Iterative Equation Research Articles

Randomness accelerates the dynamic clearing process of the COVID-19 outbreaks in China

During the implementation of strong non-pharmaceutical interventions (NPIs), more than one hundred COVID-19 outbreaks induced by different strains in China were dynamically cleared in about 40 days, which presented the characteristics of small scale clustered outbreaks with low peak levels. To address how did randomness affect the dynamic clearing process, we derived an iterative stochastic difference equation for the number of newly reported cases based on the classical stochastic SIR model and calculate the stochastic control reproduction number (SCRN). Further, by employing the Bayesian technique, the change points of SCRNs have been estimated, which is an important prerequisite for determining the lengths of the exponential growth and decline phases. To reveal the influence of randomness on the dynamic zeroing process, we calculated the explicit expression of the mean first passage time (MFPT) during the decreasing phase using the relevant theory of first passage time (FPT), and the main results indicate that random noise can accelerate the dynamic zeroing process. This demonstrates that powerful NPI measures can rapidly reduce the number of infected people during the exponential decline phase, and enhanced randomness is conducive to dynamic zeroing, i.e. the greater the random noise, the shorter the average clearing time is. To confirm this, we chose 26 COVID-19 outbreaks in various provinces in China and fitted the data by estimating the parameters and change points. We then calculated the MFPTs, which were consistent with the actual duration of dynamic zeroing interventions.

Model-free policy iteration approach to NCE-based strategy design for linear quadratic Gaussian games

This paper presents a novel completely model-free method to solve continuous-time linear quadratic (LQ) mean field games (MFGs). We consider the setting where agents can only access their own states and are coupled via their individual infinite-horizon discounted costs. For such games, a set of Nash certainty equivalence-based strategies, determined by two gain matrices and the aggregate population effect, i.e. mean field (MF) state, are developed, providing an ɛ-Nash equilibrium under certain assumptions. Note that the gain matrices are solved from two algebraic Riccati equations (AREs), respectively, and the MF state is given by the linear ordinary differential equation (ODE). All the solutions of these three equations require the exact model of system dynamics. To implement the NCE-based strategy with completely unknown system dynamics, the proposed approach develops a single-agent policy iteration (PI) and a single-agent MF generator to solve the above parameters. Specifically, first, model-based iterative equations with guaranteed convergence are developed to approximate two gain matrices. Then, the requirement of dynamical model is removed from the iteration process via the integral reinforcement learning (IRL) technique. A model-free algorithm is conducted by using measured data from a selected agent as reinforcement signals over a certain time interval. Finally, based on the obtained gain matrices, the MF state is computed offline depending on samples collected from the same agent. A numerical example is given to verify the effectiveness of the proposed method.

Team optimal decentralized estimation by multiple agents with partial history sharing information structure

AbstractIn this paper, we study the team optimal decentralized estimation with partial history sharing information structure. There exist agents that have their own observations and share partial history observations to each other. The main contributions include two aspects: One is to give the iterative equations of the common estimation, which is the conditional expectation of state with respect to the common information for all agents and the innovation of the local information for each agent; the other is to provide the structure of the team optimal decentralized estimation, that is, the linear combination of the common estimation and the innovation of the local information. The novelty lies in that the estimation can be obtained at one time by defining an augmented state. In particular, the result is reduced to the well‐known centralized estimation when all the information is sharing for all the agents.

PROGNOSTICATION THE DYNAMICS OF THE DEVELOPMENT OF ORGANIZATIONAL AND TECHNOLOGICAL SYSTEMS

The article highlights an attempt to solve one of the most important problems of modern economic science – forecasting the creation and functioning of organizational and technological systems in domestic production. In the current military and economic conditions, the condition for the implementation of an innovative model of the development of the national economy through the functioning of organizational and technological systems in its industrial production is considered. The process of functioning of such systems consists of the continuous implementation of production cycles "money – technology – modern materials – production – sales". It is a feedback system and is described by differential equations. Non-linearity leads to the emergence of similar structures and their hierarchical construction. Since the capabilities of any organizational and technological system are limited, that is, the coupling coefficients between the output and input signals in the feedback system are proportional to the output signal, then the organizational and technological system is a nonlinear dynamic system. Such systems are described by nonlinear differential or iterative equations. It was determined that the potential income, that is, the income associated with the elimination of the monopoly of the organizational and technological system, and the real income in the free market are mutually complementary quantities (analogous to the potential and kinetic energy of a physical system). According to the principle of additionality, states in which such quantities had precisely certain values today been impossible. It follows from the above that the economy with hired labor and private property is relatively free, and the Western model cannot be considered as the only one when choosing the ways of economic development of the country. Such a model is an economy based on a market free from monopolies. It should be based on protected private property and democratic right of disposal – share-based personalized financial leasing (rent). And since nonlinear systems are characterized by cooperative behavior of their elements, it is conceivably to call this type of economy as a system of civilized cooperators, that is, market socialism.

Existence, construction and extension of continuous solutions of an iterative equation with multiplication

Existence, construction and extension of continuous solutions of an iterative equation with multiplication

An Iterative Nonconformal Domain Decomposition Surface Integral Equation Method for Scattering Analysis of PEC Objects

Domain decomposition methods (DDMs) can effectively solve electromagnetic scattering problems induced by multiscale perfectly electrically conducting (PEC) objects. However, boundary conditions between adjacent subdomains usually need to be carefully treated, especially when nonconformal meshes exist. To mitigate this issue, an iterative nonconformal domain decomposition surface integral equation (IDD-SIE) formulation is proposed. Each subdomain is independently solved with scattering fields generated from other subdomains as additional excitations. The continuity of current densities across touching interfaces is guaranteed by iteratively solving coupled fields among subdomains. Therefore, troublesome boundary transmission conditions can be avoided and nonconformal meshes are intrinsically supported. Numerical results show that the proposed IDD-SIE formulation is accurate and efficient, and has excellent convergent properties in terms of iteration numbers.

Ulam stabilities of nonlinear iterative integro-differential equations

Ulam stabilities of nonlinear iterative integro-differential equations

A study on the performance of distributed training of data-driven CFD simulations

Data-driven methods for computer simulations are blooming in many scientific areas. The traditional approach to simulating physical behaviors relies on solving partial differential equations (PDEs). Since calculating these iterative equations is highly both computationally demanding and time-consuming, data-driven methods leverage artificial intelligence (AI) techniques to alleviate that workload. Data-driven methods have to be trained in advance to provide their subsequent fast predictions; however, the cost of the training stage is non-negligible. This article presents a predictive model for inferencing future states of a specific fluid simulation that serves as a use case for evaluating different training alternatives. Particularly, this study compares the performance of only CPU, multi-GPU, and distributed approaches for training a time series forecasting deep learning model. With some slight code adaptations, results show and compare, in different implementations, the benefits of distributed GPU-enabled training for predicting high-accuracy states in a fraction of the time needed by the computational fluid dynamics solver.

A Model of Conflict Society with External Influence

We study a mathematical model of abstract society with redistribution of the social energy of individuals determined by two factors, namely, by the mutual competition and the presence of external influence. The behavior of the analyzed model is described by relatively simple iterative equations generating a dynamic system with discrete time. Fixed points of the system are determined. For some of these points that are attractors, their pools are partially described. In the general case, we prove the convergence of trajectories to the equilibrium states. In particular, it is shown that the individuals with the highest initial energy are doomed to be defeated if one of weaker individuals receives a sufficiently strong external support. Four examples are discussed to illustrate the most interesting cases of external influence on the dynamics of competition between individuals in an abstract society.

Decentralized ϵ‐Nash strategy for linear quadratic mean field games using a successive approximation approach

AbstractThis paper presents a successive approximation method for decentralized strategy design in the large‐scale linear quadratic (LQ) Gaussian game. The strategy consists of transforming the original mean field game (MFG) problem into solving decoupled ordinary differential equations (ODEs) whose numerical solutions are obtained by a new two‐loop iteration algorithm. It should be noted that we employ the augmented model technique and the LQ framework to derive these low‐dimensional solvable ODEs, which is the cornerstone of constructing the decentralized ‐Nash strategy. In addition, the quadratic ODEs contained therein are approximately solved for by a sequence of iterative linear ordinary differential equations (LODEs) with guaranteed convergence. A numerical example is given to show the effectiveness of the proposed algorithm.

Architecture-Driven Physics-Informed Deep Learning for Temperature Prediction in Laser Powder Bed Fusion Additive Manufacturing With Limited Data

Abstract Physics-informed deep learning (PIDL) is one of the emerging topics in additive manufacturing (AM). However, the success of previous PIDL approaches is generally significantly dependent on the existence of massive datasets. As the data collection in AM is usually challenging, a novel Architecture-driven PIDL structure named APIDL based on the deep unfolding approach for limited data scenarios has been proposed in the current study for predicting thermal history in the laser powder bed fusion process. The connections in this machine learning architecture are inspired by iterative thermal model equations. In other words, each iteration of the thermal model is mapped to a layer of the neural network. The hyper-parameters of the APIDL model are tuned, and its performance is analyzed. The APIDL for 1000 points with 80:20 split ratio achieves testing mean absolute percentage error (MAPE) of 2.8% and R2 value of 0.936. The APIDL is compared with the artificial neural network, extra trees regressor (ETR), support vector regressor, and long short-term memory algorithms. It was shown that the proposed APIDL model outperforms the others. The MAPE and R2 of APIDL are 55.7% lower and 15.6% higher than the ETR, which had the best performance among other pure machine learning models.

An iterative identification method of joint properties and its applications in an end-toothed connection structure

Accurate identification of mechanical connections plays a significant role in predicting the dynamic behaviors of assembled structures. Traditional identification methods based on frequency response function (FRF) decoupling may not be suitable for the joints with many interface degrees of freedom (DOFs), especially in the case of insufficient measured FRF data and large experimental noise. In this paper, a new iterative method is proposed to solve this problem. First, the incremental basic formulas on the measured DOFs and pseudo-measured DOFs are derived, respectively. The former avoids the measurements of rotational or interface DOFs whereas the latter is suitable for the identification of the joint with many interface DOFs. Then, the joint properties expressed in terms of mass, stiffness and damping matrices are updated from the solution of the linear equation system combined with the iterative basic identification equations at different frequencies. Thus, the original problem is transformed into estimating the parameters iteratively by minimizing the differences between predicted responses and measured ones, which effectively improves the identification accuracy and numerical stability. Moreover, the iterative method is extended to the identification of nonlinear joints based on describing function theory. The validity and superiority of the proposed method are verified by a simple lumped parameter system. Then, a numerical example of a structure connected with bolted joints, which has a large number of interface DOFs, is simulated to verify the convergence and the robustness of the method to noise. Finally, the joint dynamic properties of an end-toothed connection under different contact states are identified, and the assembly responses are also accurately predicted. The effectiveness and the applicability of the proposed iterative identification method to the real structures are well validated.

Self-consistent Sierpinski iteration of toughening mechanism in elastomer undergoing scaled segment-chain-network

Understanding working principles and toughening mechanisms of soft elastomers has been a huge challenge due to their significant scaling effects from molecules to bulk polymer. In this study, by combining Sierpinski fractal and scaling theory, an extended Kelvin model is developed to investigate the mechanical behaviour of the soft elastomer undergoing scaled segment-chain-network. According to the scaling theory, a radial distribution function was initially introduced to explain the diffusion and relaxation behaviours of molecular segments with the Sierpinski fractal features. The self-consistent iteration of the Sierpinski fractal is then used to describe the scaling effects of segments and networks. Rubber elasticity of the polymer network is further formulated based on the self-consistent iteration equation and scaling theory. A constitutive stress–strain relationship is also derived to explore the toughness mechanism and working principle in the polymer elastomer. Finally, the effectiveness of the proposed model is verified using finite-element analysis and experimental results reported in the literature, to explore a scaling insight into toughening mechanisms of elastomers governed by the self-consistent Sierpinski iteration.

Shadow Enhancement Using 2D Dynamic Stochastic Resonance for Hyperspectral Image Classification

With the development of remote sensing technology, classification has become a meaningful way to explore the rich information in hyperspectral images (HSIs). However, various environmental factors may cause noise and shadow areas in HSIs, resulting in weak signals and difficulties in fully utilizing information. In addition, classification methods based on deep learning have made considerable progress, but features extracted from most networks have much redundancy. Therefore, a method based on two-dimensional dynamic stochastic resonance (2D DSR) shadow enhancement and convolutional neural network (CNN) classification combined with an attention mechanism (AM) for HSIs is proposed in this paper. Firstly, to protect the spatial correlation of HSIs, an iterative equation of 2D DSR based on the pixel neighborhood relationship was derived, which made it possible to perform matrix SR in the spatial dimension of the image, instead of one-dimensional vector resonance. Secondly, by using the noise in the shadow area to generate resonance, 2D DSR can help increase the signals in the shadow regions by preserving the spatial characteristics, and enhanced HSIs can be obtained. Then, a 3DCNN embedded with two efficient channel attention (ECA) modules and one convolutional block attention module (CBAM) was designed to make the most of critical features that significantly affect the classification accuracy by giving different weights. Finally, the performance of the proposed method was evaluated on a real-world HSI, and comparative studies were carried out. The experimental results showed that the proposed approach has promising prospects in HSIs’ shadow enhancement and information mining.

Cryptanalysis of a Multiround Image Encryption Algorithm Based on 6D Self-Synchronizing Chaotic Stream Cipher

Recently, a chaotic secure video communication system based on FPGA is proposed, which essentially encrypts the original image data of each frame in the video in multiple rounds by using the 6D self-synchronizing chaotic stream cipher. In this paper, the security performance of the multiround encryption algorithm is analyzed based on divide-and-conquer attack. In the case of one round encryption, the keys [Formula: see text] [Formula: see text] can be deciphered by setting 30 appropriate initial conditions according to known-plaintext attack. After that, according to chosen-ciphertext attack, the chaotic iterative equation of the decryption end is degenerated into a linear one if ciphertexts are all set to zero. Under this condition, the equations between the unknown keys and the plaintext-ciphertext pairs can be obtained by setting 24 appropriate initial conditions, and the keys [Formula: see text] [Formula: see text] can be deciphered by solving the equations in Mathematica. Secondly, in order to reduce the multiplication times of the unknown keys in the case of two-round encryption, the number of pixels in an image can be selected as little as possible. According to chosen-ciphertext attack, when the number of the pixels in an image is only one, the keys [Formula: see text] [Formula: see text] can be deciphered by setting 30 appropriate initial conditions. Then, when the number of pixels in an image is two, the keys [Formula: see text] [Formula: see text] can be deciphered by setting 24 appropriate initial conditions. However, in the case that the encryption round is three or more, the keys cannot be deciphered by the attack method proposed in this paper. The analysis results show that the multiround image encryption algorithm is not secure when the number of encryption rounds is one or two.

Improved teaching-learning-based optimization algorithm based on fusion difference mutation1

Teaching-learning-based optimization algorithm (TLBO) is a swarm intelligence optimization algorithm that simulates classroom teaching phenomenon. In order to solve the problem that TLBO algorithm is easy to fall into local optimum and has poor stability, an improved teaching-learning-based optimization algorithm based on fusion difference mutation (IDMTLBO) is proposed. Firstly, adaptive teaching factors are introduced. Secondly, in the teaching stage, each student studies according to the gap between himself and the teacher, which improves the convergence speed and convergence accuracy of the algorithm. Finally, in the learning stage, students are divided into two levels according to their learning level, and two students are randomly selected to improve the iterative equation in the learning stage with the difference mutation strategy, It improves the disadvantage that the algorithm is easy to fall into local optimum. Numerical experiments show that the convergence speed and convergence accuracy of the algorithm are obviously better than TLBO algorithm, DMTLBO algorithm, DSTLBO algorithm.

Application of the New Iterative Method (NIM) to the Generalized Burgers–Huxley Equation

In this paper, we propose the new iterative method (NIM) for solving the generalized Burgers–Huxley equation. NIM provides an approximate solution without the need for discretization and is based on a set of iterative equations. We compared the NIM with other established methods, such as Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), and the exact solution, and found that it is efficient and easy to use. NIM has the advantage of quick convergence, easy implementation, and handling of a wide range of initial conditions. The comparison of the present symmetrical results with the existing literature is satisfactory.

AerSett v1.0: a simple and straightforward model for the settling speed of big spherical atmospheric aerosols

Abstract. This study introduces AerSett v1.0 (AERosol SETTling version 1.0), a model giving the settling speed of big spherical aerosols in the atmosphere without going through an iterative equation resolution. We prove that, for all spherical atmospheric aerosols with diameter D up to 1000 µm, this direct and explicit method including the drag coefficient formulation of Clift and Gauvin (1971) and the Davies (1945) slip correction factor gives results within 2 % of the exact solution obtained from the numerical resolution of a non-linear fixed-point equation. This error is acceptable considering the uncertainties on the drag coefficient formulations themselves. For D<100 µm, the error is below 0.5 %. We provide a Fortran implementation of this simple and straightforward model, hoping that more chemistry–transport models (CTMs) and general circulation models will be able to take into account large-particle drag correction to the settling speed of big spherical aerosol particles in the atmosphere, without performing an iterative and time-consuming calculation.

The dismantling method of wheel-spoke cable-strut tension structures based on experimental and numerical study

Several studies have reported that building demolition restricts urban development in many countries worldwide. Unlike traditional demolition, building deconstruction is economical since deconstructed materials can be reused, resold or repurposed. In addition, deconstruction combats the negative environmental effects of demolition by reducing the amount of waste materials. Therefore, deconstruction should be embraced more than demolition. In this study, the dismantling method (DM) of the wheel-spoke cable-strut tension structures was investigated to promote building deconstruction. Two novel dismantling methods were proposed, the relaxation of cables and drop of inner rings (DM-RCDIR) and the integral relaxation of the structure (DM-IRS), and experimental and numerical studies were conducted. The experimental model adopted the wheel-spoke cable-strut tension structural system with a maximum span of 12.675 m, and the numerical simulation method was established based on vector mechanics. The experimental and numerical results indicated that the proposed dismantling methods are feasible, and the method established based on the Newton iterative equation is effective to calculate the relaxation length of the active cable according to the mechanical state of the strut to be removed. This study also conducted a comparative analysis and found that DM-RCDIR has more advantages on efficiency, safety and economy especially when relaxing lower cables in batches, while DM-IRS has wider applicability. Therefore, this study is conducive to promoting the popularization and application of building deconstruction for prestressed structures.

Intelligent Analysis of Energy-Saving and Environment-Friendly Concrete Materials Based on Finite Element Simulation and Supercomputer

In this paper, based on the finite element simulation method of supercomputer, a three-dimensional intelligent modeling analysis of energy-saving and environmentally friendly concrete materials is carried out. First, the basic principles of the finite element iterative integral equation are introduced. The various characteristics of concrete materials and the interaction between steel and concrete, through calculation, people have a more comprehensive and in-depth grasp of the properties of reinforced concrete structures. In the process of nonlinear finite element analysis of reinforced concrete, physical models, numerical methods and programming techniques are involved. The final formation of the analysis program is an important part of it. By using the balanced grid division strategy of graph partition theory and sparse matrix reordering technology, the performance and scalability of the program are improved to 97.2%