Showing papers in "Computational & Applied Mathematics in 2011"

An inexact subgradient algorithm for Equilibrium Problems

Abstract: We present an inexact subgradient projection type method for solving a nonsmooth Equilibrium Problem in a finite-dimensional space. The proposed algorithm has a low computational cost per iteration. Some numerical results are reported.

Operational Tau approximation for a general class of fractional integro-differential equations

Abstract: In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, errorestimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method. Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

Derivative-free methods for nonlinear programming with general lower-level constraints

Abstract: Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martinez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. The form of our main algorithm allows us to employ well established derivative-free subalgorithms for solving lower-level constrained subproblems. Numerical experiments are presented.

Differential transformation method for solving one-space-dimensional telegraph equation

Abstract: In this research, the Differential Transformation Method (DTM) has been utilized to solve the hyperbolic Telegraph equation. This method can be used to obtain the exact solutions of this equation. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method. Mathematical subject classification: 35Lxx, 35Qxx.

An SLP algorithm and its application to topology optimization

Abstract: We introduce a globally convergent sequential linear programming method for nonlinear programming. The algorithm is applied to the solution of classic topology optimization problems, as well as to the design of compliantmechanisms. The numerical results suggest that the new algorithm is faster than the globally convergent version of the method of moving asymptotes, a popular method for mechanical engineering applications proposed by Svanberg.

Active-set strategy in Powell's method for optimization without derivatives

Abstract: In this article we present an algorithm for solving bound constrained optimization problems without derivatives based on Powell's method [38] for derivative-free optimization. First we consider the unconstrained optimization problem. At each iteration a quadratic interpolation model of the objective function is constructed around the current iterate and this model is minimized to obtain a new trial point. The whole process is embedded within a trust-region framework. Our algorithm uses infinity norm instead of the Euclidean norm and we solve a box constrained quadratic subproblem using an active-set strategy to explore faces of the box. Therefore, a bound constrained optimization algorithm is easily extended. We compare our im_ plementation with NEWUOA and BOBYQA, Powell's algorithms for unconstrained and bound constrained derivative free optimization respectively. Numerical experiments show that, in general, our algorithm require less functional evaluations than Powell's algorithms.

Encoding through generalized polynomial codes

Abstract: This paper introduces novel constructions of cyclic codes using semigroup rings instead of polynomial rings. These constructions are applied to define and investigate the BCH, alternant, Goppa, and Srivastava codes. This makes it possible to improve several recent results due to Andrade and Palazzo [1].

Solving of time varying quadratic optimal control problems by

Abstract: In this paper, linear quadratic optimal control problems are solved by applying least square method based on B\'{e}zier control points. We divide the interval which includes $t$, into $k$ subintervals and approximate the trajectory and control functions by B\'{e}zier curves. We have chosen the B\'{e}zier curves as piacewise polynomials of degree three, and determined B\'{e}zier curves on any subinterval by four control points. By using least square method, we introduce an optimization problem and compute the control points by solving this optimization problem. Numerical experiments are presented to illustrate the proposed method.

Solving of time varying quadratic optimal control problems by using Bézier control points

Abstract: In this paper, linear quadratic optial control probles are solved by applying least square method based on Bezier control points. We divide the interval which includes t, into k subintervals and approximate the trajectory and control functions by Bezier curves. We have chosen the Bezier curves as piacewise polynomials of degree three, and determined Bezier curves on any subinterval by four control points. By using least square ethod, e introduce an optimization problem and compute the control points by solving this optimization problem. Numerical experiments are presented to illustrate the proposed method.

A semi-analytical computation of the Kelvin kernel for potential flows with a free surface

Abstract: A semi-analytical computation of the three dimensional Green function for seakeeping flow problems is proposed. A potential flow model is assumed with an harmonic dependence on time and a linearized free surface boundary condition. The multiplicative Green function is expressed as the product of a time part and a spatial one. The spatial part is known as the Kelvin kernel, which is the sum of two Rankine sources and a wave-like kernel, being the last one written using the Haskind-Havelock representation. Numerical efficiency is improved by an analytical integration of the two Rankine kernels and the use of a singularity subtractive technique for the Haskind-Havelock integral, where a globally adaptive quadrature is performed for the regular part and an analytic integration is used for the singular one. The proposed computation is employed in a low order panel method with flat triangular elements. As a numerical example, an oscillating floating unit hemisphere in heave and surge modes is considered, where analytical and semi-analytical solutions are taken as a reference.

Weak Allee effect in a predator-prey system involving distributed delays

Abstract: In this paper we study the influence of weak Allee effect in a predator-prey system model. This effect is included in the prey equation and we determine conditions for the occurrence of Hopf bifurcation. The stability properties of the system and the biological issues of the memory and Allee models on the coexistence of the two species are studied. Finally we present some simulations which allow one to verify the analytical conclusions obtained. Mathematical subject classification: Primary: 34C25; Secondary: 92B05.

Relaxation approaches to the optimal control of the Euler equations

Abstract: The treatment of control problems governed by systems of conservation laws poses serious challenges for analysis and numerical simulations. This is due mainly to shock waves that occur in the solution of nonlinear systems of conservation laws. In this article, the problem of the control of Euler flows in gas dynamics is considered. Numerically, two semi-linear approximations of the Euler equations are compared for the purpose of a gradient-based algorithm for optimization. One is the Lattice-Boltzmann method in one spatial dimension and five velocities (D1Q5 model) and the other is the relaxation method. An adjoint method is used. Good results are obtained even in the case where the solution contains discontinuities such as shock waves or contact discontinuities.

Accuracy of analytical-numerical solutions of the Michaelis-Menten equation

Abstract: It is the aim of this paper to investigate a suitable approach to compute solutions of the powerful Michaelis-Menten enzyme reaction equation with less computational effort. We obtain analytical-numerical solutions using piecewise finite series by means of the differential transformation method, DTM. In addition, we compute a global analytical solution by a modal series expansion. The Michaelis-Menten equation considered here describes the rate of depletion of the substrate of interest and in general is a powerful approach to describe enzyme processes. A comparison of the numerical solutions using DTM, Adomian decomposition and Runge-Kutta methods is presented. The numerical results show that the DTM is accurate, easy to apply and the obtained solutions retain the positivity property of the continuous model. It is concluded that the analytic form of the DTM and global modal series solutions are accurate, and require less computational effort than other approaches thus making them more convenient.

A global linearization approach to solve nonlinear nonsmooth constrained programming problems

Abstract: In this paper we introduce a new approach to solve constrained nonlinear non-smooth prograing probles ith any desirable accuracy even hen the objective function is a non-smooth one. In this approach for any given desirable accuracy, all the nonlinear functions of original problem (in objective function and in constraints) are approximated by a piecewise linear functions. We then represent an efficient algorithm to find the global solution of the later problem. The obtained solution has desirable accuracy and the error is completely controllable. One of the main advantages of our approach is that the approach can be extended to problems with non-smooth structure by introducing a novel definition of Global Weak Differentiation in the sense of L1 norm. Finally some numerical examples are given to show the efficiency of the proposed approach to solve approximately constraints nonlinear non-smooth programming problems.

Solution of a truss topology bilevel programming problem by means of an inexact restoration method

Abstract: We formulate a truss topology optimization problem as a bilevel programming problem and solve it by means of a line search type inexact restoration algorithm. We discuss details of the implementation and show results of numerical experiments.

Mimetic finite difference methods in image processing

Abstract: We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and restoration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference). We propose a new approach for the numerical solution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catte-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems. Mathematical subject classification: Primary: 68U10; Secondary: 65L12.

Block triangular preconditioner for static Maxwell equations

Abstract: In this paper, we explore the block triangular preconditioning techniques applied to the iterative solution of the saddle point linear systems arising from the discretized Maxwell equations. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix arestrongly clustered. Numerical experiments are given to demonstrate the efficiency of the presented preconditioner. Mathematical subject classification: 65F10.

Arithmetic fuchsian groups and space time block codes

Abstract: In the context of space time block codes (STBCs) the theory of arithmetic Fuchsian groups is presented. Additionally, in this work we present a new class of STBCs based on arithmetic Fuchsian groups. This new class of codes satisfies the property full-diversity, linear dispersion and full-rate. Mathematical subject classification: 18B35, 94A15, 20H10.

Comparing stochastic optimization methods to solve the medium-term operation planning problem

Abstract: The Medium-Term Operation Planning (MTOP) of hydrothermal systems aims to define the generation for each power plant, minimizing the expected operating cost over the planning horizon. Mathematically, this task can be characterized as a linear, stochastic, large-scale problem which requires the application of suitable optimization tools. To solve this problem, this paper proposes to use the Nested Decomposition, frequently used to solve similar problems (as in Brazilian case), and Progressive Hedging, an alternative method, which has interesting features that make it promising to address this problem. To make a comparative analysis between these two methods with respect to the quality of the solution and the computational burden, a benchmark is established, which is obtained by solving a single Linear Programming problem (the Deterministic Equivalent Problem). An application considering a hydrothermal system is carried out.

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

Abstract: A new smoothing function of the well known Fischer-Burmeister function is given. Based on this new function, a smoothing Newton-type method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm. Mathematical subject classification: 90C25, 90C30, 90C51, 65K05, 65Y20.

Closed balls for interpolating quasi-polynomials

Abstract: The classic interpolation problem asks for polynomials to fit a set of given data. In this paper, quasi-polynomials are considered as interpolating functions passing through a set of spatial points. Existence and uniqueness is obtained by means of generalized Vandermonde determinants. By means of several estimates related to these determinants, we are also able to find closed balls for any given centers that enclose the approximating curves. By choosing proper centers based on the observed spatial points, these balls may lead us to applications such as satellite tracking and control. Mathematical subject classification: 41A05.

Dynamical behavior of a pest management model with impulsive effect and nonlinear incidence rate

Abstract: In this paper, we consider the pest management model with spraying microbial pesticide and releasing the infected pests, and the infected pests have the function similar to the microbial pesticide and can infect the healthy pests, further weaken or disable their prey function till death. By using the Floquet theory for impulsive differential equations, we show that there exists a globally asymptotically stable pest eradication periodic solution when the impulsive period τ τmax. Finally, by means of numerical simulation, we showthatwith the increaseof impulsive period, the system displays complicated behaviors.

The quotient of gamma functions by the psi function

Abstract: The aim of this paper is to construct the asymptotic series of the ratio of gamma functions by Kershaw then to deduce some sharp estimates. Mathematical subject classification: 33B15, 05A16, 26D15.

Two derivative-free methods for solving underdetermined nonlinear systems of equations

Abstract: In this paper, two different approaches to solve underdetermined nonlinear system of equations are proposed. In one of them, the derivative-free method defined by La Cruz, Martinez and Raydan for solving square nonlinear systems is modified and extended to cope with the underdetermined case. The other approach is a Quasi-Newton method that uses the Broyden update formula and the globalized line search that combines the strategy of Grippo, Lampariello and Lucidi with the Li and Fukushima one. Global convergence results for both methods are proved and numerical experiments are presented.

A modified parametric iteration method for solving nonlinear second order BVPs

Abstract: The original parametric iteration method (PIM) provides the solution of a nonlinear second order boundary value problem (BVP) as a sequence of iterations. Since the successive iterations of the PIM may be very complex so that the resulting integrals in its iterative relation may not be performed analytically. Also, the implementation of the PIM generally leads to calculation of unneeded terms, which more time is consumed in repeated calculations for series solutions. In order to overcome these difficulties, in this paper, a useful improvement of the PIM is proposed. The implementation of the modified method is demonstrated by solving several nonlinear second order BVPs. The results reveal that the new developed method is a promising analytical tool to solve the nonlinear second order BVPs and more promising because it can further be applied easily to solve nonlinear higher order BVPs with highly accurate. Mathematical subject classification: Primary: 34B15; Secondary: 41A10.

Special Issue on Nonlinear Programming dedicated to the ALIO-INFORMS Joint International Meeting 2010

Abstract: The ALIO-INFORMS 2010 focused on topics such as services, logistics and transportation, manufacturing, supply chain management, environment, natural resources, biotechnology, and healthcare, emphasizing the importance of the relationship between basic research and the practice of OR/MS. This time, the Conference also included a Cluster dedicated to Nonlinear Programming, co-chaired by Ernesto G. Birgin and José Mario Mart́ınez.

Regularity results for semimonotone operators

Abstract: We introduce the concept of ρ-semimonotone point-to-set operators in Hilbert spaces. This notion is symmetrical with respect to the graph of T, as is the case for monotonicity, but not for other related notions, like e.g. hypomonotonicity, of which our new class is a relaxation. We give a necessary condition for ρ-semimonotonicity of T in terms of Lispchitz continuity of [T + ρ-11]-1 and a sufficient condition related to expansivity of T. We also establish surjectivity results for maximal ρ-semimonotone operators.

On a nonstationary nonlinear coupled system

Abstract: In this paper, a strongly nonlinear coupled elliptic-parabolic system modelling a class of engineering problems with heat effect is studied. Existence of a weak solution is first established by Schauder fixed point theorem, where the coupled functions σ(s), k(s) are assumed to be bounded. The uniqueness of the solution is obtained by applying Meyers' theorem and assuming that σ(s), k(s) are Lipschitz continuous. The regularity of the solution is then analyzed in dimension d < 2 under the assumptions on σ(s), k(s) ∈ C2(R) and the boundedness of their derivatives of second order. Finally, the blow-up phenomena of the system are studied. Mathematical subject classification: 35J60, 35K55.

Initial values for Riccati ODEs from variational PDEs

Abstract: The recently discovered variational PDEs (partial differential equations) for finding missing boundary conditions in Hamilton equations of optimal control are applied to the extended-space transformation of time-variant linear-quadratic regulator (LQR) problems. These problems become autonomous but with nonlinear dynamics and costs. The numerical solutions to the PDEs are checked against the analytical solutions to the original LQR problem. This is the first validation of the PDEs in the literature for a nonlinear context. It is also found that the initial value of the Riccati matrix can be obtained from the spatial derivative of the Hamiltonian flow, which satisfies the variational equation. This last result has practical implications when implementing two-degrees-of freedom control strategies for nonlinear systems with generalized costs.

Solutions to the recurrence relation u n+1 = v n+1 + u n ⊗ v n in terms of Bell polynomials

Abstract: Motivated by time series analysis, we consider the problem of solving the recurrence relation un+1 = vn+1 + un ⊗ vn for n ≠ 0 and un, given the sequence vn. A solution is given as a Bell polynomial. When vn can be written as a weighted sum of nth powers, then the solution un also takes this form. Mathematical subject classification: 33E99.