scholarly journals Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states

2020 ◽  
Author(s):  
Petr Plechac ◽  
Gabriel Stoltz ◽  
Ting Wang
Keyword(s):  
Linear Response ◽  
Measure Theory ◽  
Approximation Error ◽  
Ratio Method ◽  
Integration Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Long Time ◽  
Bounded Variance ◽  
Time Average

We consider numerical schemes for computing the linear response of  steady-state averages with respect to a perturbation of the drift part of the stochastic differential equation.  The schemes are based on the Girsanov change-of-measure theory in order to reweight trajectories with factors derived from a linearization of the Girsanov weights.  The resulting estimator is the product of a time average and a martingale correlated to this time average.  We investigate both its discretization and finite time approximation error. The designed numerical schemes are shown to be of bounded variance with respect to the integration time which is a desirable feature for long time simulation.  We also show how the discretization error can be improved to second order accuracy in the time step by modifying the weight process in an appropriate way.

Long-time symplectic integration: the example of four-vortex motion

1991 ◽  
Vol 432 (1886) ◽  
pp. 481-494 ◽  
Keyword(s):  
Fourth Order ◽  
Initial Conditions ◽  
Hamiltonian Formulation ◽  
Superior Performance ◽  
Integration Time ◽  
Characteristic Time Scale ◽  
Symplectic Integration ◽  
Time Step ◽  
Long Time ◽  
Runge Kutta

Detailed comparisons are made between long-time numerical integration of the motion of four identical point vortices obtained using both a fourth-order symplectic integration method of the implicit Runge-Kutta type and a standard fourth-order explicit Runge-Kutta scheme. We utilize the reduced hamiltonian formulation of the four-vortex problem due to Aref & Pomphrey. Initial conditions which give both fully chaotic and also quasi-periodic motions are considered over integration times of order 10 6 -10 7 times the characteristic time scale of the evolution. The convergence, as the integration time step is decreased, of the Poincaré section is investigated. When smoothness of the section compared to the converged image, and the fractional change in the hamiltonian H are used as diagnostic indicators, it is found that the symplectic scheme gives substantially superior performance over the explicit scheme, and exhibits only an apparent qualitative degrading in results up to integration time steps of order the minimum timescale of the evolution. It is concluded that this performance derives from the symplectic rather than the implicit character of the method.

scholarly journals The Influence of an Integration Time Step on Dynamic Calculation of a Vehicle-Track-Bridge under High-Speed Railway

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 431
Author(s):  
Junjie Ye ◽  
Hao Sun
Keyword(s):  
High Speed ◽  
Coupled Model ◽  
Short Wave ◽  
Integration Time ◽  
Vertical Acceleration ◽  
Dynamic Calculation ◽  
Time Step ◽  
High Speed Railway ◽  
Track Irregularity ◽  
Track Bridge

In order to study the influence of an integration time step on dynamic calculation of a vehicle-track-bridge under high-speed railway, a vehicle-track-bridge (VTB) coupled model is established. The influence of the integration time step on calculation accuracy and calculation stability under different speeds or different track regularity states is studied. The influence of the track irregularity on the integration time step is further analyzed by using the spectral characteristic of sensitive wavelength. According to the results, the disparity among the effect of the integration time step on the calculation accuracy of the VTB coupled model at different speeds is very small. Higher speed requires a smaller integration time step to keep the calculation results stable. The effect of the integration time step on the calculation stability of the maximum vertical acceleration of each component at different speeds is somewhat different, and the mechanism of the effect of the integration time step on the calculation stability of the vehicle-track-bridge coupled system is that corresponding displacement at the integration time step is different. The calculation deviation of the maximum vertical acceleration of the car body, wheel-sets and bridge under the track short wave irregularity state are greatly increased compared with that without track irregularity. The maximum vertical acceleration of wheel-sets, rails, track slabs and the bridge under the track short wave irregularity state all show a significant declining trend. The larger the vibration frequency is, the smaller the range of integration time step is for dynamic calculation.

scholarly journals Comparison of numerical schemes of river flood routing with an inertial approximation of the Saint Venant equations

RBRH ◽  
2018 ◽  
Vol 23 (0) ◽  
Author(s):  
Alice César Fassoni-Andrade ◽  
Fernando Mainardi Fan ◽  
Walter Collischonn ◽  
Artur César Fassoni ◽  
Rodrigo Cauduro Dias de Paiva
Keyword(s):  
Numerical Stability ◽  
Flood Routing ◽  
Computational Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Simple Formulation ◽  
One Dimensional ◽  
Saint Venant Equations ◽  
Original Scheme ◽  
The One

ABSTRACT The one-dimensional flow routing inertial model, formulated as an explicit solution, has advantages over other explicit models used in hydrological models that simplify the Saint-Venant equations. The main advantage is a simple formulation with good results. However, the inertial model is restricted to a small time step to avoid numerical instability. This paper proposes six numerical schemes that modify the one-dimensional inertial model in order to increase the numerical stability of the solution. The proposed numerical schemes were compared to the original scheme in four situations of river’s slope (normal, low, high and very high) and in two situations where the river is subject to downstream effects (dam backwater and tides). The results are discussed in terms of stability, peak flow, processing time, volume conservation error and RMSE (Root Mean Square Error). In general, the schemes showed improvement relative to each type of application. In particular, the numerical scheme here called Prog Q(k+1)xQ(k+1) stood out presenting advantages with greater numerical stability in relation to the original scheme. However, this scheme was not successful in the tide simulation situation. In addition, it was observed that the inclusion of the hydraulic radius calculation without simplification in the numerical schemes improved the results without increasing the computational time.

scholarly journals MHDSTS: a new explicit numerical scheme for simulations of partially ionised solar plasma

2018 ◽  
Vol 615 ◽  
pp. A67 ◽  
Author(s):  
P. A. González-Morales ◽  
E. Khomenko ◽  
T. P. Downes ◽  
A. de Vicente
Keyword(s):  
Numerical Scheme ◽  
Operator Splitting ◽  
Conservation Equation ◽  
Ambipolar Diffusion ◽  
Point Of View ◽  
Integration Time ◽  
Solar Photosphere ◽  
Time Step ◽  
Diffusion Term ◽  
Fluid Approximation

The interaction of plasma with magnetic field in the partially ionised solar atmosphere is frequently modelled via a single-fluid approximation, which is valid for the case of a strongly coupled collisional media, such as solar photosphere and low chromosphere. Under the single-fluid formalism the main non-ideal effects are described by a series of extra terms in the generalised induction equation and in the energy conservation equation. These effects are: Ohmic diffusion, ambipolar diffusion, the Hall effect, and the Biermann battery effect. From the point of view of the numerical solution of the single-fluid equations, when ambipolar diffusion or Hall effects dominate can introduce severe restrictions on the integration time step and can compromise the stability of the numerical scheme. In this paper we introduce two numerical schemes to overcome those limitations. The first of them is known as super time-stepping (STS) and it is designed to overcome the limitations imposed when the ambipolar diffusion term is dominant. The second scheme is called the Hall diffusion scheme (HDS) and it is used when the Hall term becomes dominant. These two numerical techniques can be used together by applying Strang operator splitting. This paper describes the implementation of the STS and HDS schemes in the single-fluid code MANCHA3D. The validation for each of these schemes is provided by comparing the analytical solution with the numerical one for a suite of numerical tests.

Dynamic Co-Simulation Methods for Combined Transmission-Distribution System With Integration Time Step Impact on Convergence

2019 ◽  
Vol 34 (2) ◽  
pp. 1171-1181 ◽  
Author(s):  
Ramakrishnan Venkatraman ◽  
Siddhartha Kumar Khaitan ◽  
Venkataramana Ajjarapu
Keyword(s):  
Distribution System ◽  
Integration Time ◽  
Simulation Methods ◽  
Time Step

scholarly journals A 2-DIMENSIONAL CELLULAR AUTOMATON FOR AGENTS MOVING FROM ORIGINS TO DESTINATIONS

2005 ◽  
Vol 16 (12) ◽  
pp. 1849-1860 ◽  
Author(s):  
NAJEM MOUSSA
Keyword(s):  
Cellular Automaton ◽  
Cluster Size ◽  
Bimodal Distribution ◽  
Minimal Distance ◽  
Time Step ◽  
Neighbor Site ◽  
Model Dynamics ◽  
Long Time ◽  
Freely Moving ◽  
Small Clusters

We develop a two-dimensional cellular automaton (CA) as a simple model for agents moving from origins to destinations. Each agent moves towards an empty neighbor site corresponding to the minimal distance to its destination. The stochasticity or noise (p) is introduced in the model dynamics, through the uncertainty in estimating the distance from the destination. The friction parameter "μ" is also introduced to control the probability that movement of all involved agents to the same site (conflict) is denied at each time step. This model displays two states; namely the freely moving and the jamming state. If μ is large and p is low, the system is in the jamming state even if the density is low. However, if μ is large and p is high, a freely moving state takes place whenever the density is low. The cluster size and the travel time distributions in the two states are studied in detail. We find that only very small clusters are present in the freely moving state, while the jamming state displays a bimodal distribution. At low densities, agents can take a very long time to reach their destinations if μ is large and p is low (jamming state); but long travel times are suppressed if p becomes large (freely moving state).

scholarly journals IL-GLOBO (1.0) – integrated Lagrangian particle model and Eulerian general circulation model GLOBO: development of the vertical diffusion module

2014 ◽  
Vol 7 (5) ◽  
pp. 2181-2191 ◽  
Author(s):  
D. Rossi ◽  
A. Maurizi
Keyword(s):  
General Circulation Model ◽  
General Circulation ◽  
Spline Interpolation ◽  
Circulation Model ◽  
Linear Interpolation ◽  
Computational Time ◽  
Integration Time ◽  
Particle Model ◽  
Vertical Diffusion ◽  
Time Step

Abstract. The development and validation of the vertical diffusion module of IL-GLOBO, a Lagrangian transport model coupled online with the Eulerian general circulation model GLOBO, is described. The module simulates the effects of turbulence on particle motion by means of a Lagrangian stochastic model (LSM) consistently with the turbulent diffusion equation used in GLOBO. The implemented LSM integrates particle trajectories, using the native σ-hybrid coordinates of the Eulerian component, and fulfils the well-mixed condition (WMC) in the general case of a variable density profile. The module is validated through a series of 1-D offline numerical experiments by assessing its accuracy in maintaining an initially well-mixed distribution in the vertical. A dynamical time-step selection algorithm with constraints related to the shape of the diffusion coefficient profile is developed and discussed. Finally, the skills of a linear interpolation and a modified Akima spline interpolation method are compared, showing that both satisfy the WMC with significant differences in computational time. A preliminary run of the fully integrated 3-D model confirms the result only for the Akima interpolation scheme while the linear interpolation does not satisfy the WMC with a reasonable choice of the minimum integration time step.

scholarly journals ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xingjie Helen Li ◽  
Fei Lu ◽  
Felix X.-F. Ye
Keyword(s):  
Large Time ◽  
Gradient System ◽  
Numerical Schemes ◽  
Time Step ◽  
Inference Problem ◽  
Time Dynamics ◽  
Efficient Simulation ◽  
Time Flow ◽  
Ergodic Measures ◽  
Locally Lipschitz

<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>

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