Algorithm For Approximate Solution Research Articles

FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

DELAY MODELING OF MATHEMATICAL MODELS OF BIOLOGY AND IMMUNOLOGY

Systems of differential-difference equations are mathematical models of many applied problems of biology, ecology, medicine, economics. The variety of mathematical models of real dynamic processes is due to the fact that their evolution does not occur instantaneously, but with some delays that have different biological interpretations. The introduction of delay allows you to build adequate mathematical models and describe new effects and phenomena in physics, ecology, immunology and other sciences. The exact solution of differential-difference equations can be found only in the simplest cases, so algorithms for finding approximate solutions of such equations are important. In this paper, a family of difference schemes is constructed for the approximate finding of solutions to initial problems with delay. Special cases are generalized Euler difference schemes. The conditions for the convergence of the generalized explicit Euler difference scheme are established. To automate the numerical simulation of systems with delays, an application program has been developed, which is used to approximate the solutions of SIR models with two delays.

Regularization of algorithms for estimation of errors of differential equations approximate solutions

When estimating the error of numerical solutions of ordinary differential equations, it is proposed to use the regularization of error estimation algorithms for approximate solutions. Regularization is considered as the use of additional a priori information about the numerical solution. A priori information is estimated by the method of the inverse error analysis, which implements the transformation of the statement of the problem of evaluating the accuracy of the numerical solution. In this case, the problem is selected so that the approximate solution is the exact solution of the transformed problem, close to the original problem. It is also possible to add some additional restrictions to the conditions of the problem. This allows us to solve the incorrectly posed problems of estimating errors in numerical solutions, which in most cases are unstable. It is known that earlier in the numerical solution of linear algebra problems the inverse error analysis was used by Wilkinson J. and also Voevodin V.V., application of the inverse error analysis in other areas of numerical analysis can be noted. Using direct and inverse error analysis Voevodin V.V. effectively calculated majorants of rounding errors in the most important methods of linear algebra and significantly developed the method of inverse error analysis.. However, the regularization process associated with the reverse analysis of errors has not been previously studied. The article discusses the characteristics of regularization in the inverse analysis of errors in the numerical solution of ordinary differential equations with initial data. The regularization of error estimation algorithms has advantages in the numerical solution of ordinary differential equations of real models of physics and technology.

A Fragmented Model for the Problem of Land Use on Hypergraphs

The paper considers a mathematical model of the land use problem on hypergraphs. It is shown that, within the framework of this model, the problem can be formulated as an optimization problem on a fragmented structure. Moreover, the problem of finding the optimal solution itself reduces to the problem of unconditional combinatorial optimization on a set of permutations. A variant of a hybrid algorithm for finding approximate solutions to the problem based on a combination of a fragmented algorithm and an ant colony algorithm is proposed.

Convergence of a Two-Stage Proximal Algorithm for the Equilibrium Problem in Hadamard Spaces

An iterative two-stage proximal algorithm for approximate solution of equilibrium problems in Hadamard spaces is considered. This algorithm is an analog of the already studied two-stage algorithm for equilibrium problems in a Hilbert space. For Lipschitz-type pseudo-monotone bifunctions, a theorem on the weak convergence of sequences generated by the algorithm is proved.

О постановках обратных задач

Some possible options for the formulation of inverse problems are considered. The ultimate research goals in these cases determine the algorithms for the approximate solution of the inverse problem and allow one to correctly interpret these solutions. Two main statements of inverse problems considered: inverse problems of synthesis and inverse problems of measurement. It is shown that in inverse synthesis problems one should not take into account the error of the mathematical model. In addition, it is possible in these cases to synthesize approximate solution algorithms that do not have a regularizing property. Examples of practical problems considered.

A routing and consolidation decision model for containerized air-land intermodal operations

The study aims to provide the decision support for airfreight forwarders regarding the air-land intermodal operation, for which shipments are transported to gateway airports by air transportation and then delivered to final destinations by trucking service. A mixed integer programming model has been formulated to tackle the integrated routing and consolidation decision, given the various types of air containers. This study has designed an approximate solution algorithm based on Lagrangian Relaxation. In the numerical experiment, the solution algorithm can achieve an objective function value very close to optimality or the Lagrangian lower bound within an acceptable computation time.

Application of Ant Algorithm for Software Optimization

Different technologies, methods and algorithms are used when developing high-quality software systems. There are various ways to create optimal software. One of them is an ant algorithm. The ant algorithm can be attributed to the field of biomimetic. Ant algorithm is one of the most effective polynomial algorithms for finding approximate solutions, and it also solves similar route search problems in graphs. Various methods, including the ant algorithm, are used to improve software efficiency and optimize it. The essence of the concept is to analyze, use, and meta-heuristically optimize the behavior of ants in search of paths to the food source of the colony. This article analyzes the studies in this area, and explains the idea of the algorithm used, compares different ant systems, shows program code operators as ants, and applies the ant algorithm to them. As a result of applying the algorithm, the shortest path to some operators (cycle, condition) contained in the program code is found. The experiments perform good results.

Hybrid Beamforming for Active Sensing Using Sparse Arrays

This paper studies hybrid beamforming for active sensing applications, such as millimeter-wave or ultrasound imaging. Hybrid beamforming can substantially lower the cost and power consumption of fully digital sensor arrays by reducing the number of active front ends. Sparse arrays can be used to further reduce hardware costs. We consider phased arrays and employ linear beamforming with possibly sparse array configurations at both the transmitter and receiver. The quality of the acquired images is improved by adding together several component images corresponding to different transmissions and receptions. In order to limit the acquisition time of an image, we formulate an optimization problem for minimizing the number of component images subject to achieving a desired point spread function. Since this problem is not convex, we propose algorithms for finding approximate solutions in the fully digital beamforming case, as well as in the more challenging hybrid and analog beamforming cases that employ quantized phase shifters. We also determine upper bounds on the number of component images needed for achieving the fully digital solution using fully analog and hybrid architectures, and derive closed-form expressions for the beamforming weights in these cases. Simulations demonstrate that a hybrid sparse array with very few elements, and even fewer front ends, can achieve the resolution of a fully digital uniform array at the expense of a longer image acquisition time.

On the well‐posedness of periodic problems for the system of hyperbolic equations with finite time delay

AbstractA periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well‐posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.

Matrix Algorithm of Approximate Solution of Wave Equations in Inhomogeneous Magnetoactive Plasma

The problem of propagation of an electromagnetic wave in plane-stratified magnetoactive plasma is analyzed. A matrix algorithm of approximate solution of a set of wave equations is proposed. The algorithm consists in successive finding of the medium-inhomogeneity-induced corrections to the local roots of the dispersion relation and local polarization vectors. The set of field equations is reduced to a set of algebraic equations. The proposed algorithm is convenient for numerical calculations. In contrast to the classical geometrical-optics approximation, the proposed algorithm allows one to take into account the weak effect of linear mode interaction. Examples of numerical calculations of the power reflection coefficient of whistler waves incident on the ionosphere from above are presented. The proposed matrix algorithm can be useful to find the coefficients of reflection and linear transformation of waves in a smoothly inhomogeneous ionosphere.

Auxiliary Principle and Iterative Algorithm for a Generalized Nonlinear Mixed Quasi-variational-like Inequality

In this paper, we employ the auxiliary principle technique to study a generalized nonlinear mixed quasi-variational-like inequality in Hilbert spaces. Fist, we establish the existence and uniqueness of solution of the corresponding auxiliary generalized nonlinear mixed quasi-variational-like inequality by making use of minimizing sequence of a convex function. Then based on the existence result, we construct an iterative algorithm for finding approximate solution to the exact solution of the generalized nonlinear mixed quasi-variational-like inequality. Our results extend, improve and unify some known results in the literature.

One Approach to the Solution of a Nonlocal Problem for Systems of Hyperbolic Equations with Integral Conditions

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations with two independent variables. We study the solvability of the considered problem and construct algorithms aimed at finding its approximate solutions by introducing additional functional parameters. The investigated problem is reduced to an equivalent problem that consists of the Goursat problem for a system of hyperbolic equations with parameters and a boundary-value problem with integral condition for a system of ordinary differential equations for the introduced parameters. We propose algorithms for finding approximate solutions of the problem based on the algorithms used for the solution of the equivalent problem and prove their convergence to the exact solution.

On Flood Algorithm for Approximate Solution of Smooth Nonlinear Programming Problems with Linear Constraints of Large Dimension

On Flood Algorithm for Approximate Solution of Smooth Nonlinear Programming Problems with Linear Constraints of Large Dimension

WELL-POSEDNESS OF A NONLOCAL PROBLEM WITH INTEGRAL CONDITIONS FOR THIRD ORDER SYSTEM OF THE PARTIAL DIFFERENTIAL EQUATIONS

The nonlocal problem with integral conditions for the system of partial differential equations third-order is considered. The existence and uniqueness of classical solution to nonlocal problem with integral conditions for third-order system of partial differential equations are studied and the method for constructing their approximate solutions is proposed. Conditions of an unique solvability to nonlocal problem with integral conditions for third order system of partial differential equations are established. By introduction of new unknown functions, we have reduced the considered problem to an equivalent problem consisting of a nonlocal problem with integral conditions and parameters for a system of hyperbolic equations of second order and a integral relations. We have offered the algorithm for finding approximate solution to investigated problem and have proved its convergence. Sufficient conditions for the existence of unique solution to the equivalent problem with parameters are obtained. Well-posedness of the nonlocal problem with integral conditions for third order system of partial differential equations are established in the terms of well-posedness to nonlocal problem with integral conditions for system of hyperbolic equations second order. Key Words: third order partial differential equations, nonlocal problem, integral condition, system of hyperbolic equations second order, solvability, algorithm.

BEM model for radiative transport phenomena in optically thick compressible viscous fluids

The objective of this article is to develop a boundary element numerical model to solve coupled problems involving heat energy diffusion, convection and radiation in a participating medium. Radiation is modelled using two approaches for optically thick fluids: the Rosseland diffusion approximation and the P1 approximation.The governing Navier–Stokes equations are written in the velocity–vorticity formulation for the kinematics and kinetics of the fluid motion. The approximate numerical solution algorithm is based on boundary element numerical model in its macro-element formulation.The developed algorithm is validated by comparing results of radiative heat transfer for different optical thicknesses with benchmark data. Furthermore, the developed algorithm is tested by simulating natural convection and radiation heat transfer under large temperature differences where compressible flow solution is required.

Viable Algorithmic Options for Designing Reactive Robot Swarms

A central problem in swarm robotics is to design a controller that will allow the member robots of the swarm to collectively perform a given task. Of particular interest in massively distributed applications are reactive controllers with severely limited computational and sensory abilities. In this article, we give the results of the first computational complexity analysis of the reactive swarm design problem. Our core results are derived relative to a generalization of what is arguably the simplest possible type of reactive controller, the so-called computation-free controller proposed by Gauci et al., which operates in grid-based environments in a noncontinuous manner. We show that the design of a generalized computation-free swarm for an arbitrary given task in an arbitrary given environment is not polynomial-time solvable either in general or by the most desirable types of approximation algorithms (including evolutionary algorithms with high probabilities of producing correct solutions) but is solvable in effectively polynomial time relative to several types of restrictions on swarms, environments, and tasks. All of our results hold for the design of several more complex types of generalized computation-free swarms. Moreover, all of our intractability and inapproximability results hold for the design of any type of reactive swarm (including those based on the popular feed-forward neural network and Brooks-style subsumption controllers) operating in grid-based environments in a noncontinuous manner whose member robots satisfy two simple conditions. As such, our results give the first theoretical survey of the types of efficient exact and approximate solution algorithms that are and are not possible for designing several types of reactive swarms.

Быстрый алгоритм приближенного нахождения решения задачи Коши для ОДУ

Быстрый алгоритм приближенного нахождения решения задачи Коши для ОДУ

Traffic Events Oriented Dynamic Traffic Assignment Model for Expressway Network: A Network Flow Approach

In this paper, we focus on the dynamic traffic assignment (DTA) problem on large-scale expressway networks especially under the condition of traffic events (such as severe weather, large traffic accidents etc.). We formulate the expressway network DTA problem as a nonconvex optimization problem. Considering the difficulty of solving the problem, we put forward an approximate solution algorithm based on the network flow theory with high computational efficiency and strong applicability. Specifically, a machine learning approach and a series of customization strategies of network flow model (including multi-origin and multidestination split, flow discretization and parallel edges addition methods) are proposed to transform the DTA problem into a resoluble network flow problem. After that, in order to improve the operational efficiency of an expressway network after traffic events occurring, the proposed network flow based two-layer DTA model (NTDM) is separated into two parts-traffic flow limitation model based on maximum flow algorithm and traffic assignment model based on minimum cost flow algorithm. Experiments on real expressway network data in some provinces of China show that our method can make a significant 26.03% reduction in saturation rate in a particular scenario. In efficiency test, the NTDM is at least 600 times faster than the traditional analytic method in a network with thousands of nodes.

Haar wavelets operational matrix based algorithm for computational modelling of hyperbolic type wave equations

PurposeThe purpose of this study is to develop an algorithm for approximate solutions of nonlinear hyperbolic partial differential equations.Design/methodology/approachIn this paper, an algorithm based on the Haar wavelets operational matrix for computational modelling of nonlinear hyperbolic type wave equations has been developed. These types of equations describe a variety of physical models in nonlinear optics, relativistic quantum mechanics, solitons and condensed matter physics, interaction of solitons in collision-less plasma and solid-state physics, etc. The algorithm reduces the equations into a system of algebraic equations and then the system is solved by the Gauss-elimination procedure. Some well-known hyperbolic-type wave problems are considered as numerical problems to check the accuracy and efficiency of the proposed algorithm. The numerical results are shown in figures and Linf, RMS and L2 error forms.FindingsThe developed algorithm is used to find the computational modelling of nonlinear hyperbolic-type wave equations. The algorithm is well suited for some well-known wave equations.Originality/valueThis paper extends the idea of one dimensional Haar wavelets algorithms (Jiwari, 2015, 2012; Pandit et al., 2015; Kumar and Pandit, 2014, 2015) for two-dimensional hyperbolic problems and the idea of this algorithm is quite different from the idea for elliptic problems (Lepik, 2011; Shi et al., 2012). Second, the algorithm and error analysis are new for two-dimensional hyperbolic-type problems.