Iranian Journal of Numerical Analysis and Optimization
Volume:12 Issue: 2, Summer and Autumn 2022

We estimate a function f with N independent observations by using Leg-endre wavelets operational matrices. The function f is approximated with the solution of a special minimization problem. We introduce an explicit expression for the penalty term by Legendre wavelets operational matrices. Also, we obtain a new upper bound on the approximation error of a differentiable function f using the partial sums of the Legendre wavelets. The validity and ability of these operational matrices are shown by several examples of real-world problems with some constraints. An accurate ap-proximation of the regression function is obtained by the Legendre wavelets estimator. Furthermore, the proposed estimation is compared with a non-parametric regression algorithm and the capability of this estimation is illustrated.

‎‎shifted Legendre orthonormal polynomials (SLOPs) are used to approximate the numerical solutions of fractional optimal control problems‎. ‎To do so‎, ‎first‎, ‎the operational matrix of the Caputo fractional derivative‎, ‎the SLOPs‎, ‎and Lagrange multipliers are used to convert such problems into algebraic equations‎. ‎Also‎, ‎the method is proposed for solving multidimensional problems. ‎We obtained the error bound of the operational matrix in fractional derivatives and proved the convergence of the method‎. ‎Then‎, ‎this is tested on some nonlinear examples‎. ‎‎Comparison of our results with those obtained by other techniques in previous studies revealed the accuracy of the proposed technique for nonlinear and multidimensional problems‎.

Let $A$ be an $n$-by-$n$ matrix with index $\alpha>0$ and $b \in \mathbb{C}^n$.  In this paper, the problem of stagnation of the DGMRES method for the singular linear system $Ax=b$ is considered. We show that DGMRES$(A, b, \alpha)$ has partial stagnation of order at least $k$ if and only if  $(0, \ldots, 0)$ belongs to the the joint numerical range of matrices  ${B^{\alpha+1}, \ldots, B^{\alpha+k}},$ where $B$ is a compression of $A$ to the range of $A^{\alpha}.$ Also, we characterize nonsingular part of a matrices $A$ such that DGMRES$(A, b, \alpha)$ does not stagnate for all $b \in \mathbb{C}^n$.  Moreover, a sufficient condition for non-existence of real stagnation vectors $b \in \mathcal{R}(A^{\alpha}) $ for DGMRES method is presented and the DGMRES stagnation of special matrices are studied.

There is significant interest in studying security games for defense op-timization and reducing the effects of attacks on various security systems involving vital infrastructures, financial systems, security, and urban safe-guarding centers. Game theory can be used as a mathematical tool to maximize the efficiency of limited security resources. In a game, players are smart, and it is natural for each player (defender or attacker) to try to deceive the opponent using various strategies in order to increase his payoff. Defenders can use deception as an effective means of enhancing security protection by giving incorrect information, hiding specific security resources, or using fake resources. However, despite the importance of de-ception in security issues, there is no considerable research on this field, and most of the works focus on deception in cyber environments. In this paper, a mixed-integer linear programming problem is proposed to allocate forces efficiently in a security game with multiple attackers using game the-ory analysis. The important subjects of information are their credibility and reliability. Especially when the defender uses deceptive defense forces, there are more ambiguity and uncertainty. Security game with Z-number payoffs is considered to apply both ambiguities in the payoffs and the reli-ability of earning these payoffs. Finally, the proposed method is illustrated by some numerical examples.

We propose a two-phase algorithm for solving continuous rank-one quadratic knapsack problems (R1QKPs). In particular, we study the solution structure of the problem without the knapsack constraint. In fact, an $O(n\log n)$ algorithm is suggested in this case. We then use the solution structure to propose an $O(n^2\log n)$ algorithm that finds an interval containing the optimal value of the Lagrangian dual of R1QKP. In the second phase, we solve the Lagrangian dual problem using a traditional single-variable optimization method. We perform a computational test on random instances and compare our algorithm with the general solver CPLEX.

This paper aims to apply and investigate the compact finite difference methods for solving integer-order and fractional-order Riccati differential equations. The fractional derivative in the fractional case is described in the Caputo sense. In solving the Riccati equation, we first approximate first-order derivatives using the approach of compact finite difference. In this way, the system of nonlinear equations is obtained, which solves the Riccati equation. In addition, we examine the convergence analysis of the proposed approach for the fractional and nonfractional cases and prove that the methods are convergent under some suitable conditions. Examples are also given to illustrate the efficiency of our method compared to other methods.

In this paper, a two-dimensional time-fractional telegraph equation is considered with derivative in the sense of Caputo and $ 1<\beta<2$. The aim of this work is to extend the Crank--Nicolson method for this time-fractional telegraph equation. The stability and convergence of the numerical method are investigated. Also, the accuracy and efficiency of the proposed method are demonstrated by numerical experiments.

The complexity of solving differential equations in real-world applications motivates researchers to extend numerical methods. Among different numerical and semi-analytical methods for solving initial and boundary value problems, the differential transform method (DTM) has received no-table attention. It has developed and experienced generalizations for implementing other types of mathematical problems such as optimal control, calculus of variations, and integral equations. This review aims to provide insight into DTM. History, theoretical base, applications, computational aspects, and its revisions are reviewed. The present study helps to understand the theory, capabilities, and features of the DTM, as well as its drawbacks and limitations.

In this paper, we introduce two new schemes based on the global Hessen-berg processes for computing approximate solutions to low-rank Sylvester tensor equations. We first construct bases for the matrix and extended matrix Krylov subspaces by applying the global and extended global Hes-senberg processes. Then the initial problem is projected into the matrix or extended matrix Krylov subspaces with small dimensions. The reduced Sylvester tensor equation obtained by the projection methods can be solved by using a recursive blocked algorithm. Furthermore, we present the upper bounds for the residual tensors without requiring the computation of the approximate solutions in any iteration. Finally, we illustrate the perfor-mance of the proposed methods with some numerical examples.

In this paper, we present an efficient method to solve linear time-delay optimal control problems with a quadratic cost function. In this regard, first, by employing the Pontryagin maximum principle to time-delay systems, the original problem is converted into a sequence of two-point boundary value problems (TPBVPs) that have both advance and delay terms. Then, using the continuous Runge–Kutta (CRK) method, the resulting sequences are recursively solved by the shooting method to obtain an optimal control law. This obtained optimal control consists of a linear feedback term, which is obtained by solving a Riccati matrix differential equation, and a forward term, which is an infinite sum of adjoint vectors, that can be obtained by solving sequences of delay TPBVPs by the shooting CRK method. Finally, numerical results and their comparison with other available results illustrate the high accuracy and efficiency of our proposed method.

A linearly implicit difference scheme for the space fractional coupled nonlinear Schrödinger equation is proposed. The resulting coefficient matrix of the discretized linear system consists of the sum of a complex scaled identity and a symmetric positive definite, diagonal-plus-Toeplitz, matrix. An efficient block Gauss–Seidel over-relaxation (BGSOR) method has been established to solve the discretized linear system. It is worth noting that the proposed method solves the linear equations without the need for any system solution, which is beneficial for reducing computational cost and memory requirements. Theoretical analysis implies that the BGSOR method is convergent under a suitable condition. Moreover, an appropriate approach to compute the optimal parameter in the BGSOR method is exploited. Finally, the theoretical analysis is validated by some numerical experiments.

The focus of this article is on the study of discrete optimal control problems (DOCPs) governed by time-varying systems, including time-varying delays in control and state variables. DOCPs arise naturally in many multi-stage control and inventory problems where time enters discretely in a natural fashion. Here, the Euler--Lagrange formulation (which are two-point boundary values with time-varying multi-delays) is employed as an efficient technique to solve DOCPs with time-varying multi-delays. The main feature of the procedure is converting the complex version of the discrete-time optimal control problem into a simple form of differential equations. Since the main problem is in discrete form, then the Euler--Lagrange equation changes to an algebraic system with initial and final conditions. The graphic representation of numerical simulation results shows that the proposed method can effectively and reliably solve DOCPs with time-varying multi-delays.