arXiv subjects

Benjamin Seibold

Publications and source records attributed to Benjamin Seibold.

At least 37 records · Page 2Linked to original sources

arXiv2021

Bringing discrete-time Langevin splitting methods into agreement with thermodynamics

In light of the recently published complete set of statistically correct Gronbech-Jensen (GJ) methods for discrete-time thermodynamics, we revise a differential operator splitting method for the Langevin equation in order to comply with the basic GJ thermodynamic sampling features, namely the Boltzmann distribution and Einstein diffusion, in linear systems. This revision, which is based on the introduction of time scaling along with flexibility of a discrete-time velocity attenuation parameter, provides a direct link between the ABO splitting formalism and the GJ methods. This link brings about the conclusion that any GJ method has at least weak second order accuracy in the applied time step. It further helps identify a novel half-step velocity, which simultaneously produces both correct kinetic statistics and correct transport measures for any of the statistically sound GJ methods. Explicit algorithmic expressions are given for the integration of the new half-step velocity into the GJ set of methods. Numerical simulations, including quantum-based molecular dynamics (QMD) using the QMD suite LATTE, highlight the discussed properties of the algorithms as well as exhibit the direct application of robust, time step independent stochastic integrators to quantum-based molecular dynamics.

physics.comp-ph↗

arXiv2021

Integrated Framework of Vehicle Dynamics, Instabilities, Energy Models, and Sparse Flow Smoothing Controllers

This work presents an integrated framework of: vehicle dynamics models, with a particular attention to instabilities and traffic waves; vehicle energy models, with particular attention to accurate energy values for strongly unsteady driving profiles; and sparse Lagrangian controls via automated vehicles, with a focus on controls that can be executed via existing technology such as adaptive cruise control systems. This framework serves as a key building block in developing control strategies for human-in-the-loop traffic flow smoothing on real highways. In this contribution, we outline the fundamental merits of integrating vehicle dynamics and energy modeling into a single framework, and we demonstrate the energy impact of sparse flow smoothing controllers via simulation results.

eess.SY↗

arXiv2020

Infinite Time Solutions of Numerical Schemes for Advection Problems

This paper addresses the question whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. The motivation is that such methods may serve as building blocks for long-time accurate solutions in more complex advection-dominated problems. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods cannot meet this convergence criterion. Then we present a new numerical methodology, based on a nonlinear jet scheme framework. We show that these methods do satisfy the new convergence criterion, thus establishing that numerical methods exist that converge on an infinite time horizon, and demonstrate the long-time accuracy gains incurred by this property.

math.NA↗

arXiv2020

DIRK Schemes with High Weak Stage Order

Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples.

math.NA↗

arXiv2020

The Challenge of Stochastic Størmer-Verlet Thermostats Generating Correct Statistics

In light of the recently developed complete GJ set of single random variable stochastic, discrete-time Størmer-Verlet algorithms for statistically accurate simulations of Langevin equations, we investigate two outstanding questions: 1) Are there any algorithmic or statistical benefits from including multiple random variables per time-step, and 2) are there objective reasons for using one or more methods from the available set of statistically correct algorithms? To address the first question, we assume a general form for the discrete-time equations with two random variables and then follow the systematic, brute-force GJ methodology by enforcing correct thermodynamics in linear systems. It is concluded that correct configurational Boltzmann sampling of a particle in a harmonic potential implies correct configurational free-particle diffusion, and that these requirements only can be accomplished if the two random variables per time step are identical. We consequently submit that the GJ set represents all possible stochastic Størmer-Verlet methods that can reproduce time-step-independent statistics of linear systems. The second question is thus addressed within the GJ set. Based in part on numerical simulations of complex molecular systems, and in part on analytic scaling of time, we analyze the apparent difference in stability between different methods. We attribute this difference to the inherent time scaling in each method, and suggest that this scaling may lead to inconsistencies in the interpretation of dynamical and statistical simulation results. We therefore suggest that the method with the least inherent time-scaling, the GJ-I/GJF-2GJ method, be preferred for statistical applications where spurious rescaling of time is undesirable.

cond-mat.stat-mech↗

arXiv2020

Quantitative Relationship between Population Mobility and COVID-19 Growth Rate based on 14 Countries

This study develops a framework for quantification of the impact of changes in population mobility due to social distancing on the COVID-19 infection growth rate. Using the Susceptible-Infected-Recovered (SIR) epidemiological model we establish that under some mild assumptions the growth rate of COVID-19 deaths is a time-delayed approximation of the growth rate of COVID-19 infections. We then hypothesize that the growth rate of COVID-19 infections is a function of population mobility, which leads to a statistical model that predicts the growth rate of COVID-19 deaths as a delayed function of population mobility. The parameters of the statistical model directly reveal the growth rate of infections, the mobility-dependent transmission rate, the mobility-independent recovery rate, and the critical mobility, below which COVID-19 growth rate becomes negative. We fitted the proposed statistical model on publicly available data from 14 countries where daily death counts exceeded 100 for more than 3 days as of May 6th, 2020. The publicly available Google Mobility Index (GMI) was used as a measure of population mobility at the country level. Our results show that the growth rate of COVID-19 deaths can be accurately estimated 20 days ahead as a quadratic function of the transit category of GMI (adjusted R-squared = 0.784). The estimated 95% confidence interval for the critical mobility is in the range between 36.1% and 47.6% of the pre-COVID-19 mobility. This result indicates that a significant reduction in population mobility is needed to reverse the growth of COVID-19 epidemic. Moreover, the quantitative relationship established herein suggests that a readily available, population-level metric such as GMI can be a useful indicator of the course of COVID-19 epidemic.

q-bio.PE↗

arXiv2020

Massively Parallel Stencil Strategies for Radiation Transport Moment Model Simulations

The radiation transport equation is a mesoscopic equation in high dimensional phase space. Moment methods approximate it via a system of partial differential equations in traditional space-time. One challenge is the high computational intensity due to large vector sizes (1600 components for P39) in each spatial grid point. In this work, we extend the calculable domain size in 3D simulations considerably, by implementing the StaRMAP methodology within the massively parallel HPC framework NAStJA, which is designed to use current supercomputers efficiently. We apply several optimization techniques, including a new memory layout and explicit SIMD vectorization. We showcase a simulation with 200 billion degrees of freedom, and argue how the implementations can be extended and used in many scientific domains.

physics.comp-ph↗

arXiv2020

High-order Methods for a Pressure Poisson Equation Reformulation of the Navier-Stokes Equations with Electric Boundary Conditions

Pressure Poisson equation (PPE) reformulations of the incompressible Navier-Stokes equations (NSE) replace the incompressibility constraint by a Poisson equation for the pressure and a suitable choice of boundary conditions. This yields a time-evolution equation for the velocity field only, with the pressure gradient acting as a nonlocal operator. Thus, numerical methods based on PPE reformulations, in principle, have no limitations in achieving high order. In this paper, it is studied to what extent high-order methods for the NSE can be obtained from a specific PPE reformulation with electric boundary conditions (EBC). To that end, implicit-explicit (IMEX) time-stepping is used to decouple the pressure solve from the velocity update, while avoiding a parabolic time-step restriction; and mixed finite elements are used in space, to capture the structure imposed by the EBC. Via numerical examples, it is demonstrated that the methodology can yield at least third order accuracy in space and time.

math.NA↗

arXiv2019

Structural Properties of the Stability of Jamitons

It is known that inhomogeneous second-order macroscopic traffic models can reproduce the phantom traffic jam phenomenon: whenever the sub-characteristic condition is violated, uniform traffic flow is unstable, and small perturbations grow into nonlinear traveling waves, called jamitons. In contrast, what is essentially unstudied is the question: which jamiton solutions are dynamically stable? To understand which stop-and-go traffic waves can arise through the dynamics of the model, this question is critical. This paper first presents a computational study demonstrating which types of jamitons do arise dynamically, and which do not. Then, a procedure is presented that characterizes the stability of jamitons. The study reveals that a critical component of this analysis is the proper treatment of the perturbations to the shocks, and of the neighborhood of the sonic points.

physics.soc-ph↗

arXiv2019

Are commercially implemented adaptive cruise control systems string stable?

In this article, we assess the string stability of seven 2018 model year adaptive cruise control (ACC) equipped vehicles that are widely available in the US market. Seven distinct vehicle models from two different vehicle makes are analyzed using data collected from more than 1,200 miles of driving in car-following experiments with ACC engaged by the follower vehicle. The resulting dataset is used to identify the parameters of a linear second order delay differential equation model that approximates the behavior of the black box ACC systems. The string stability of the data-fitted model associated with each vehicle is assessed, and the main finding is that all seven vehicle models have string unstable ACC systems. For one commonly available vehicle model that offers ACC as a standard feature on all trim levels, we validate the string stability finding with a multi-vehicle platoon experiment in which all vehicles are the same year, make, and model. In this test, an initial disturbance of 6 mph is amplified to a 25 mph disturbance, at which point the last vehicle in the platoon is observed to disengage the ACC. The data collected in the driving experiments is made available, representing the largest publicly available comparative driving dataset on ACC equipped vehicles.

eess.SY↗

arXiv2019

Unconditional Stability for Multistep ImEx Schemes: Theory

This paper presents a new class of high order linear ImEx multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable-coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed implicit-explicit matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.

math.NA↗

arXiv2019

Comparison of Modern Langevin Integrators for Simulations of Coarse-Grained Polymer Melts

For a wide range of phenomena, current computational ability does not always allow for fully atomistic simulations of high-dimensional molecular systems to reach time scales of interest. Coarse-graining (CG) is an established approach to alleviate the impact of computational limits while retaining the same algorithms used in atomistic simulations. It is of importance to understand how algorithms such as Langevin integrators perform on non-trivial CG molecular systems, and in particular how large of an integration time step can be used without introducing unacceptable amounts of error into averaged quantities of interest. To investigate this, we examined three different Langevin integrators on a CG polymer melt: the recently developed BAOAB method by Leimkuhler and Matthews, the Gronbech-Jensen and Farago method, or G-JF, and the frequently used Brunger-Brooks-Karplus integrator, also known as BBK. We compute and analyze key statistical properties for each. Our results indicate that the three integrators perform similarly when using a small friction parameter; however, outside of this regime the use of large integration steps produces significant deviations from the predicted diffusivity and steady-state distributions for all integration methods examined with the exception of G-JF.

physics.comp-ph↗

arXiv2018

Unconditional Stability for Multistep ImEx Schemes: Practice

This paper focuses on the question of how unconditional stability can be achieved via multistep ImEx schemes, in practice problems where both the implicit and explicit terms are allowed to be stiff. For a class of new ImEx multistep schemes that involve a free parameter, strategies are presented on how to choose the ImEx splitting and the time stepping parameter, so that unconditional stability is achieved under the smallest approximation errors. These strategies are based on recently developed stability concepts, which also provide novel insights into the limitations of existing semi-implicit backward differentiation formulas (SBDF). For instance, the new strategies enable higher order time stepping that is not otherwise possible with SBDF. With specific applications in nonlinear diffusion problems and incompressible channel flows, it is demonstrated how the unconditional stability property can be leveraged to efficiently solve stiff nonlinear or nonlocal problems without the need to solve nonlinear or nonlocal problems implicitly.

math.NA↗

arXiv2017

Off-Ramp Coupling Conditions Devoid of Spurious Blocking and Re-Routing

When modeling vehicular traffic flow on highway networks via macroscopic models, suitable coupling conditions at the network nodes are crucial. Frequently, the evolution of traffic flow on each network edge is described in a lane-averaged fashion using a single-class Lighthill-Whitham-Richards model. At off-ramps, split ratios (i.e., what percentage of traffic exits the highway) are prescribed that can be drawn from historic data. In this situation, classical FIFO coupling conditions yield unrealistic results, in that a clogged off-ramp yields zero flux through the node. As a remedy, non-FIFO conditions have been proposed. However, as we demonstrate here, those lead to spurious re-routing of vehicles. Then, a new coupling model, FIFOQ, is presented that preserves the desirable properties of non-FIFO models, while not leading to any spurious re-routing.

physics.soc-ph↗

arXiv2017

A Comparative Study of Limiting Strategies in Discontinuous Galerkin Schemes for the $M_1$ Model of Radiation Transport

The $M_1$ minimum entropy moment system is a system of hyperbolic balance laws that approximates the radiation transport equation, and has many desirable properties. Among them are symmetric hyperbolicity, entropy decay, moment realizability, and correct behavior in the diffusion and free-streaming limits. However, numerical difficulties arise when approximating the solution of the $M_1$ model by high order numerical schemes; namely maintaining the realizability of the numerical solution and controlling spurious oscillations. In this paper, we extend a previously constructed one-dimensional realizability limiting strategy to 2D. In addition, we perform a numerical study of various combinations of the realizability limiter and the TVBM local slope limiter on a third order Discontinuous Galerkin (DG) scheme on both triangular and rectangular meshes. In several test cases, we demonstrate that in general, a combination of the realizability limiter and a TVBM limiter is necessary to obtain a robust and accurate numerical scheme. Our code is published so that all results can be reproduced by the reader.

math.NA↗

arXiv2017

Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments

Traffic waves are phenomena that emerge when the vehicular density exceeds a critical threshold. Considering the presence of increasingly automated vehicles in the traffic stream, a number of research activities have focused on the influence of automated vehicles on the bulk traffic flow. In the present article, we demonstrate experimentally that intelligent control of an autonomous vehicle is able to dampen stop-and-go waves that can arise even in the absence of geometric or lane changing triggers. Precisely, our experiments on a circular track with more than 20 vehicles show that traffic waves emerge consistently, and that they can be dampened by controlling the velocity of a single vehicle in the flow. We compare metrics for velocity, braking events, and fuel economy across experiments. These experimental findings suggest a paradigm shift in traffic management: flow control will be possible via a few mobile actuators (less than 5%) long before a majority of vehicles have autonomous capabilities.

eess.SY↗

arXiv2017

Traffic Flow Control and Fuel Consumption Reduction via Moving Bottlenecks

Moving bottlenecks, such as slow-driving vehicles, are commonly thought of as impediments to efficient traffic flow. Here, we demonstrate that in certain situations, moving bottlenecks---properly controlled---can actually be beneficial for the traffic flow, in that they reduce the overall fuel consumption, without imposing any delays on the other vehicles. As an important practical example, we study a fixed bottleneck (e.g., an accident) that has occurred further downstream. This new possibility of traffic control is particularly attractive with autonomous vehicles, which (a) will have fast access to non-local information, such as incidents and congestion downstream; and (b) can execute driving protocols accurately.

physics.soc-ph↗

arXiv2017

A Collapsed Generalized Aw-Rascle-Zhang Model and Its Model Accuracy

This work presents a collapsed generalized Aw-Rascle-Zhang (CGARZ) model, which fits into a generic second order model (GSOM) framework. GSOMs augment the evolution of the traffic density by a second state variable characterizing a property of vehicles or drivers. A cell transmission model for the numerical solution of GSOMs is derived, which is based on analyzing the sending and receiving functions of the traffic density and total property. The predictive accuracy of the CGARZ model is then compared to the classical first-order LWR and four second-order models. To that end, a systematic approach to calibrate model parameters from sensor flow-density data is introduced and applied to all models studied. The comparative model validation is conducted using two types of field data: vehicle trajectory data, and loop detector data. It is shown that the CGARZ model provides an intriguing combination of simple model dynamics in the free-flow regime and a representation of the spread of flow-density data in the congested regime, while possessing a competitive prediction accuracy.

physics.soc-ph↗