Discrete Approximation Method Research Articles

A discrete approximation method for modeling interval-censored multistate data.

Many longitudinal studies are designed to monitor participants for major events related to the progression of diseases. Data arising from such longitudinal studies are usually subject to interval censoring since the events are only known to occur between two monitoring visits. In this work, we propose a new method to handle interval-censored multistate data within a proportional hazards model framework where the hazard rate of events is modeled by a nonparametric function of time and the covariates affect the hazard rate proportionally. The main idea of this method is to simplify the likelihood functions of a discrete-time multistate model through an approximation and the application of data augmentation techniques, where the assumed presence of censored information facilitates a simpler parameterization. Then the expectation-maximization algorithm is used to estimate the parameters in the model. The performance of the proposed method is evaluated by numerical studies. Finally, the method is employed to analyze a dataset on tracking the advancement of coronary allograft vasculopathy following heart transplantation.

On the convergence of the trajectories of the dynamical Moudafi’s viscosity approximation system

AbstractWe study the asymptotic behavior of trajectories of the continuous dynamical system(CDS) associated with the discrete viscosity approximation method for fixed point problem of nonexpansive mapping (DDS) which was introduced by Moudafi (J Math Anal Appl 241:46–55, 2000). We establish that the trajectories x(t) of the continuous dynamical system (CDS) has an asymptotic behavior similar to the behavior of the sequences $$(x_{n})$$ ( x n ) generated by the discrete viscosity approximation (DDS)

Duality in the problems of optimal control described by Darboux-type differential inclusions

AbstractThis paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the discrete approximation method to get sufficient conditions of optimality for the convex problem given by Darboux differential inclusions and the polyhedral problem for a hyperbolic differential inclusion with state constraint. We formulate the adjoint inclusions in the Euler-Lagrange inclusion and Hamiltonian forms. Then, we construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraint and prove so-called duality results. Moreover, we show that each pair of primal and dual problem solutions satisfy duality relations and that the optimal values in the primal convex and dual concave problems are equal. Finally, we establish the dual problem to the polyhedral Darboux problem and provide an example to demonstrate the main constructions of our approach.

Joint Models for Longitudinal and Time-to-event Data with JM R Package

Even if the value of the risk factor evolves continuously in the extended Cox model, the inference may be somewhat biased because it employs a discrete approximation method. As a result, if the risk factor’s value not only fluctuates constantly but also contains measurement error, a model that substitutes the average value rather than the measured value of the risk factor might be considered. Such a model is known as a joint model. After introducing the most widely used the present-value model among joint models, this model was extended to various types of data. In addition, several residuals for model diagnosis were introduced, methods for predicting the probability of event occurrence and the value of the longitudinal risk factor by application of the estimation model were introduced, and an index that can evaluate how well the longitudinal risk factors divide patients into high-risk and low-risk groups was introduced. Finally, all the statistical inference methods introduced in this study were implemented using the JM R package and the source codes were supplied.

Power analysis of approximation methods for parameter estimation in Cox regression model with longitudinal covariate and tied survival times

Cox regression model (CRM) with time-varying covariates is widely used in medical and health related studies to investigate the effects of longitudinal variables on survival. In this paper, tied survival refers to units in which different subjects have the same survival time, while the term interaction indicates the relationship between longitudinal covariate and survival time. Unlike previous studies, we calculate the statistical power of Wald χ 2 statistics to test the interaction term based on Monte Carlo (MC) simulations when Breslow, Efron, Discrete, and Exact approximation methods are used for handling tied survival times. A linear mixed effect model (LMM) is used to generate longitudinal covariate such as time-varying covariate in simulations. A numerical example is provided to illustrate the CRM with the interaction term between longitudinal covariate and survival time. Using extensive MC simulations under different conditions for censored proportion, type I error, and number of subjects, statistical power of Wald χ 2 statistics is calculated using four different methods in the CRM. Statistical power value calculated using Breslow method is usually lower than statistical power values calculated when the other three methods are used. The proportion of censored observations in survival analysis has an important effect on power calculations.

Polyhedral optimization of discrete and partial differential inclusions of parabolic type

ABSTRACT The purpose of this paper is to look into the optimization of the first mixed boundary value problems for partial differential inclusions of the parabolic type. More specifically, we discuss a constructive approach to the study and solution of optimization problems for partial differential inclusions based on the discrete-approximate method. We formulate necessary and sufficient conditions of optimality by reducing the original first mixed boundary value problems to finite-dimensional problems in mathematical programming. All of our conditions are presented through the notion of Euler-Lagrange inclusions and transversality conditions. The effectiveness of our approach and its superiority over well-known methods are demonstrated by the example of the linear optimal control problem. Then, by passing formally to the limit as the discrete steps approach zero, we derive sufficient conditions of optimality for parabolic differential inclusions in the discrete-approximate problem.

Secure Transmission by Leveraging Multiple Intelligent Reflecting Surfaces in MISO Systems

Recent advance of Intelligent Reflecting Surface (IRS) introduces a new dimension for secure communications by reconfiguring the transmission environments. In this paper, we devise a secure transmission scheme for multi-user MISO systems by leveraging multiple collaborative IRSs. Specifically, to guarantee the worst-case achievable secrecy rate among multiple legitimate users, we formulate a max-min problem that can be solved by an alternating optimization method to decouple it into multiple sub-problems. Based on semidefinite relaxation and successive convex approximation, each sub-problem can be further converted into convex problem and easily solved. Extensive experimental results demonstrate that our proposed scheme can adapt to complex scenarios for multiple users and achieve significant gain in terms of achievable secrecy rate. Compared to the traditional single IRS scheme, the proposed scheme can achieve better performance at the range of 2.4-6.4 bps/Hz with the increase in the number of reflecting elements in the multi-user scenarios. We also evaluate the gap between the secrecy rate for our proposed scheme under continuous phase shift/amplitude control and discrete phase shift/amplitude control, and our results show that the secrecy rate obtained from discrete approximation method converges to that achieved from the proposed scheme when increasing the discretization granularity.

Discrete Stochastic Optimization for Public Health Interventions with Constraints

Many public health threats exist, motivating the need to find optimal intervention strategies. Given the stochastic nature of the threats (e.g., the spread of pandemic influenza, the occurrence of drug overdoses, and the prevalence of alcohol-related threats), deterministic optimization approaches may be inappropriate. In this paper, we implement a stochastic optimization method to address aspects of the 2009 H1N1 and the COVID-19 pandemics, with the spread of disease modeled by the open-source Monte Carlo simulations, FluTE, and Covasim, respectively. Without testing every possible option, the objective of the optimization is to determine the best combination of intervention strategies so as to result in minimal economic loss to society. To reach our objective, this application-oriented paper uses the discrete simultaneous perturbation stochastic approximation method (DSPSA), a recursive simulation-based optimization algorithm, to update the input parameters in the disease simulation software so that the output iteratively approaches minimal economic loss. Assuming that the simulation models for the spread of disease (FluTE for H1N1 and Covasim for COVID-19 in our case) are accurate representations for the population being studied, the simulation-based strategy we present provides decision makers a powerful tool to mitigate potential human and economic losses from any epidemic. The basic approach is also applicable in other public health problems, such as opioid abuse and drunk driving.

The Skill Assessment of Weather and Research Forecasting and WAVEWATCH-III Models During Recent Meteotsunami Event in the Persian Gulf

This study aims to use a fully realistic high-resolution mesoscale atmospheric and wave model to reproduce met-ocean conditions during a meteotsunami in the Persian Gulf. The atmospheric simulations were performed with the Weather and Research Forecasting (WRF) model by varying planetary boundary layer, microphysics, cumulus, and radiations parameterizations. The atmospheric results were compared to the meteorological observations (e.g., air pressure and wind speed) from the coastal and island synoptic and buoy stations of the nearest area to the meteotsunami event. The results show that using Mellor-Yamada-Nakanishi-Niino (MYNN) scheme for planetary boundary and surface layer had the best performance for stations over the water, whereas applying Mellor-Yamada-Janjic scheme for planetary boundary and Eta similarity surface layer had the best performance for stations over the land. For wave simulations, the WAVEWATCH-III model was employed with the well-known WAM-Cycle4 formulation and a more recent ST6 package. Six WRF experiments and ERA5 wind data were used to force the wave models. The new error parameter was introduced to identify the optimum wind data for wave simulation. EXP4 configuration which uses the MYNN scheme for planetary boundary and surface layer was led to minimum error, while ERA5 severely underestimated Hs and Tp parameters. For the first time, the Gaussian Quadrature Method (GQM) was implemented in the WAVEWATCH-III model and combined with a depth scale to be used in the Persian Gulf. This method is more accurate for non-linear wave-wave interaction than the default Discrete Interaction Approximation (DIA) method. Lower coefficients for dissipation term were required for GQM and the resulted bulk wave parameters were improved compared to the DIA method. The calibrated ST6 formulation with GQM resulted in a more realistic prediction of wave spectrum than the default settings of the WAVEWATCH-III.

Dynamic modeling and characteristic analysis of the double-reactor and quadruple-turbine nuclear power system based on steam main-pipeline distribution characteristics

Motivated by the insufficient insights on the dynamic characteristics of the double-reactor and quadruple-turbine nuclear power system (DQNPS) and the lack of the distributed-parameter model of the steam main-pipeline (SMP), the full-system dynamic model of the DQNPS is developed based on the conservation of mass, energy, and momentum in this work, where the SMP is modeled using discrete approximate distributed-parameter modelling method. Dynamic behavior analysis is then carried out on the validated model, through which, the dynamic characteristics of key variables, such as the steam pressures in the SMP are investigated in detail under the four typical step disturbances. The simulation results show that the DQNPS is of high nonlinearity and strongly coupled among multi-variables, and that the steam pressures in the SMP exhibit obvious distributed-characteristic which is the root cause of load redistribution in the DQNPS. Furthermore, it can be obtained from the simulations that the steam generators and the SMP in the DQNPS has the strong ability of energy-storage, which is valuable for the control of the system. The simulation results in this work can provide a useful reference for research on modeling and control of multi-reactor nuclear power system.

Learning to cluster for rendering with many lights

We present an unbiased online Monte Carlo method for rendering with many lights. Our method adapts both the hierarchical light clustering and the sampling distribution to our collected samples. Designing such a method requires us to make clustering decisions under noisy observation, and making sure that the sampling distribution adapts to our target. Our method is based on two key ideas: a coarse-to-fine clustering scheme that can find good clustering configurations even with noisy samples, and a discrete stochastic successive approximation method that starts from a prior distribution and provably converges to a target distribution. We compare to other state-of-the-art light sampling methods, and show better results both numerically and visually.

PROBLEM OF A DISCRETE DATA ARRAY APPROXIMATION BY A SET OF ELEMENTARY GEOMETRIC ALGORITHMS

Context. In this paper, a problem of a discrete data array approximation by a set of elementary geometric algorithms and a recognition model representation in a form of algorithmic classification tree has been solved. The object of the present study is a concept of a classification tree in a form of an algorithm trees. The subject of this study are the relevant models, methods, algorithms and schemes of different classification tree construction. 
 Objective. The goal of this work is to create a simple and efficient method and algorithmic scheme of building the tree-like recognition and classification models on the basis of the algorithm trees for training selections of large-volume discrete information characterized by a modular structure of independent recognition algorithms assessed in accordance with the initial training selection data for a wide class of applied tasks. 
 Method. A scheme of classification tree (algorithm tree) synthesis has been suggested being based on the data array approximation by a set of elementary geometric algorithms that constructs a tree-like structure (the ACT model) for a preset initial training selection of arbitrary size. The latter consists of a set of autonomous classification/recognition algorithms assessed at each step of the ACT construction according to the initial selection. A method of the algorithmic classification tree construction has been developed with the basic idea of step-by-step arbitrary-volume and structure initial selection approximation by a set of elementary geometric classification algorithms. When forming a current algorithm tree vertex, node and generalized attribute, this method provides alignment of the most effective and high-quality elementary classification algorithms from the initial set and complete construction of only those paths in the ACT structure, where the most of classification errors occur. The scheme of synthesizing the resulting classification tree and the ACT model developed allows one to reduce considerably the tree size and complexity. The ACT construction structural complexity is being assessed on the basis of a number of transitions, vertices and tiers of the ACT structure that allows the quality of its further analysis to be increased, the efficient decomposition mechanism to be provided and the ACT structure to be built in conditions of fixed limitation sets. The algorithm tree synthesis method allows one to construct different-type tree-like recognition models with various sets of elementary classifiers at the preset accuracy for a wide class of artificial intelligence theory problems. 
 Results. The method of discrete training selection approximation by a set of elementary geometric algorithms developed and presented in this work has received program realization and was studied and compared with those of logical tree classification on the basis of elementary attribute selection for solving the real geological data recognition problem. 
 Conclusions. Both general analysis and experiments carried out in this work confirmed capability of developed mechanism of constructing the algorithm tree structures and demonstrate possibility of its promising use for solving a wide spectrum of applied recognition and classification problems. The outlooks of the further studies and approbations might be related to creating the othertype algorithmic classification tree methods with other initial sets of elementary classifiers, optimizing its program realizations, as well experimental studying this method for a wider circle of applied problems.

Optimization of fully controlled sweeping processes

The paper is devoted to deriving necessary optimality conditions in a general optimal control problem for dynamical systems governed by controlled sweeping processes with hard-constrained control actions entering both polyhedral moving sets and additive perturbations. By using the first-order and mainly second-order tools of variational analysis and generalized differentiation, we develop a well-posed method of discrete approximations, obtain optimality conditions for solutions to discrete-time control systems, and then establish by passing to the limit verifiable necessary optimality conditions for local minimizers of the original controlled sweeping process that are expressed entirely in terms of its given data. The efficiency of the obtained necessary optimality conditions for the sweeping dynamics is illustrated by solving three nontrivial examples of their own interest.

DISCRETE APPROXIMATION OF CONTINUOUS OBJECTS WITH MATLAB

This work is dedicated to the study of various discrete approximation methods for continuous links, which is theobligatory step in the digital control systems synthesis for continuous dynamic objects and the guidelines development for performing these opera-tions using the MATLAB programming system. The paper investigates such sampling methods as pulse-, step-, and linearly invariant Z-transformations, substitution methods based on the usage of numerical integration various methods and the zero-pole correspond-ence method. The paper presents examples of using numerical and symbolic instruments of the MATLABto perform these opera-tions, offers an m-function improved version for continuous systems discretization by the zero-pole correspondence method, which allows this method to approach as step-invariant as linearly invariant Z-transformations; programs for continuous objects discrete approximation in symbolic form have been developed, which allows to perform comparative analysis of sampling methods and sys-tems synthesized with their help and to study quantization period influence on sampling accuracy by analytical methods. A compari-son of discrete transfer functions obtained by different methods and the corresponding reactions in time to different signalsis per-formed. Usingof the developed programs it is determined that the pulse-invariant Z-transformationcan be usedonly when the input of a continuous object receives pulse signals, and the linear-invariant transformation should be used for intermittentsignals at the input. The paper also presents an algorithm for applying the Tustin method, which corresponds to the replacement of analogueinte-gration by numerical integrationusingtrapezoidal method. It is shown that the Tustin method is the most suitable for sampling of first-order regulators with output signal limitation. The article also considers the zero-pole correspondence method and shows that it has the highest accuracy among the roughmethods of discrete approximation.Based on the performed research, recommendations for the use of these methods in the synthesis of control systems for continuous dynamic objects are given.

Lambda modes comparison for different approximations of the neutron transport equation: Diffusion, SN and SP3

The methods presented in this paper solve the Simplified Spherical Harmonics approximation to the multidimensional neutron transport equation. 1D, 2D and 3D systems were modeled with Cartesian geometry using the finite difference method to discretize the spatial variables. The method is able to simulate any energy group discretization, including up-scattering terms. The Krylov Shur method was used to calculate the solution of the steady-state equation by solving a generalized eigenvalue problem. This methodology has the capability to calculate any number of eigenfunctions. A formulation review of the Simplified Spherical Harmonics is explained in this work, as well as, a study of the boundary conditions for different approaches of the finite difference method. The results calculated by this methodology are compared with the discrete ordinates and diffusion approximation methods, all of them, using the same spatial discretization in order to show the different accuracy of each method without influence of the method used for discretizing the spatial variable. The results show the validity of each method for different benchmark problems.

DIRECT METHOD OF LIE-ALGEBRAIC DISCRETE APPROXIMATIONS FOR SOLVING BACKWARD HEAT EQUATION

A direct method of Lie-algebraic discrete approximation for numerical solving the Cauchy problem for the backward heat equation is proposed in this paper. The key idea of direct method of Lie-algebraic discrete approximations is using analytical approaches, in particular the method of small parameter or Taylor series expansion, to construct analytical approximation of the solution for the problem in the form of power series with respect to the time variable. The conditions for convergence of analytical series are studied in particular. By means of small parameter method the recurrence relation for evaluation of each member of a sequence is provided. This approach enables fast computation and signicant reduction of computational cost in compare to Generalized method of Lie-algebraic discrete approximations which performs complete discretization by all variables. Thereafter, the discrete match of recurrence relation is built using quasi-representations of the Lie-algebra basis elements, which means, that each dierential operator is replaced by its analogue matrix which is quasi-representation of dierential operator in nite dimensional space. It is proved that computational scheme has a factorial rate of convergence. The proposed approach is applied to model case and obtained results are compared with nite dierence method, classical method of Lie-algebraic discrete approximations and Generalized method of Lie-algebraic discrete approximation. The convergence rates for all of these methods are compared in dierent functional spaces. In addition, we study the count of arithmetical operations for equal set of nodes. Demonstrated a possibility for reusability of the numerical scheme for heat equation.

On the optimal control of rate-independent soft crawlers

Existence of optimal solutions and necessary optimality conditions for a controlled version of Moreau's sweeping process are derived. The control is a measurable ingredient of the dynamics and the constraint set is a polyhedron. The novelty consists in considering time periodic trajectories, adding the requirement that the control has zero average, and considering an integral functional that lacks weak semicontinuity. A model coming from the locomotion of a soft-robotic crawler, that motivated our setting, is analysed in detail. In obtaining necessary conditions, a variant of the method of discrete approximations is used.

Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems

Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems, see Rockafellar and Wets (Math. Oper. Res. 16:119–147, 1991), Ruszczynski and Shapiro (Stochastic Programming, Handbook in OR & MS, North-Holland Publishing Company, Amsterdam, 2003) and Ruszczynski (Math. Program. 79:333–353, 1997). In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. This is achieved by introducing nonanticipativity constraints for the first stage decision variables, rearranging the minimax problem through Lagrange decomposition and applying the well-known primal-dual hybrid gradient (PDHG) method to the new minimax problem. The algorithmic framework does not depend on specific structure of the ambiguity set. To extend the algorithm to the case that the underlying random variables are continuously distributed, we propose a discretization scheme and quantify the error arising from the discretization in terms of the optimal value and the optimal solutions when the ambiguity set is constructed through generalized prior moment conditions, the Kantorovich ball and $$\phi$$ -divergence centred at an empirical probability distribution. Some preliminary numerical tests show the proposed decomposition algorithm featured with parallel computing performs well.

Time-consistent multiperiod mean semivariance portfolio selection with the real constraints

In this paper, a new multiperiod mean semivariance portfolio selection with the transaction costs, borrowing constraints, threshold constraints and cardinality constraints is proposed. In the model, the return and risk of assets are characterized by mean value and semivariance, respectively. Because the semivariance operator is not separable, the optimal solution of the model is not time-consistent. The time-consistent strategy for this model can be obtained by using game approach. The time-consistent strategy, which is a mix integer dynamic optimization problem with path dependence, is approximately turned into a dynamic programming problem by approximate dynamic programming method. A novel discrete approximate iteration method is designed to obtain the optimal time-consistent strategy, and is proved linearly convergent. Finally, the comparison analysis of trade-off parameters is given to illustrate the idea of our model and the effectiveness of the designed algorithm.

Modelling and Dynamic Behaviour Analysis of the Steam Supply System for Double-Reactor and Multi-Turbine Nuclear Power Plant

This paper proposes a first principle dynamic model of the Nuclear Steam Supply System (NSSS) for the Double-Reactor and Multi-Turbine Nuclear Power Plant (DMNPP). The main devices within the nuclear steam supply system, such as Once Through Steam Generator (OTSG), Steam Main Pipe (SMP)and Steam Turbine (ST) are modelled using the movable boundary approach and discrete approximate distributed modelling method. Dynamic behaviour analysis is then carried out on the developed model, through which, the dynamic characteristics of key variables, such as the distribution pressures in the SMP are investigated in detail. The simulation results show that when the opening degree of the ST regulating valve or the OTSG steam valve change, the fluctuations of the pressure at each point in the SMP are different. The maximum fluctuation value of the pressure is 0.12 MPa when the opening degree of the third ST regulating valve is increased by 20 % and is 0.015 MPa when the opening degree of the second OTSG steam valve is increased by 10 %. It is discovered that the steam pressure of the SMP has a self-stabilizing ability and the NSSS of the DMNPP is of highly nonlinearity and strongly coupled among multi-variables.