A modified Hestenes–Stiefel conjugate gradient method with an optimal property

In this paper, based on the numerical efficiency of Hestenes–Stiefel (HS) method, a new modified HS algorithm is proposed for unconstrained optimization. The new direction independent of the line search satisfies in the sufficient descent condition. Motivated by theoretical and numerical features of three-term conjugate gradient (CG) methods proposed by Narushima et al., similar to Dai and Kou approach, the new direction is computed by minimizing the distance between the CG direction and the direction of the three-term CG methods proposed by Narushima et al. Under some mild conditions, we establish global convergence of the new method for general functions when the standard Wolfe line search is used. Numerical experiments on some test problems from the CUTEst collection are given to show the efficiency of the proposed method.     

An efficient adaptive three-term extension of the Hestenes–Stiefel conjugate gradient method

A new three-term Hestenes–Stiefel-type conjugate gradient method is proposed in which the search direction can satisfy the sufficient descent condition as well as an adaptive conjugacy condition. It is notable that search directions of the method are dynamically adjusted between that of the Newton method and the 3HS+ method, accelerating the convergence or reducing the condition number of iteration matrix. Under mild conditions, we show that the proposed method converges globally for general objective functions. Numerical experiments indicate that the method is practically promising.     

An efficient modified Polak–Ribière–Polyak conjugate gradient method with global convergence properties

The conjugate gradient (CG) method is one of the most popular methods for solving large-scale unconstrained optimization problems. In this paper, a new modified version of the CG formula that was introduced by Polak, Ribière, and Polyak is proposed for problems that are bounded below and have a Lipschitz-continuous gradient. The new parameter provides global convergence properties when the strong Wolfe-Powell (SWP) line search or the weak Wolfe-Powell (WWP) line search is employed. A proof of a sufficient descent condition is provided for the SWP line search. Numerical comparisons between the proposed parameter and other recent CG modifications are made on a set of standard unconstrained optimization problems. The numerical results demonstrate the efficiency of the proposed CG parameter compared with the other CG parameters.     

A nonsmooth nonlinear conjugate gradient method for interactive contact force problems

Interactive rigid body simulation is important for robot simulation and virtual design. A vital part of the simulation is the computation of contact forces. This paper addresses the contact force problem, as used in interactive simulation. The contact force problem can be formulated in the form of a nonlinear complementarity problem (NCP), which can be solved using an iterative splitting method, such as the projected Gauss–Seidel (PGS) method. We present a novel method for solving the NCP problem by applying a Fletcher–Reeves type nonlinear nonsmooth conjugate gradient (NNCG) type method. We analyze and present experimental convergence behavior and properties of the new method. Our results show that the NNCG method has at least the same convergence rate as PGS, and in many cases better.     

A computational method for optimal control problems with a free final time using the modified quasilinearization and the gradient method†

This paper presents a computational method for the solution of a class of optimal control problems with a free final time. The method is based on the combination of the modified quasilinearization and the gradient method. A numerical example is included for illustration.     

A new conjugate gradient algorithm with cubic Barzilai–Borwein stepsize for unconstrained optimization

In this paper, a new conjugate gradient (CG) algorithm in Dai–Liao (DL) family is presented for solving unconstrained optimization problems. The proposed algorithm tries to adjust positive values for the so-called DL parameter by using quadratic and/or cubic models of the objective function. More precisely, the cubic regularization model of the objective function is properly employed when the non-positive curvature is detected. Besides, the CG parameter is introduced so that the generated CG directions are descent. Under some standard assumptions, we establish the convergence property of the new proposed algorithm. Numerical results on some test problems are reported. The results show that the new algorithm performs well and is competitive with CG_DESCENT method.     

Convergence of the modified SOR–Newton method for non-smooth equations

Motivated by Chen [On the convergence of SOR methods for nonsmooth equations. Numer. Linear Algebra Appl. 9 (2002), pp. 81–92], in this paper, we further investigate a modified SOR–Newton (MSOR–Newton) method for solving a system of nonlinear equations F(x)=0, where F is strongly monotone and locally Lipschitz continuous but not necessarily differentiable. The convergence interval of the parameter in the MSOR–Newton method is given. Compared with that of the SOR–Newton method, this interval can be enlarged. Furthermore, when the B-differential of F(x) is difficult to compute, a simple replacement can be used, which can reduce the computational load. Numerical examples show that at the same cost of computational complexity, this MSOR–Newton method can converge faster than the corresponding SOR–Newton method by choosing a suitable parameter.     

A note on the variational approach to the Benjamin–Bona–Mahony equation using He's semi-inverse method

He's semi-inverse method is used to obtain variational formulation of the Benjamin–Bona–Mahony (BBM) equation. Using the Ritz method, the solitary solutions can be easily obtained.     

A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method

In this work we construct a modified trigonometrically fitted symplectic Runge Kutta Nyström method based on the fourth order five stages method of Calvo and Sanz-Serna (1994). We apply the new method on the numerical integration of the two-dimensional harmonic oscillator, the two-body problem, a perturbed two-body problem and two two-dimensional nonlinear oscillatory Hamiltonian systems.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

Jacobi wavelet collocation method for the modified Camassa–Holm and Degasperis–Procesi equations

Matrices representations of integrations of wavelets have a major role to obtain approximate solutions of integral, differential and integro-differential equations. In the present work, operational matrix representation of rth integration of Jacobi wavelets is introduced and to find these operational matrices, all details of the processes are demonstrated for the first time. Error analysis of offered method is also investigated in present study. In the planned method, approximate solutions are constructed with the truncated Jacobi wavelets series. Approximate solutions of the modified Camassa–Holm equation and Degasperis–Procesi equation linearized using quasilinearization technique are obtained by presented method. Applicability and accuracy of presented method is demonstrated by examples. The proposed method is also convergent even when a minor number of grid points. The numerical results obtained by offered technique are compatible with those in the literature.

     

A modified predictor-corrector scheme for the Klein–Gordon equation

A numerical method based on a three-time level finite-difference scheme has been proposed for the solution of the two forms of the Klein–Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor-corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector is modified by considering, as known, the already evaluated corrected values instead of the predictor ones. This modified scheme is applied to problems possessing periodic, kinks and soliton waves. The accuracy as well as the long-time behaviour of the proposed scheme is discussed and comparisons with the relevant known in the bibliography schemes are given.     

A semi-automatic method for burn scar delineation using a modified Chan–Vese model

We propose a novel semi-automatic method for burn scar delineation from Landsat imagery using a modified Chan–Vese model. Burn scars appear reddish-brown in band 742 false-colour composite of Landsat 7 images. This property is used in our algorithm to delineate burn scars. Firstly, we visually choose sample pixels from the burn scar. From these pixels, a discrimination function for burn scars is determined by the principal component analysis and interval estimation. Then we define a modified Chan–Vese functional. The minimizer of the functional corresponds to the boundary of the burn scar. In order to minimize this functional, the corresponding contour evolution equation is given. We use the discrimination function to locate an initial contour that is near the boundary of the burn scar. The evolving curve then efficiently converges to the desired boundary. A Landsat image over Russia is used to examine our algorithm. The result shows that the algorithm is effective.     

Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints

In this paper, the Wei–Yao–Liu (WYL) conjugate gradient projection algorithm will be studied for nonlinear monotone equations with convex constraints, which can be viewed as an extension of the WYL conjugate gradient method for solving unconstrained optimization problems. These methods can be applied to solving large-scale nonlinear equations due to the low storage requirement. We can obtain global convergence of our algorithm without requiring differentiability in the case that the equation is Lipschitz continuous. The numerical results show that the new algorithm is efficient.     

A spectrum slicing method for the Kohn–Sham problem

Solving the Kohn–Sham equation, which arises in density functional theory, is a standard procedure to determine the electronic structure of atoms, molecules, and condensed matter systems. The solution of this nonlinear eigenproblem is used to predict the spatial and energetic distribution of electronic states. However, obtaining a solution for large systems is computationally intensive because the problem scales super-linearly with the number of atoms. Here we demonstrate a divide and conquer method that partitions the necessary eigenvalue spectrum into slices and computes each partial spectrum on an independent group of processors in parallel. We focus on the elements of the spectrum slicing method that are essential to its correctness and robustness such as the choice of filter polynomial, the stopping criterion for a vector iteration, and the detection of duplicate eigenpairs computed in adjacent spectral slices. Some of the more prominent aspects of developing an optimized implementation are discussed.     

A modified conditional Metropolis–Hastings sampler

A modified conditional Metropolis–Hastings sampler for general state spaces is introduced. Under specified conditions, this modification can lead to substantial gains in statistical efficiency while maintaining the overall quality of convergence. Results are illustrated in two settings: a toy bivariate Normal model and a Bayesian version of the random effects model.     

A denoising–classification–retracking method to improve spaceborne estimates of the water level–surface–volume relation over the Urmia Lake in Iran

ABSTRACT

Decreasing the volume of the Urmia Lake, as the largest inland water body in Iran, is one of the current environmental and water resource management concerns. This study obtains a reliable spaceborne water level (WL)–area–volume relationship for the Urmia Lake using terrestrial, aerial and satellite-based data. The aim of this study is to improve Urmia Lake’s WL derived from satellite altimetry and, consequently, to more accurately estimate the volume of the lake for the last decade. To this end, improved WL is obtained from the Satellite with Argos and Altika (SARAL/AltiKa) and Jason-2 altimetry missions by performing a post-processing method. The post-processing method includes a denoising, a classification and appropriate retracking algorithms. The results are validated against in situ gauge data and also compared with results from Prototype Innovant de Système de Traitement pour les Applications Côtières et l’Hydrologie (PISTACH) and Prototype on AltiKa for Coastal, Hydrology and Ice (PEACHI) products. The Denoising–Classification–Retracking (DCR) method improves the root mean square error (RMSE) of WL with respect to those of PISTACH and PEACHI by 54% and 24%, respectively. The surface area of the lake is determined from Landsat 7 Enhanced Thematic Mapper Plus (ETM+) images based on calculating normalized difference water index (NDWI). The results are validated against the surface area obtained from aerial photogrammetry and Cartosat high resolution image. Moreover, based on bathymetric map a Look-up table including surface area and volume of the lake at specific levels are formed. The obtained surface area is then compared with the values of the Look-up table. The normalized root mean square error between surface extent obtained from proposed method and corresponding values is about 11%. The estimated lake’s volume is compared with the level-volume curve from the bathymetric data. The result showed the RMSE of this comparison is about 0.12 km³. Our validated results show that the lake has lost 75% of its volume from late 2008 to early 2016 but continued with an increase in its volume in May 2017 twice as much as in early 2016. Our results support urgent or long-term restoration plan of Lake Urmia and highlight the important role of spaceborne sensors for hydrological applications.     

A particle swarm–BFGS algorithm for nonlinear programming problems

This article proposes a hybrid optimization algorithm based on a modified BFGS and particle swarm optimization to solve medium scale nonlinear programs. The hybrid algorithm integrates the modified BFGS into particle swarm optimization to solve augmented Lagrangian penalty function. In doing so, the algorithm launches into a global search over the solution space while keeping a detailed exploration into the neighborhoods. To shed light on the merit of the algorithm, we provide a test bed consisting of 30 test problems to compare our algorithm against two of its variations along with two state-of-the-art nonlinear optimization algorithms. The numerical experiments illustrate that the proposed algorithm makes an effective use of hybrid framework when dealing with nonlinear equality constraints although its convergence cannot be guaranteed.