Discrete Time Steps Research Articles

DIGITAL IMAGE REDUCTION FOR EFFICIENT COMPUTATION TOPOLOGICAL CHANGES IN THE ROCK MATRIX

The article describes the application of the digital image reduction algorithm to speed up the calculation of persistent diagrams that describe changes in the topology of the pore space of the rock matrix during the dissolution process. The dependence of the efficiency of the reduction algorithm on the properties of the rock sample and the value of the discrete time step is shown.

Numerical analysis of a bi-modal covid-19 SITR model

This study presents a structure preserving nonstandard finite difference scheme to analyze a susceptible-infected-treatment-recovered (SITR) dynamical model of coronavirus 2019 (covid-19) with bimodal virus transmission in susceptible population. The underlying model incorporates the possible treatment measures as the emerging scenario of covid-19 vaccines. Keeping in view the fact that the real time data for covid-19 is updated at discrete time steps, we propose a new structure preserving numerical scheme for the proposed model. The proposed numerical scheme produces realistic solutions of the complex bi-modal SITR nonlinear model, converges unconditionally to steady states and reflects dynamical consistency with continuous sense of the model. The analysis of the model reveals that the model remains stable at the steady state points. The basic reproduction number Rcovid falls less than 1 when treatment rate is increased and disease will die out. On the other hand, it predicts that human population may face devastating effects of pandemic if the treatment measures are not strictly implemented.

Optimizing the cellular automata finite element model for additive manufacturing to simulate large microstructures

This study proposes an implementation of the cellular automata finite element (CAFE) model in order to optimize its performance for simulating the solidification of additively manufactured (AM) materials. The translating melt pool in AM leads to a highly localized region of activity at any given time. The time scale associated with the evolving temperature field is much larger than the time scale associated with grain solidification. A separation of temporal scales is proposed such that solidification analysis is treated independently as sub-cycles within each discrete time step of the scanning laser. At each laser time step, all of the domain is idle except a small portion, which is the region where the material is undercooled but has yet to be solidified. This region is identified, and a second partition of the computational resources is established such that the solidification computations are evenly balanced. Numerical studies demonstrate that the proposed implementation achieves dramatic improvement in parallel scalability than previously reported and thus advances the state-of-the-art of CAFE modeling for AM in terms of computational efficiency. This improved computational performance is necessary to simulate the large 3D polycrystalline microstructures required to evaluate texture and grain morphology, and two of such simulations are presented. The model is validated for laser powder bed fusion 316L stainless steel, where the polycrystalline grain morphology and crystallographic texture of the simulated microstructure closely match experimental characterization data. A second simulation is performed for a different scan pattern in order to highlight and evaluate its effect on microstructural features.

State-space based discretize-then-differentiate adjoint sensitivity method for transient responses of non-viscously damped systems

In this paper, a new design sensitivity analysis (DSA) method for transient responses of non-viscously damped systems is developed. The damping force of the non-viscously damped systems depends on the past history of motion and is represented by convolution integrals over suitable kernel functions. The equations of motion are first transformed into a state-space formulation and the time-history analysis of the responses relies on a modified precise integration method (MPIM). The residual equation in each discrete time step is achieved by a recurrence formula of the MPIM. The sensitivities of the transient responses are calculated with an adjoint variable method (AVM). In particular, a discretize-then-differentiate approach is adopted for the DSA. The accuracy, consistency, implementation effort and computational complexity are discussed. Two numerical examples are illustrated to show the performances of the proposed method compared with other two methods. The results indicate that, when calculating the sensitivities of the transient responses for non-viscously damped systems based on the MPIM, the order of differentiation and discretization has no obvious effect on the consistency. The proposed state-space based discretize-then-differentiate AVM is more efficient than other two methods.

New alternative numerical approaches for solving the glioma model and their efficiencies

In this article, the numerical solution of glioma, or glioblastomas, which is one of the most aggressive forms of cancer is considered. A heterogeneous nonlinear diffusion logistic density model is taken as the main focus. To obtain the numerical results, three different discretization techniques: Pseudospectral method (PSM) using Chebyshev–Gauss–Lobatto collocation points, method of lines (MoL), and cubic B-splines (cBS) are employed on the spatial domain, whereas 4th-order Runge–Kutta (RK4) is considered on the time domain. Adapting cBS and PSM discretization to the glioma model is studied at first in this study. In addition to the theoretical convergence results, detailed comparative computational results are presented. All these methods are compared in terms of their efficiencies in varying time step and mesh discretization not only to one another, but also with the methods given in the literature.

Numerical simulation of quantum nonequilibrium phase transitions without finite-size effects

Classical $(1+1)D$ cellular automata, as for instance Domany-Kinzel cellular automata, are paradigmatic systems for the study of non-equilibrium phenomena. Such systems evolve in discrete time-steps, and are thus free of time-discretisation errors. Moreover, information about critical phenomena can be obtained by simulating the evolution of an initial seed that, at any finite time, has support only on a finite light-cone. This allows for essentially numerically exact simulations, free of finite-size errors or boundary effects. Here, we show how similar advantages can be gained in the quantum regime: The many-body critical dynamics occurring in $(1+1)D$ quantum cellular automata with an absorbing state can be studied directly on an infinite lattice when starting from seed initial conditions. This can be achieved efficiently by simulating the dynamics of an associated one-dimensional, non-unitary quantum cellular automaton using tensor networks. We apply our method to a model introduced recently and find accurate values for universal exponents, suggesting that this approach can be a powerful tool for precisely classifying non-equilibrium universal physics in quantum systems.

Evaluating SWMM capabilities to simulate closed pipe transients

One of the most used 1D tools to model collection systems is the Storm Water Management Model (SWMM). Solving the full form of the Saint Venant equations, this model represents typical unsteady flow conditions in sewer systems. However, it may be insufficient to address fast transient flow conditions that can be present during extreme events or unscheduled operational conditions. SWMM version 5.1.013 was implemented the Preissmann slot as an alternative method to handle pressurization. To date, no studies were found analysing the applicability of SWMM to represent closed-pipe fast transient flows applying the Preissmann slot pressurization algorithm. Therefore, the present work investigates the ability to model different fast transients conditions in SWMM under various spatial and temporal discretizations along with variations in the Preissmann slot algorithm. Using an alternative implementation of the SLOT pressurization algorithm, along with artificial spatial discretization and routing time-steps estimated by the Courant stability condition, it is shown that SWMM is capable to perform satisfactory certain types of closed pipe transient simulations.

UAV Trajectory Planning With Probabilistic Geo-Fence via Iterative Chance-Constrained Optimization

Chance-constrained optimization provides a promi- sing framework for solving control and planning problems with uncertainties, due to its modeling capability to capture randomness in real-world applications. In this paper, we consider a UAV trajectory planning problem with probabilistic geo-fence, building on the chance-constrained optimization approach. In the considered problem, randomness of the model, such as the uncertain boundaries of geo-fences, is incorporated in the formulation. By solving the formulated chance-constrained optimization with a novel sampling based solution method, the optimal UAV trajectory is achieved while limiting the probability of collision with geo-fences to a prefixed threshold. Furthermore, to obtain a totally collision-free trajectory, i.e., avoiding the collision not only at the discrete time-steps but also within the entire time horizon, we build on the idea of an iterative scheme. That is, to iterate the solving of the chance-constrained optimization until the collision with probabilistic geo-fence is avoided at any time within the time horizon. At last, we validate the effectiveness of our method via numerical simulations.

The effect of time discretization on the propeller hydrodynamic performance simulation in self-propulsion and open water conditions

Numerical simulation technology is widely used in the field of naval architect and marine engineering. The flow induced by the propeller is unsteady. In order to demonstrate the effect of time discretization format and time step on the simulation of the flow, this paper takes an oil tanker and its propeller as the research objects, and simulations have been conducted by using OpenFoam. By comparing the thrust, torque and the vortex after the propeller, the effects of the time discretization format and time step on the numerical simulations are analyzed. This paper believes that changing the discretization format of the time term while keep the time step unchanged has little effect on the calculation results in the condition of open water test simulation. It is beneficial to simulate of the vortex after the propeller hub by using a small time step. The first-order backward Euler discretization format in the condition of self propulsion simulation is recommended. If the error of thrust compared with experiment data is larger than expected, the second-order discretization format should be considered. In order to improve the accuracy of numerical simulation, a smaller time step could be considered.

Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

AbstractThis paper introduces a semi‐discrete implicit Euler (SDIE) scheme for the Allen‐Cahn equation (ACE) with fidelity forcing on graphs. The continuous‐in‐time version of this differential equation was pioneered by Bertozzi and Flenner in 2012 as a method for graph classification problems, such as semi‐supervised learning and image segmentation. In 2013, Merkurjev et. al. used a Merriman‐Bence‐Osher (MBO) scheme with fidelity forcing instead, as heuristically it was expected to give similar results to the ACE. The current paper rigorously establishes the graph MBO scheme with fidelity forcing as a special case of an SDIE scheme for the graph ACE with fidelity forcing. This connection requires the use of the double‐obstacle potential in the ACE, as was already demonstrated by Budd and Van Gennip in 2020 in the context of ACE without a fidelity forcing term. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the discrete time step converges to zero. In the second part of the paper we develop the SDIE scheme as a classification algorithm. We also introduce some innovations into the algorithms for the SDIE and MBO schemes. For large graphs, we use a QR decomposition method to compute an eigendecomposition from a Nyström extension, which outperforms the method used by, for example, Bertozzi and Flenner in 2012, in accuracy, stability, and speed. Moreover, we replace the Euler discretization for the scheme's diffusion step by a computation based on the Strang formula for matrix exponentials. We apply this algorithm to a number of image segmentation problems, and compare the performance with that of the graph MBO scheme with fidelity forcing. We find that while the general SDIE scheme does not perform better than the MBO special case at this task, our other innovations lead to a significantly better segmentation than that from previous literature. We also empirically quantify the uncertainty that this segmentation inherits from the randomness in the Nyström extension.

A User-Centric Mechanism for Sequentially Releasing Graph Datasets under Blowfish Privacy

In this article, we present a privacy-preserving technique for user-centric multi-release graphs. Our technique consists of sequentially releasing anonymized versions of these graphs under Blowfish Privacy. To do so, we introduce a graph model that is augmented with a time dimension and sampled at discrete time steps. We show that the direct application of state-of-the-art privacy-preserving Differential Private techniques is weak against background knowledge attacker models. We present different scenarios where randomizing separate releases independently is vulnerable to correlation attacks. Our method is inspired by Differential Privacy (DP) and its extension Blowfish Privacy (BP). To validate it, we show its effectiveness as well as its utility by experimental simulations.

A posteriori choice of time-discretization step in finite difference methods for solving ill-posed Cauchy problems in Hilbert space

Abstract Finite difference semidiscretization methods for solving an ill-posed Cauchy problem in a Hilbert space are investigated. The problems involve linear positively definite selfadjoint operators. We justify an a posteriori scheme for the choice of the time-discretization step and establish accuracy estimates in terms of the error level of input data.

Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models

We consider the problem of parameter estimation for a class of continuous-time state space models (SSMs). In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function, as described, e.g., in [P. Del Moral, A. Doucet, and S. S. Singh, M$2$AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time, where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward-type Feynman--Kac formula in continuous time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.

Dynamics of interdependent critical infrastructures - a prioritised distribution model

Previous research articles about interdependent critical infrastructures developed different mathematical models describing the dynamic direct and indirect interactions of these infrastructures. Many of these models use discrete time steps to show how a reduced operability of one infrastructure C at time t will lead at time t + 1 to reductions in all infrastructures to which C distributes its output; and many of them use a standard mode of distribution which will be called proportional. According to a recent result, with this proportional distribution a reduced operability in only one infrastructure will lead to an enduring reduction of all operabilities even after full repairs of all damages. This research replaces this proportional mode by a newly defined prioritised one. It shows that a system with prioritised distribution mode leads in nearly all practical cases to an increased behaviour of the whole system: after repair of all damages, all infrastructures will return automatically to full operability.

Hamiltonian Monte Carlo method for estimating variance components.

A Hamiltonian Monte Carlo algorithm is a Markov chain Monte Carlo method, and the method has a potential to improve estimating parameters effectively. Hamiltonian Monte Carlo is based on Hamiltonian dynamics, and it follows Hamilton's equations, which are expressed as two differential equations. In the sampling process of Hamiltonian Monte Carlo, a numerical integration method called leapfrog integration is used to approximately solve Hamilton's equations, and the integration is required to set the number of discrete time steps and the integration stepsize. These two parameters require some amount of tuning and calibration for effective sampling. In this study, we applied the Hamiltonian Monte Carlo method to animal breeding data and identified the optimal tunings of leapfrog integration for normal and inverse chi-square distributions. Then, using real pig data, we revealed the properties of the Hamiltonian Monte Carlo method with the optimal tuning by applying models including variance explained by pedigree information or genomic information. Compared with the Gibbs sampling method, the Hamiltonian Monte Carlo method had superior performance in both models. We have provided the source codes of this method written in the Fortran language at https://github.com/A-ARAKAWA/HMC.

A minimizing Movement approach to a class of scalar reaction–diffusion equations

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction–diffusion equations of the form [see formula in PDF] with parameters Λ, Σ > 0 and no-flux boundary condition [see formula in PDF] which is built on their gradient-flow-like structure in the space [see formula in PDF] of finite nonnegative Radon measures on [see formula in PDF], endowed with the recently introduced Hellinger-Kantorovich distance HKΛ,Σ. It is proved that, under natural general assumptions on [see formula in PDF] and [see formula in PDF], the Minimizing Movement scheme [see formula in PDF] for [see formula in PDF] yields weak solutions to the above equation as the discrete time step size τ ↓ 0. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance HKΛ,Σ, which will play an important role in this context, is established in the general setting of a separable Hilbert space; that result will constitute a starting point for the study of the differentiability of HKΛ,Σ along absolutely continuous curves which will be carried out in a subsequent paper.

Effects of fuel blending on first stage and overall ignition processes

The effects of blending ratio on mixtures of an alcohol-to-jet (ATJ) fuel and a conventional petroleum-derived fuel on first stage ignition and overall ignition delay are examined at engine-relevant ambient conditions. Experiments are conducted in a high-temperature pressure vessel that maintains a small flow of dry air at the desired temperature (825 K and 900 K) and pressure (6 MPa and 9 MPa) for fuel injections from a custom single-hole, axially-oriented injector, representing medium (7.5 mg) and high (10 mg) engine loading. Formaldehyde, imaged using planar laser-induced fluorescence, is measured at discrete time steps throughout the first and second stage ignition process and is used as a marker of unburned short-chain hydrocarbons formed after the initial breakdown of the fuel. The formaldehyde images are used to calculate the first stage ignition delay for each ambient and fuel loading condition. Chemiluminescence imaging of excited hydroxyl radical at 75 kHz is used to determine the overall ignition delay. At all conditions, increased volume fraction of ATJ resulted in longer, but non-linearly increasing, overall ignition delay. Across all of the blends, first stage ignition delay accounted for about 15% of the increase in overall ignition delay compared to the military's aviation kerosene, F-24, which is Jet A with additives, while extended first stage ignition duration accounted for 85% of the increase. It is observed that blends consisting of 0–60% by volume of the low cetane number ATJ fuel produced nearly identical first stage ignition delays. These results will inform the development of ignition models that can capture the non-linear effects of fuel blending on ignition processes.

Integrating machine learning with Markov chain and cellular automata models for modelling urban land use change

Modelling urban land use change is of profound concern to environmental scientists who have found cellular automata models very attractive for simulating urban dynamics. The quest for suitable predictive models to improve realistic simulation of urban land use change has resulted in the use of several notable cellular automata calibrations. Cellular automata model has become very attractive and one of the strongest models for urban growth simulation due to its simplicity and possibility of evolution. However, the inability of cellular automata to include driving forces of urban growth in the simulation process has warranted further cellular automata calibrations to minimize this weakness. To address this problem, and contrary to previous cellular automata calibrations, this research presents a novel integration of support vector machine, Markov chain and cellular automata for urban change modelling. Support vector machine is introduced as a machine learning technique to mine the impact of the explanatory variables that drive urban change. Markov chain is employed to mine the urban transition probabilities between the various urban epochs while cellular automata are used to implement the incremental discrete time steps based on neighbourhood interaction from an initial time to a future time. This modelling is implemented using Landsat data acquired in 1984, 2000 and 2015 over Lagos in Nigeria, Africa’s most populous city. Urban transitions (1984–2000 and 2000–2015) are used to simulate future urban state in 2030 and validation metrics include McNemar's test. The introduction of stochasticity into the model helps create the typical randomness inherent in the real world for deriving future urban forms through discrete cellular automata iterations. The high accuracy obtained in this experiment implies a substantial fit between the predicted and reference data. This outcome proves the robustness of this method for modelling urban change.

Discrete event optimization of a vehicle charging station with multiple sockets

The relevance and presence of Electric Vehicles (EVs) are increasing all over the world since they seem an effective way to fight pollution and greenhouse gas emissions, especially in urban areas. One of the main issues related to EVs is the necessity of modifying the existing infrastructure to allow the installation of new charging stations (CSs). In this scenario, one of the most important problems is the definition of smart policies for the sequencing and scheduling of the vehicle charging process. The presence of intermittent energy sources and variable execution times represent just a few of the specific features concerning vehicle charging systems. Even though optimization problems regarding energy systems are usually considered within a discrete time setting, in this paper a discrete event approach is proposed. The fundamental reason for this choice is the necessity of limiting the number of the decision variables, which grows beyond reasonable values when a short time discretization step is chosen. The considered optimization problem regards the charging of a series of vehicles by a CS connected with a renewable energy source, a storage element, and the main grid. The objective function to be minimized results from the weighted sum of the (net) cost for purchasing energy from the external grid, the weighted tardiness of the services provided to the customers, and a cost related to the occupancy of the socket during the charging. The approach is tested on a real case study. The limited computational burden allows also the implementation in real-case applications.

Low-Complexity Decentralized Algorithm for Aggregate Load Control of Thermostatic Loads

Thermostatically controlled loads such as refrigerators are exceptionally suitable as a flexible demand resource. This paper derives a decentralised load control algorithm for refrigerators. It is adapted from an existing continuous time control approach, with the aim to achieve low computational complexity and an ability to handle discrete time steps of variable length -- desirable features for embedding in appliances and high-throughput simulations. Simulation results of large populations of heterogeneous appliances illustrate the accurate aggregate control of power consumption and high computational efficiency. Tracking accuracy is quantified as a function of population size and time step size, and correlations in the tracking error are investigated. The controller is shown to be robust to errors in model specification and to sudden perturbations in the form of random refrigerator door openings.