert spaces, is to first obtain best fit Gaussian approximations, which are more amenable to numerical approximation. These Gaussians can then be used to accelerate sampling algorithms. This begs the questions of how one should measure optimality and how the optimizers can be obtained. Here, we consider the problem of minimizing the distance with respect to relative entropy. We examine this minimization problem by seeking roots of the first variation of relative entropy, taken with respect to the mean of the Gaussian, leaving the covariance fixed. Adapting a convergence analysis of Robbins-Monro to the infinite dimensional setting, we can justify the application of this algorithm and highlight necessary assumptions to ensure convergence, not only in the context of relative entropy minimization, but other infinite dimensional problems as well. Numerical examples in path space, showing the robustness of this method with respect to dimension, are provided.

math.NA↗