Discrete Time Steps Research Articles

Convergence of regular spiking and intrinsically bursting Izhikevich neuron models as a function of discretization time with Euler method

This study investigates the trade-off between computational efficiency and accuracy of Izhikevich neuron models by numerically quantifying their convergence to provide design guidelines in choosing the limit time steps during a discretization procedure. This is important for bionic engineering and neuro-robotic applications where the use of embedded computational resources requires the introduction of optimality criteria. Specifically, the regular spiking (RS) and intrinsically bursting (IB) Izhikevich neuron models are evaluated with step inputs of various amplitudes. We analyze the convergence of spike sequences generated under different discretization time steps (10 µs to 10 ms), with respect to an ideal reference spike sequence approximated with a discretization time step of 1 µs. The differences between the ideal reference and the computed spike sequences were quantified by Victor–Purpura (VPd) and van Rossum (VRd) distances. For each distance, we found two limit discretization times (lower dt1 and upper dt2), as a function of the applied input and thus firing rate, beyond which the convergence is lost for each neuron model. The estimated limit time steps were found to be consistent regardless of metric used (VPd and VRd) and neuron type (RS and IB), but also to depend on the average inter-spike interval (ISI) produced by the neurons. However, in most cases, a good trade-off between the quality of the convergence of the models dynamics and the computational load required to simulate them numerically was found for values of the discretization time step between dt1 ∈ (0.1 ms, 1 ms) and dt2 ∈ (2 ms, 3 ms).

Collision-Free Encoding for Chance-Constrained Nonconvex Path Planning

The path planning methods based on nonconvex constrained optimization, such as mixed-integer linear programming (MILP), have found various important applications, ranging from unmanned aerial vehicles (UAVs) and autonomous underwater vehicles (AUVs) to space vehicles. Moreover, their stochastic extensions have enabled risk-aware path planning, which explicitly limits the probability of failure to a user-specified bound. However, a major challenge of those path planning methods is constraint violation between discrete time steps. In the existing approach, a path is represented by a sequence of waypoints and the safety constraints (e.g., obstacle avoidance) are imposed on waypoints. Therefore, the trajectory between waypoints could violate the safety constraints. A naive continuous-time extension results in unrealistic computation cost. In this paper, we propose a novel approach to ensure constraint satisfaction between waypoints without employing a continuous-time formulation. The key idea is to enforce that the same inequality constraint is satisfied on any two adjacent time steps, under assumptions of polygonal obstacles and straight line trajectory between waypoints. The resulting problem encoding is MILP, which can be solved efficiently by commercial solvers. Thus, we also introduce novel extensions to risk-allocation path planners with improved scalability for real-world scenarios and run-time performance. While the proposed encoding approach is general, the particular emphasis of this paper is placed on the chance-constrained, nonconvex path-planning problem (CNPP). We provide extensive simulation results on CNPP to demonstrate the path safety and scalability of our encoding and related path planners.

Rare event simulation for steady-state probabilities via recurrency cycles.

We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space Rd and discrete time steps (i.e., a discrete-time Rd-valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate models. The numerical experiments show that RMS can boost the computational efficiency by several orders of magnitude compared to the Monte Carlo method.

A unified gas-kinetic scheme for continuum and rarefied flows VI: Dilute disperse gas-particle multiphase system

In this paper, a unified gas-kinetic scheme (UGKS) for multiphase dilute gas-particle system is proposed. The UGKS multiphase (UGKS-M) is a finite volume method, which captures flow physics in the regimes from collisionless multispecies transport to the two-fluid hydrodynamic Navier-Stokes (NS) solution with the variation of Knudsen number, and from granular flow regime to dusty gas dynamics with the variation of Stokes number. The reason for preserving the multiscale nature in UGKS-M is mainly coming from the direct modeling of the flow physics in the scales of discrete cell size and time step, where the ratio of the time step to the particle collision time determines flow behavior in different regimes. For the particle phase, the time evolution solution of the kinetic model equation is used in the construction of numerical flux, which takes into account the particle transport, collision, and acceleration. The gas phase is assumed to be in the continuum flow regime and evolves numerically by the gas-kinetic scheme (GKS), which is a subset of the UGKS for the Navier-Stokes solutions. The interaction between the gas and particle phase is calculated based on a velocity space mapping method, which solves accurately the kinetic acceleration process. The stability of UGKS-M is determined by the CFL condition only. With the inclusion of the material temperature evolution equation of solid particles, once the total energy loss in inelastic collision transfers into particle material thermal energy, the UGKS-M conserves the total mass, momentum, and energy for the whole multiphase system. In the numerical tests, the UGKS-M shows multiscale property in capturing the particle trajectory crossing (PTC), particle wall reflecting phenomena, and vortex-induced segregation of inertial particles under different Stokes numbers. The scheme is also applied to simulate shock induced fluidization problem, where the simulation results agree well with experimental measurements.

Reactive contaminant transport simulation using the analytic element method, random walk particle tracking and kernel density estimator

In this study, we propose a simulation model to solve the two-dimensional groundwater flow and advection-dispersion-reaction equation (ADRE) by coupling analytic element method (AEM), random walk particle tracking (RWPT) and kernel density estimator (KDE). In the AEM-RWPT-KDE model, AEM solves the groundwater flow equation, RWPT solves the advection-dispersion-reaction equation and KDE enhances the accuracy and computational efficiency of RWPT. AEM generates a continuous velocity distribution which is suitable for RWPT. An analytic expression is modified to simulate radioactive decay for discrete transport time steps and is embedded in the RWPT model. Linear adsorption is also incorporated in the RWPT model by tracking particles with a retarded velocity. Unlike Eulerian transport models such as finite element or finite difference methods, RWPT is completely free from numerical dispersion. The comparison of results illustrates the superiority of the RWPT model over accurate Eulerian-Lagrangian models with reference to the analytical solution. Further, the AEM-RWPT-KDE model is used to simulate the transport processes of Radium-228 and Trichloroethene (TCE) in two hypothetical aquifer cases. Both case studies reflect the practical applicability of the proposed methodology.

Numerical study and comparison of alternative time discretization schemes for an ultrasonic guided wave propagation problem coupled with fluid–structure interaction

Discretization is a crucial step in the numerical treatment of coupled multiphysics problems. In the present paper, we focus on the numerical solution of ultrasonic guided waves (UGWs) propagation problem coupled with fluid–structure interaction. We track UGWs propagation through fluid, solid and their interface, where the structure is in motion. Discretization is done via a Finite Element method. For this, we follow the Rothe method, in which discretization in time is followed by space discretization, solving the resulting problem according to the monolithic approach. Further, we provide a systematic analysis of alternative discretization schemes, time-steps, and mesh refinements. Specifically, we contrast one first-order time discretization scheme (cf. backward Euler) and three second-order schemes (cf. Crank–Nicolson, shifted Crank–Nicolson, and fractional-step-θ). First, we establish the robustness of the convergence result. Second, we estimate the experimental order of convergence for alternative schemes and space–time discretization combinations.

Brownian Motion using a Piezo Actuated Microscope Stage

Single particle tracking (SPT) is an important collection of techniques and algorithms for the study of biomolecular transport and understanding the role of the dynamics of small molecules, proteins, and other organic material in health and disease on a cellular level. However successful these techniques are, it is difficult to compare them directly, especially as one is most interested in their performance on a specific experimental setup. Here, we demonstrate through Monte-Carlo simulation and experiment, the ability of a piezo actuated microscope stage to replicate faithfully a discrete-time sampled Brownian motion trajectory. Simulations consisted of an exact discrete model of the piezo stage including closed-loop control, the input Brownian motion trajectory, and feed forward model inverse control. A range of stage response times, discrete time step durations, and diffusion constants were modeled. The output of the system was compared to the input using mean squared displacement (MSD), Kullback-Leibler divergence comparing the single time step displacement distributions, and stage tracking error. We found that MSD was preserved over a large range of stage response times, discrete time step durations, and diffusion constant values. Physical experiments consisted of an epifluorescent microscope taking widefield images of quantum dots fixed to a microscope slide moved using the piezo actuated microscope stage following a Brownian motion trajectory control input. We envision this work will allow for the characterization and evaluation of SPT microscopes and localization algorithms by moving fixed fluorophores through a known Brownian motion trajectory to provide a known ground truth in an experimental setting. Future work includes using other biologically relevant motion models such as confined diffusion and elastic tethering as the motion model, and a comparison of SPT microscope configurations and estimation algorithms through simulated motion with a piezo stage.

Gen.parRep: A first implementation of the Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems

Metastability is one of the major encountered obstacles when performing long molecular dynamics simulations, and many methods were developed to address this challenge. The “Parallel Replica”(ParRep) dynamics is known for allowing to simulate very long trajectories of metastable Langevin dynamics in the materials science community, but it relies on assumptions that can hardly be transposed to the world of biochemical simulations. The later developed “Generalized ParRep” variant solves those issues, but it was not applied to significant systems of interest so far.In this article, we present the program gen.parRep, the first publicly available implementation of the Generalized Parallel Replica method (BSD 3-Clause license), targeting frequently encountered metastable biochemical systems, such as conformational equilibria or dissociation of protein–ligand complexes. It will be shown that the resulting C++ implementation exhibits a strong linear scalability, providing up to 70% of the maximum possible speedup on several hundreds of CPUs. Program summaryProgram Title: gen.parRepProgram Files doi:http://dx.doi.org/10.17632/dgp8wdgtgt.1Licensing provisions: BSD 3-clauseProgramming language: C++ (mostly), C and LuaNature of problem: Molecular dynamics simulations of chemical and biological systems usually encounter the problem of metastability, because of the timescale separation between the time discretization step used for dynamics and the usual mean time between conformational changes. The use of Accelerated dynamics [1] methods is usually necessary in order to address this challenge.Solution method: The Generalized Parallel Replica method [2] accelerates the exit from metastable states, providing a linear speedup of N, N being the number of replicas of the system running in parallel. This C++ implementation, the first available so far, exhibits a strong linear scaling on hundreds of CPUs, therefore ready for production studies on High Performance Computing (HPC) machines.Additional comments: Git repository: https://gitlab.inria.fr/parallel-replica/gen.parRep

Unbalanced multi-drawing urn with random addition matrix

In this paper, we consider a two color multi-drawing urn model. At each discrete time step, we draw uniformly at random a sample of m balls (m≥1) and note their color, they will be returned to the urn together with a random number of balls depending on the sample’s composition. The replacement rule is a 2 × 2 matrix depending on bounded discrete positive random variables. Using a stochastic approximation algorithm and martingales methods, we investigate the asymptotic behavior of the urn after many draws.

Limitation principle for computational fluid dynamics

Theoretical gas dynamics uses the physical Knudsen number $$\mathrm {Kn}_\mathrm{p}$$, which is defined as the ratio of the particle mean free path $$\lambda $$ to the characteristic length scale L, to categorize the flow into different regimes. The Boltzmann equation is the fundamental equation for dilute gases, while the Navier–Stokes (NS) equations are used for the description of continuum flow at $$\mathrm {Kn}_\mathrm{p}\le 10^{-3}$$. For computational fluid dynamics (CFD), the numerical resolution is limited by the discrete cell size and time step. Therefore, we can define a cell Knudsen number $$\mathrm {Kn}_\mathrm{c}$$ as the ratio of the particle mean free path $$\lambda $$ to the cell size $$\Delta x$$. In CFD, the numerical solution and the corresponding numerical flow regime are fully controlled by a numerical Knudsen number $$\mathrm {Kn}_\mathrm{n}$$, which is a function of the physical Knudsen number $$\mathrm {Kn}_\mathrm{p}$$ and the cell Knudsen number $$\mathrm {Kn}_\mathrm{c}$$. The limitation principle relates to the connections between $$\mathrm {Kn}_\mathrm{n}$$, $$\mathrm {Kn}_\mathrm{p}$$, and $$\mathrm {Kn}_\mathrm{c}$$. In this paper, based on the relationship between the modeling equation, cell resolution, and the physical structure thickness, we propose the division of numerical flow regimes. According to the limitation principle, the range of validity of the NS equations is extended to $$\max (\mathrm {Kn}_\mathrm{p},\mathrm {Kn}_\mathrm{c})\le 10^{-3}$$. During a mesh refinement process, in some cases the NS equations alone may not be able to capture the flow physics once the large gradients and high-frequency modes are resolved by numerical mesh size and time step. In order to obtain a physical solution in the corresponding numerical scale efficiently, a multiscale method is preferred to identify the flow physics in the corresponding cell Knudsen number $$\mathrm {Kn}_\mathrm{c}$$, such as capturing hydrodynamic wave propagation in the coarse mesh resolution case and the kinetic particle transport in the fine mesh case. The unified gas-kinetic scheme (UGKS) is such a multiscale method for providing continuum, near-continuum, and non-equilibrium solutions with a variation of cell Knudsen number. Numerical examples with different physical Knudsen numbers are calculated under different cell Knudsen numbers. These results show the mesh size effect on the numerical representation of a physical solution. In comparison with the NS and direct Boltzmann solvers, the multiscale UGKS is able to capture flow physics in different regimes seamlessly with a variation of numerical resolution.

Spatial analysis of modal time evolution in room acoustics.

Time-domain simulation methods allow the observation of the state of the simulation domain at each discrete time step. In this work an approach to analyze the progress of the sound field in the simulation domain using time-windowing and spectrum analysis is presented. The method makes it possible to analyze the effect of geometric structures into the spatiotemporal distribution of energy in the domain at a frequency range of interest. Several examples are presented.

A New Operator Splitting Method for the Euler Elastica Model for Image Smoothing

Euler's elastica model has a wide range of applications in Image Processing and Computer Vision. However, the non-convexity, the non-smoothness and the nonlinearity of the associated energy functional make its minimization a challenging task, further complicated by the presence of high order derivatives in the model. In this article we propose a new operator-splitting algorithm to minimize the Euler elastica functional. This algorithm is obtained by applying an operator-splitting based time discretization scheme to an initial value problem (dynamical flow) associated with the optimality system (a system of multivalued equations). The sub-problems associated with the three fractional steps of the splitting scheme have either closed form solutions or can be handled by fast dedicated solvers. Compared with earlier approaches relying on ADMM (Alternating Direction Method of Multipliers), the new method has, essentially, only the time discretization step as free parameter to choose, resulting in a very robust and stable algorithm. The simplicity of the sub-problems and its modularity make this algorithm quite efficient. Applications to the numerical solution of smoothing test problems demonstrate the efficiency and robustness of the proposed methodology.

A Simulation Technique for Radiation Properties of Time-Varying Media Based on Frequency-Domain Solvers

A novel simulation technique is proposed for the radiation properties of time-varying media. This method divides the time-variation process into finite discrete time steps. The radiation field of the media at each instant in time is simulated by using commercially available frequency-domain solvers, and then these results are processed and synthesized to obtain the radiation field of the time-varying media. In other words, by using this technique, the radiation field of the time-varying media can be obtained based on the frequency-domain simulation results. The proposed technique is introduced with a time-varying grating, and the scattered wave and radiation pattern are illustrated. Moreover, an analytical model of the time-varying grating is also introduced to further explore the proposed technique. This simulation technique can be used to analyze many applications such as the radiation properties of time-modulated media, especially space-time-modulated metamaterials. It fills the gap of existing simulation methods and provides a new way for direct EM simulation of time-varying structures.

A method for determining information diffusion cascades on social networks

Information diffusion on social networks has many potential real-world applications such as online marketing, e-government campaigns, and predicting large social events. Modeling information diffusion is therefore a crucial task in order both to understand its diffusion mechanism and to better control it. Our research aims at finding what factors might influence people in adopting a piece of information that is being shared on a social network. In this study, the traditional independent cascade model for information diffusion is extended with discrete time steps. The proposed model is capable of incorporating three different sources of diffusion influence: user-user influence, user-content preference, and external influence. Specifically, these sources of influence are quantified into real values of diffusion probability. To calculate user-user influence, we adopt and extend the disease transmission model according to the role of the user who diffuses the content. User-content preference, which measures the correlation between user preference and the adopted contents, is calculated based on a topic-based model. External influence is detected in a diffusion time step and is quantified and incorporated into our model for the next diffusion time step by applying and solving a logistic function. Moreover, the process of information diffusion is characterized by constructing a tree of information adoption and the diffusion scale is quantified by predicting the number of infected nodes. It is found that these sources of influence, especially external influence, play a significant role in information diffusion and eventually affect the shape and size of the diffusion cascade. The model is validated on both synthetic and real-world datasets. Experimental results confirm the advantage of our proposed method, which significantly improves over the previous models in terms of prediction accuracy

Interval methods for robust gain scheduling controllers

A novel interval-based approach for a gain-scheduled controller synthesis is presented in this paper which aims at stabilizing continuous-time dynamic systems of finite dimension in a guaranteed manner over predefined finitely long time horizons. The fundamental aim of the design is the temporal adaptation of the feedback gain in combination with a reduction of the interval widths which characterize outer enclosures of the states that are reachable in the worst case at a certain time instant while ensuring asymptotic stability of the closed-loop dynamics. For that purpose, feedback gains are computed first for an initial state enclosure. Second, they are tested for validity in such a way that the parameterization of the controller shall be applicable over the whole time discretization step. In case of the verification failing, the gain is modified after determining a bounding box for those states that can be reached over the considered prediction window. That means, an offline calculation of controller gains is possible so that prespecified performance indicators with respect to the closed-loop dynamics are guaranteed to be satisfied. The robust and/or optimal control problem is solved efficiently using linear matrix inequality (LMI) techniques, while methods from interval analysis are furthermore employed during the underlying reachability analysis. Additionally, efficient approaches are presented which reduce overestimation due to the unavoidable wrapping effect. Hence, the proposed design method aims, simultaneously, at the reduction of overestimation and a guaranteed stability verification. To conclude this paper, the resulting control strategy is verified numerically for an inverted pendulum as a prototypical benchmark application.

Dynamics of interdependent critical infrastructures – A mathematical model with unexpected results

The article develops a formal mathematical model which describes the dynamic interactions of interdependent critical infrastructures in discrete time steps, i.e. the direct and indirect consequences (cascading effects) of a disrupted infrastructure to depending infrastructures. The main result here is that a disturbed infrastructure with only a reduced performance level draws (on the long run) all other infrastructures to the same reduced performance level. Also the development of all infrastructures after a full repair of the disturbances is investigated. The related result is that even after a full repair the infrastructures cannot return to full performance without external support. Both results are proved as mathematical theorems. Visualized examples are shown to demonstrate the results.

Limiting behavior of 3-color excitable media on arbitrary graphs

Fix a simple graph $G=(V,E)$ and choose a random initial 3-coloring of vertices drawn from a uniform product measure. The 3-color cycle cellular automaton is a process in which at each discrete time step in parallel, every vertex with color $i$ advances to the successor color $(i+1)\bmod 3$ if in contact with a neighbor with the successor color, and otherwise retains the same color. In the Greenberg–Hastings model, the same update rule applies only to color 0, while other two colors automatically advance. The limiting behavior of these processes has been studied mainly on the integer lattices. In this paper, we introduce a monotone comparison process defined on the universal covering space of the underlying graph, and characterize the limiting behavior of these processes on arbitrary connected graphs. In particular, we establish a phase transition on the Erdős–Renyi random graph. On infinite trees, we connect the rate of color change to the cloud speed of an associated tree-indexed walk. We give estimates of the cloud speed by generalizing known results to trees with leaves.

CEASE: A Collaborative Event-Triggered Average-Consensus Sampled-Data Framework With Performance Guarantees for Multi-Agent Systems

The paper proposes a distributed framework for collaborative, event-triggered, average consensus, sampled data (CEASE) algorithms for undirected networked multi-agent systems with two classes of performance guarantees. Referred to as the E-CEASE algorithm, the first approach ensures an exponential rate of convergence and derives associated conditions and optimal design parameters using the Lyapunov stability theorem. The second approach provides a structured tradeoff between the number of transmissions and rate of consensus convergence based on a guaranteed cost and is referred to as the G-CEASE. The distributed implementations of CEASE are event-driven in the sense that agents transmit within their respective neighborhoods only on the triggering of an event. To reduce communication and processing, the triggering condition in CEASE is monitored at discrete-time steps. Monte-Carlo simulations on randomized networks quantify the effectiveness of the proposed approaches.

LES for pedestrian level wind around an idealized building array—Assessment of sensitivity to influencing parameters

Large eddy simulation (LES) is increasingly recognized as an important approach in the assessment of pedestrian level wind (PLW) conditions, while its numerical performance is sensitive to a few key parameter settings. This study aims to further identify the sensitivities to these main influencing parameters in modeling wind flow around a building array using LES. This is achieved by quantitatively comparing mean wind velocities from the LES predictions and those from a benchmark wind tunnel experiment. The parameters investigated are the mesh resolution, discretization time step and sampling period, vortices’ number of the inlet flow, the upstream distance of the computational domain, and the sub-grid scale (SGS) models. Results are quantified using four validation indices. Specifically, the correlation coefficient R between the predicted results using SGS model with the dynamic Smagorinsky-Lilly method and the experimental results is 0.89, which is higher than those between the experiment results and other three SGS models. Based on the sensitivity tests, it is recommended that the normalized discretization time step be set below 0.09, and that the normalized sampling period and vortices’ number be above 219 and 190, respectively, in the LES simulation of PLW flows around a building array.

Integral approach to energy planning and electric grid assessment in a renewable energy technology integration for a 50/50 target applied to a small island

This paper presents an energy planning, a grid assessment, and an economic analysis, considering three growing scenarios (Low, Base and High) in the electricity consumption, to supply the energy demand for a hybrid power system (Photovoltaics/Wind/Diesel/Battery) on a small island by 2050. The main aim of this study is to present a methodology to optimize and reduce the backup time of the battery bank, included from the hybrid power generation system selected. Also, it will compare four different battery technologies, simultaneously, without changes in the renewable energy targets settled in 50% until 2050 and without changes in the safe continuous operation of the grid. The methodology includes a grid assessment analysis to obtain a reliable, strong and safe operation response based on the grid code parameters, even in case of disturbance.In the proposed methodology the analysis is developed on the basis of the use of two simulation model tools. The First simulation model tool determines the optimal values of variables that the system designer controls, such as the mix of components (Photovoltaics/Wind/Diesel/Battery) that make up the system and the size or quantity of each variable. This model uses the multiyear analysis based on a time-domain simulation run at the energy-flow level with discrete time-steps of 1 h. The Second simulation model tool assumes all the variables and parameters on the grid as constants during the period of the time analyzed. The power flow is analyzed through a programming language command script function and reflects the system response at a specific time with given specific variables and parameters. The final technical proposal and its financial analysis are obtained applying and validating this methodology on a small island, as well as, the selection of the system to be installed for the renewable electricity generation. The electric grid modifications and reinforcements through the years until 2050, according to the grid code and the renewable energy targets settled for the island’s electric power system are included.