Discrete Time Steps Research Articles

Temporally and spatially segregated discretization for a coupled electromechanical myocardium model

Abstract In this paper, we propose a novel temporally and spatially segregated numerical scheme to discretize the coupled electromechanical model of myocardium. We perform several numerical experiments with activation of a myocardial slab with structural inhomogeneity and evaluate the dependence of numerical errors on the size of spatial and temporal discretization steps. In our study, we show that the spatial step for the mechanical equations h m ⩽2.5 mm yields reasonable results with noticeable errors only in the region of myocardial inhomogeneity. We also show that time step τ m ⩽1 ms can be used for temporal discretization of mechanical equations, and the propagation velocity of the activation and contraction fronts differs from the reference one by no more than 1.3%for such time step. Finally, we show that the increase of time discretization steps of the mechanical equations τ m and the monodomain equation τ e leads to phase errors with opposite signs, and we can compensate these errors by tuning the relationship between the time steps.

Domain decomposition with local time discretization for the nonlinear Stokes–Biot system

This work presents a domain decomposition method for the fluid-poroelastic structure interaction (FPSI) system, which utilizes local time integration for subproblems. To derive the domain decomposition scheme, we introduce a Lagrange multiplier and define time-dependent Steklov–Poincaré-type operators based on the interface conditions. These operators are employed to transform the coupled system into an evolutionary nonlinear interface problem, which is then solved using an iterative algorithm. This approach provides the flexibility to use different time discretization schemes and step sizes in subdomains, making it an efficient method for simulating multiphysics systems. We present numerical tests for both non-physical and physical problems to demonstrate the accuracy and efficiency of this method.

A new lattice approach for risk-minimization hedging under generalized autoregressive conditional heteroskedasticity models

This paper explores the calculation of risk-minimization hedging strategies, specifically local and global risk minimization strategies for contingent claims under affine and non-affine GARCH models with the known closed forms of its first four moments across times under the physical measure. A unified and efficient willow tree method is introduced for various GARCH models. Unlike methods that provide option values and hedging ratios solely at the inception time, the proposed willow tree method generates a comprehensive table of option values and hedging ratios at each discrete time step across possible asset prices. Additionally, the method showcases robust performance in hedging at lower frequencies than the underlying asset’s modeling frequency (e.g., weekly or monthly hedging using a daily GARCH model). Lastly, the willow tree method outperforms the Monte Carlo method, offering greater efficiency, accuracy, and flexibility in solving risk-minimization hedging problems.

The complexity of growing a graph

We study a new algorithmic process of graph growth which starts from a single initial vertex and operates in discrete time-steps, called slots. In every slot, the graph grows via two operations (i) vertex generation and (ii) edge activation. The process completes at the last slot where a (possibly empty) subset of the edges of the graph are removed. Removed edges are called excess edges. The main problem investigated in this paper is: Given a target graph G, design an algorithm that outputs a process that grows G, called a growth schedule. Additionally, we aim to minimize the total number of slots k and of excess edges ℓ used by the process. We provide both positive and negative results, with our main focus being either schedules with sub-linear number of slots or with no excess edges.

Intrinsic Universality in Automata Networks III: On Symmetry versus Asynchrony

An automata network is a finite assembly of interconnected entities endowed with a set of local maps defined over a common finite alphabet. These relationships are represented through an interaction graph. Together with the local functions, an assignment known as an update scheme directs the evolution of the network by updating specific subsets of entities at discrete time steps. Despite the scrutiny of interaction graphs and update schemes, their profound impact on automata network dynamics remains largely open. This work investigates the intricate interplay between these aspects, with a focus on how update schemes can counterbalance constraints stemming from symmetric local interactions. This paper is the third of a series about intrinsic universality, a notion that assesses both dynamical and computational complexity, encompassing transient behaviors, attractors, and prediction or reachability problems. We consider four update modes—parallel, block-sequential, local clocks, and general periodic— along with several families of symmetric signed conjunctive boolean networks defined by local constraints on signs. Our main result is to show a diagonal complexity leap in this two-dimensional landscape: the stronger the local constraints the higher the level of asynchrony required to obtain intrinsic universality or increase in complexity. We also show how in some cases asynchronism allows to simulate directed interactions from undirected ones with the same local rules.

A Review of Differential Equations that is applicable to Differential Difference Equations

Mathematical modeling relies heavily on differential equations because they offer a robustframework for studying and understanding dynamical systems. However, the study of differential differenceequations (DDEs) has evolved as a subfield within differential equations due to the increasing prevalence ofsystems with discrete time steps and delayed effects. The purpose of this work is to present a synopsisof research on differential equations that may be used in the analysis of differential difference equations.The review starts with a thorough introduction to ODEs and PDEs (ordinary and partial differentialequations). Fundamental ideas, analytical approaches, and numerical strategies for solving thesecontinuous-time problems are discussed. ODEs and PDEs are highlighted for their importance insimulating a wide range of physical, biological, and engineering systems.

Visually communicating pathological changes: A case study on the effectiveness of phong versus outline shading

In this paper, we investigate the suitability of different visual representations of pathological growth and shrinkage using surface models of intracranial aneurysms and liver tumors. By presenting complex medical information in a visually accessible manner, audiences can better understand and comprehend the progression of pathological structures. Previous work in medical visualization provides an extensive design space for visualizing medical image data. However, determining which visualization techniques are appropriate for a general audience has not been thoroughly investigated.We conducted a user study (n = 40) to evaluate different visual representations in terms of their suitability for solving tasks and their aesthetics. We created surface models representing the evolution of pathological structures over multiple discrete time steps and visualized them using illumination-based and illustrative techniques. Our results indicate that users’ aesthetic preferences largely coincide with their preferred visualization technique for task-solving purposes. In general, the illumination-based technique has been preferred to the illustrative technique, but the latter offers great potential for increasing the accessibility of visualizations to users with color vision deficiencies.

Discrete-time optimal control problems with time delay argument: New discrete-time Euler–Lagrange equations with delay

Discrete-time optimal control problems are a crucial type of control problems that deal with a dynamic system evolving in discrete time-steps. This paper introduces a new technique for solving linear discrete-time optimal control problems with state delays, applicable to both finite and infinite time horizons. Our method employs a Riccati matrix equation, optimizing control strategies and ensuring system stability through bounded control inputs. We adopt a Bolza problem for the performance index, which guides the classification of control issues. The technique simplifies problems into manageable Riccati matrix equations using the Euler–Lagrange equations and Pontryagin maximum principle, ensuring stability and necessary condition compliance. The paper validates the approach with numerical examples from quantum mechanics and classical physics, demonstrating its practicality and potential for broader application.

Krylov-based adaptive-rank implicit time integrators for stiff problems with application to nonlinear Fokker-Planck kinetic models

We propose a high order adaptive-rank implicit integrators for stiff time-dependent PDEs, leveraging extended Krylov subspaces to efficiently and adaptively populate low-rank solution bases. This allows for the accurate representation of solutions with significantly reduced computational costs. We further introduce an efficient mechanism for residual evaluation and an adaptive rank-seeking strategy that optimizes low-rank settings based on a comparison between the residual size and the local truncation errors of the time-stepping discretization. We demonstrate our approach with the challenging Lenard-Bernstein Fokker-Planck (LBFP) nonlinear equation, which describes collisional processes in a fully ionized plasma. The preservation of the equilibrium state is achieved through the Chang-Cooper discretization, and strict conservation of mass, momentum and energy via a Locally Macroscopic Conservative (LoMaC) procedure. The development of implicit adaptive-rank integrators, demonstrated here up to third-order temporal accuracy via diagonally implicit Runge-Kutta schemes, showcases superior performance in terms of accuracy, computational efficiency, equilibrium preservation, and conservation of macroscopic moments. This study offers a starting point for developing scalable, efficient, and accurate methods for high-dimensional time-dependent problems.

Accuracy of discrete- and continuous-time mean-field theories for epidemic processes on complex networks.

Discrete- and continuous-time approaches are frequently used to model the role of heterogeneity on dynamical interacting agents on the top of complex networks. While, on the one hand, one does not expect drastic differences between these approaches, and the choice is usually based on one's expertise or methodological convenience, on the other hand, a detailed analysis of the differences is necessary to guide the proper choice of one or another approach. We tackle this problem by investigating both discrete- and continuous-time mean-field theories for the susceptible-infected-susceptible (SIS) epidemic model on random networks with power-law degree distributions. We compare the discrete epidemic link equations(ELE) and continuous pair quenched mean-field (PQMF) theories with the corresponding stochastic simulations, both theories that reckon pairwise interactions explicitly. We show that ELE converges to the PQMF theory when the time step goes to zero. We performed an epidemic localization analysis considering the inverse participation ratio (IPR). Both theories present the same localization dependence on network degree exponent γ: for γ<5/2 the epidemics are localized on the maximum k-core of networks with a vanishing IPR in the infinite-size limit while, for γ>5/2, the localization happens on hubs that do not form a densely connected set and leads to a finite value of the IPR. However, the IPR and epidemic threshold of ELE depend on the time-step discretization such that a larger time step leads to more localized epidemics. A remarkable difference between discrete- and continuous-time approaches is revealed in the epidemic prevalence near the epidemic threshold, in which the discrete-time stochastic simulations indicate a mean-field critical exponent θ=1 instead of the value θ=1/(3-γ) obtained rigorously and verified numerically for the continuous-time SIS on the same networks.

Degree evolution in a general growing network

We consider the preferential attachment model introduced by Deijfen and Lindholm (2009) in which, at every discrete time step: (i) either we add a vertex and connect it to an older vertex; or (ii) we add an edge between two random vertices; or (iii) we delete one edge. We show that, when the deletion probability equals 1/3, the expected degree of any given vertex grows logarithmically, thus correcting a statement made in Lindholm and Vallier (2011). Moreover we show that, when the deletion probability is strictly less than 1/3, then the function which scales the expected degree of a given vertex, identified in Lindholm and Vallier (2011), also guarantees almost sure convergence for the degree process of a given vertex.

Image segmentation with Cellular Automata

Image segmentation is a computer vision technique that involves dividing an image into distinct and meaningful regions or segments. The objective was to partition the image into areas that share similar visual characteristics. Noise and undesirable artifacts introduce inconsistencies and irregularities in image data. These inconsistencies severely affect the ability of most segmentation algorithms to distinguish between true image features, leading to less reliable and lower-quality results. Cellular Automata (CA) is a computational concept that consists of a grid of cells, each of which can be in a finite number of states. These cells evolve over discrete time steps based on a set of predefined rules that dictate how a cell's state changes according to its own state and the states of its neighboring cells. In this paper, a new segmentation approach based on the CA model was introduced. The proposed approach consisted of three phases. In the initial two phases of the process, the primary objective was to eliminate noise and undesirable artifacts that can interfere with the identification of regions exhibiting similar visual characteristics. To achieve this, a set of rules is designed to modify the state value of each cell or pixel based on the states of its neighboring elements. In the third phase, each element is assigned a state that is chosen from a set of predefined states. These states directly represent the final segmentation values for the corresponding elements. The proposed method was evaluated using different images, considering important quality indices. The experimental results indicated that the proposed approach produces better-segmented images in terms of quality and robustness.

Modeling the deterioration of the stiffness and of the collagen fibril distribution in a discrete model of the cornea microstructure

We present a stochastic approach to model the mechanical deterioration of the reinforcing microstructure of the human cornea. The fundamental structural micro-components of the stroma, collagen and crosslinks, are assembled deterministically into an elementary trusswork cell, multiply repeated and distorted to form a three-dimensional shell with the shape of a cornea. The spatial orientation of the collagen-like elements of each cell is thus characterized stochastically with a non correlated random angle field, obeying an assigned probability density function, leading to a non-deterministic structural stiffness. It follows that the mechanical response of the model to the action of deterministic forces equivalent to the intraocular pressure is stochastic due to the random spatial orientation of collagen fibers. The deterioration of the mechanical stiffness of the collagen components is described through a scalar variable field, evolving in space and in time, representative of a progressive damage which causes heterogeneity and asymmetric behavior. The damage variable acts in two ways on the global stiffness: (i) by reducing the stiffness of the collagen components; (ii) by modifying the dispersion coefficient of the probability density function. The equilibrium equations of the damaging model are solved at discrete time steps, with a fully explicit solution scheme, by means of the stochastic finite element improved perturbation method. The results show that when the collagen fibril stiffness reduces to 10% of the healthy value, as expected in the case of the vision-impairing condition known as keratoconus, the displacement field due to intraocular pressure is significantly affected in terms of both average and variance distributions. This effect confers a typical conical shape to the cornea. In particular, the analysis shows that high values of the response variances are confined in the keratoconus area, which agrees with a high level of uncertainties due to loss of fibril organization and thickness reduction under pathologic conditions.

An encoder-decoder ConvLSTM surrogate model for simulating geological CO2 sequestration with dynamic well controls

In Geological Carbon Sequestration (GCS), effectively managing the project requires predicting state variables such as pressure and saturation. However, numerical simulation of multiphase flow in subsurface porous media involves solving large linear algebra systems, resulting in a substantial computational burden. This can make it impractical for real-time history matching or optimization. Meanwhile, deep learning-based surrogate models are emerging as fast and accurate approximators. This study delves into the application of the Encoder-Decoder Convolutional Long Short-Term Memory (ED-ConvLSTM) neural network for predicting the complex evolution of state variables under dynamic CO2 injection schemes. Inception blocks enhanced with light-weighted attention modules are introduced in the encoder to extract high-dimensional input features. ConvLSTM is employed to propagate spatial temporal information in the low-dimensional latent space. Further, progressive upsampling blocks are used to reconstruct the latent features for the desired output. Instead of taking discrete time steps as an input feature, the proposed network captures the dynamic dependencies with the inherent ConvLSTM cell. The network has access to data at only portion of the initial time steps during training stage, while it is used to predict the state variables at unseen time steps during testing stage. Results show that the network can produce excellent predictions for both pressure and saturation, even at unseen future time steps. The remarkable generalizability to different geological permeability fields is also evaluated. ED-ConvLSTM outperforms the standard U-Net by far, especially when predicting beyond the training time period. These numerical experiments demonstrate the advantages of ED-ConvLSTM in terms of prediction accuracy, extrapolability and generalizability. This study highlights the importance of incorporating recurrent connections into the deep neural networks for simulating time-dependent multiphase flow problems. The proposed methodology has great potential in GCS surrogate modeling and offers a possible approach for real-time optimization of CO2 injection.

Factorisation in the semiring of finite dynamical systems

Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS A is cancellative if one can divide by it unambiguously, i.e. AX=AY implies X=Y. We prove that an FDS A is cancellative if and only if it has a fixed point. Secondly, we prove that if an FDS A has a k-th root (i.e. B such that Bk=A), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs.

Stochastic Calculus and Its Impact On Analyzing Option Pricing

This research study the behaviour of the correction of Black-Scholes portfolios based on historical stock price data. We build a model that simulates one sample path of the stock price stochastic process at discrete time steps and track the correction over a time interval as it relates to the change in stock price over time. We also study the effect that relaxing this assumption has on the self-financing property of the replicating portfolio. We show that as the frequency of the time steps increases, the correction is more likely to be close to 0. We also show that the majority of historical stock return series that studied have caused the replicated portfolio to have a positive correction. We conclude that the Black-Scholes model can be used to find the no-arbitrage rational price for an option, a financial instrument that derives its value from the value of an underlying asset.

Methodology for Determining the Parameters of a Mathematical Model of the Dynamics of the Psychophysiological State of a Metallurgical Equipment Operator

Introduction. Mathematical modelling is effective in the analysis of industrial safety at metallurgical plants, in particular for tracking problems of the man — machine system. To introduce the time factor, recurrence relations (in a discrete model) and differential relations (in a continuous model) are used. However, it is also necessary to solve the problem of linking the model parameters to the real conditions of the production environment and to the human factor.The aim of this study is to create a method for determining the parameters of simulation mathematical models of the dynamics of the operator’s psychophysiological indicators affecting the work.Materials and Methods. The operator’s psychophysiological state (PPS) was assessed by performance, fatigue levels, and error rate. The data were collected by the Digital Correction Task (DCT) test. Based on the obtained results, the experimental values of the operator’s PPS indicators, which were reduced to the normalized scale [0, 1], were calculated. These indicators for a particular respondent, the mathematical model and the developed algorithm were used to determine the numerical values of the model parameters. In order to interpret the indicators of performance, fatigue and error rate, we introduced scales with five gradations.Results. The use of the authors’ modified version of the mathematical model showed a significant improvement in its prognostic properties. Out of 10 participants the best result was shown by respondent no. 7, the worst result was shown by respondent no. 8. During the first working hour (from 9.00 to 10.00) their performance increased almost equally, from 0.5–0.55 to almost 0.6. Then the score of respondent no. 7 increased and remained well above the “good” level until the end of the day. The score of respondent no. 8 dropped and was below average from 14.00 to 15.00. The difference was largely determined by the operators’ chronotypes. Their chronophysiological characteristics also affected fatigue and error rate. The model’s quality varied for different participants in the experiments. In one case it was excellent (mean relative error ≤5%), in three cases it was good (≤10%) and in four it was satisfactory (≤15%).Discussion and Conclusion. The proposed approach allows us to obtain the dynamic profiles of psychophysiological characteristics for every individual, to assess their interrelationships and to perform a prediction on the basis of a modified mathematical model. However, in order to extend the functionality of the models to the real working conditions of the metallurgical plant operator, it is necessary to increase the sample size, reduce the discrete time step and conduct studies for different working conditions, considering technological, climatic, environmental, psychological and other factors.

Online state estimation of Li-ion batteries using continuous-discrete nonlinear Kalman filters based on a nonlinear simplified electrochemical model

Online monitoring of Lithium-ion batteries (LIBs) during the charge and discharge periods is imperative to characterize their performance, assess their safety, and mitigate their degradation. One way to achieve this goal is to resort to model-based state estimations. Electrochemical based modelling has demonstrated significant potential to predict both the microscopic and the macroscopic battery dynamics. Existing electrochemical models are however continuous-time systems while measurements are performed at discrete time steps. Yet, most estimators are designed based on a fully discretized model. In this study, various types of continuous-discrete nonlinear Kalman filters are developed not only to address this challenge but also to compare their performance in utilizing highly nonlinear and meaningful physical variables. The filters rest on a simplified electrochemical model that accounts for the nonlinearities that stem from the dependence of the parameters on the electrolyte concentration. The performance of the estimators in adaptive and nonadaptive modes is tested for two different driving schedule loads (FUDS and Synthetic random higher C-rate). The results reveal that the estimators can track the real states' values if the filter model matches the real battery, even in the presence of 30 % error in the states' initial conditions. Furthermore, unlike the continuous-discrete extended Kalman filter (CD-EKF), the suggested continuous-discrete unscented Kalman filter (CD-UKF) effectively predicts the actual state values, even in instances where a significant structural mismatch exists between the embedded model and the synthetically generated measurement signal.

Predicting the spatio-temporal dynamics of Popillia japonica populations

AbstractWe developed a reaction–diffusion model to describe the spatio-temporal dynamics of the Japanese beetle (Popillia japonica Newman), based on adult abundance data collected by the Regional Phytosanitary Service during the monitoring activities in the infested area in Lombardy (northern Italy), from 2015 to 2021. The model simulated the abundance of the pest, with a discrete time step of one year along linear trajectories departing from an initial point of establishment. The model allowed the determination of the rate of expansion (i.e. the speed at which the leading edge of a population wave moves over time) of the pest along 13 different trajectories, ranging from 4.5 to 13.8 km/y, with a mean value of 8.2 km/y. Finally, we developed a land suitability index that summarises the effect of land use on the trajectory-specific rate of expansion of P. japonica. Specifically, the model revealed an increase in the rate of expansion of 260 m per year for each additional percentage point in the land suitability index. The model presented and the knowledge acquired in this work represent an important step forward in the comprehension of P. japonica population dynamics, and they represent important elements for the development of a decision support tool for pest risk managers to design and implement scientifically driven management actions.

Audio-Based Emotion Recognition Using Self-Supervised Learning on an Engineered Feature Space.

Emotion recognition models using audio input data can enable the development of interactive systems with applications in mental healthcare, marketing, gaming, and social media analysis. While the field of affective computing using audio data is rich, a major barrier to achieve consistently high-performance models is the paucity of available training labels. Self-supervised learning (SSL) is a family of methods which can learn despite a scarcity of supervised labels by predicting properties of the data itself. To understand the utility of self-supervised learning for audio-based emotion recognition, we have applied self-supervised learning pre-training to the classification of emotions from the CMU Multimodal Opinion Sentiment and Emotion Intensity (CMU- MOSEI)'s acoustic data. Unlike prior papers that have experimented with raw acoustic data, our technique has been applied to encoded acoustic data with 74 parameters of distinctive audio features at discrete timesteps. Our model is first pre-trained to uncover the randomly masked timestamps of the acoustic data. The pre-trained model is then fine-tuned using a small sample of annotated data. The performance of the final model is then evaluated via overall mean absolute error (MAE), mean absolute error (MAE) per emotion, overall four-class accuracy, and four-class accuracy per emotion. These metrics are compared against a baseline deep learning model with an identical backbone architecture. We find that self-supervised learning consistently improves the performance of the model across all metrics, especially when the number of annotated data points in the fine-tuning step is small. Furthermore, we quantify the behaviors of the self-supervised model and its convergence as the amount of annotated data increases. This work characterizes the utility of self-supervised learning for affective computing, demonstrating that self-supervised learning is most useful when the number of training examples is small and that the effect is most pronounced for emotions which are easier to classify such as happy, sad, and angry. This work further demonstrates that self-supervised learning still improves performance when applied to the embedded feature representations rather than the traditional approach of pre-training on the raw input space.