Modified Conjugate Gradient Method Research Articles

Simultaneous estimation of heat source strength of hot wire and thermal conductivity of material

The hot-wire (HW) technique is a widely used method for measuring the thermal conductivity (k) of insulation materials. However, there are some limitations to the applicability. For instance, inaccuracies may arise when the wire is assumed to be infinitely thin and long, or when the sample material is assumed to be an infinite medium. Additionally, the accuracy of k measurements relies on accurately specifying the heat source strength (QW ) generated by the HW. The modified conjugate gradient method (CGM) provides a solution to these limitations of the HW technique. In the modified CGM technique, a scaling factor based on the Levenberg-Marquardt method is utilized to address the low derivative and divergence issues of the traditional CGM. The current work simultaneously estimates QW generated by the HW and k of the sample materials by using modified CGM. The investigation looked into the effect of sensor positioning relative to the HW and the introduction of artificial Gaussian error (noise) with a standard deviation of 0.01 °C, 0.02 °C and 0.1 °C into the simulated temperature measurements. The modified CGM estimates the unknown parameters accurately compared to the traditional CGM. The measurement location should be closer to the HW to obtain accurate estimation results. The noise level of 0.1 °C results in low accuracy of estimation of k and QW for Plexiglas with one sensor, which can be enhanced by employing two sensors. Overall, the modified CGM algorithm proved to be a robust inverse algorithm that could be employed to estimate unknown parameters.

A Modified Gradient Method for Distributionally Robust Logistic Regression over the Wasserstein Ball

In this paper, a modified conjugate gradient method under the forward-backward splitting framework is proposed to further improve the numerical efficiency for solving the distributionally robust Logistic regression model over the Wasserstein ball, which comprises two phases: in the first phase, a conjugate gradient descent step is performed, and in the second phase, an instantaneous optimization problem is formulated and solved with a trade-off minimization of the regularization term, while simultaneously staying in close proximity to the interim point obtained in the first phase. The modified conjugate gradient method is proven to attain the optimal solution of the Wasserstein distributionally robust Logistic regression model with nonsummable steplength at a convergence rate of 1/T. Finally, several numerical experiments to validate the effectiveness of theoretical analysis are conducted, which demonstrate that this method outperforms the off-the-shelf solver and the existing first-order algorithmic frameworks.

Compressed Imaging Reconstruction Based on Block Compressed Sensing with Conjugate Gradient Smoothed l0 Norm

Compressed imaging reconstruction technology can reconstruct high-resolution images with a small number of observations by applying the theory of block compressed sensing to traditional optical imaging systems, and the reconstruction algorithm mainly determines its reconstruction accuracy. In this work, we design a reconstruction algorithm based on block compressed sensing with a conjugate gradient smoothed l0 norm termed BCS-CGSL0. The algorithm is divided into two parts. The first part, CGSL0, optimizes the SL0 algorithm by constructing a new inverse triangular fraction function to approximate the l0 norm and uses the modified conjugate gradient method to solve the optimization problem. The second part combines the BCS-SPL method under the framework of block compressed sensing to remove the block effect. Research shows that the algorithm can reduce the block effect while improving the accuracy and efficiency of reconstruction. Simulation results also verify that the BCS-CGSL0 algorithm has significant advantages in reconstruction accuracy and efficiency.

Two modified conjugate gradient methods for solving unconstrained optimization and application

Conjugate gradient methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems as they do not require the storage of any matrices. In order to obtain a theoretically effective and numerically efficient method, two modified conjugate gradient methods (called the MCB1 and MCB2 methods) are proposed. In which the coefficientβkin the two proposed methods is inspired by the structure of the conjugate gradient parameters in some existing conjugate gradient methods. Under the strong Wolfe line search, the sufficient descent property and global convergence of the MCB1 method are proved. Moreover, the MCB2 method generates a descent direction independently of any line search and produces good convergence properties when the strong Wolfe line search is employed. Preliminary numerical results show that the MCB1 and MCB2 methods are effective and robust in minimizing some unconstrained optimization problems and each of these modifications outperforms the four famous conjugate gradient methods. Furthermore, the proposed algorithms were extended to solve the problem of mode function.

A modified PRP-type conjugate gradient algorithm with complexity analysis and its application to image restoration problems

In this paper, a modified conjugate gradient method is proposed for nonconvex optimization. This method possesses the sufficient descent property independent of any line search. The global convergence property of the algorithm is established under the Wolfe line search strategy or the Armijo line search condition, respectively. Additionally, the complexity analysis of the proposed algorithm is investigated. To reach the point with the norm of the gradient below ɛ, the worst-case complexity bound matches that of the gradient method. Moreover, the obtained numerical results demonstrate that the modified method is effective for large-scale optimization problems and image restoration problems.

Optimal investigation of the reheating furnace based on a 3D numerical model with the formation of both oxide scale layer and skid marks

The performance of reheating furnace is highly dependent on the slab temperatures which further determine energy consumption. In general, the oxide scale layer and skid marks have a huge impact on the temperature distributions of the slab. Thus, an optimal control problem with a 3-dimensional (3D) numerical model is proposed to reduce the generation of skid marks, oxide scale layer and energy losses for the reheating furnace. Firstly, a 3D numerical Partial Difference Equation (PDE) model is built by considering the formation of the oxide scale layer and skid marks. The proposed model is implied by the parallel implementation strategy of GPU to reduce the computational time. Secondly, the 3D PDE-constrained optimal control problem is developed for the reheating furnace and solved by the proposed modified conjugate gradient (MCG) method. Finally, the accuracy of the proposed 3D model is verified by the industry measurement data, and the fast convergence performance of the MCG method is also validated. This work can significantly reduce the generation of oxide scale layer and energy consumption.

The Cauchy problem for Laplace's equation via a modified conjugate gradient method and energy space approaches

In the present work, we focus on the resolution of the Cauchy Laplace problem using an energetic variational minimization approach in the framework of a finite element method. A new strategy of regularization, called a filtering procedure regularization, is developed. The advantage of using this new regularization is that it does not require a regularization parameter and is easy to implement. An optimal a priori error estimate is proven, for the first time up to our knowledge, in the context of the finite element method. Some numerical results are presented to illustrate the performance of our approach.

A Derivative-Free MZPRP Projection Method for Convex Constrained Nonlinear Equations and Its Application in Compressive Sensing

Nonlinear systems of equations are widely used in science and engineering and, therefore, exploring efficient ways to solve them is paramount. In this paper, a new derivative-free approach for solving a nonlinear system of equations with convex constraints is proposed. The search direction of the proposed method is derived based on a modified conjugate gradient method, in such a way that it is sufficiently descent. It is worth noting that, unlike many existing methods that require a monotonicity assumption to prove the convergence result, our new method needs the underlying function to be pseudomonotone, which is a weaker assumption. The performance of the proposed algorithm is demonstrated on a set of some test problems and applications arising from compressive sensing. The obtained results confirm that the proposed method is effective compared to some existing algorithms in the literature.

Nonstationary fuzzy neural network based on FCMnet clustering and a modified CG method with Armijo-type rule

Nonstationary fuzzy inference systems (NFISs) model the variation in opinions of individual experts and expert groups. They have the capability similar to type-2 fuzzy systems in some regards with less computational costs. This paper presents a nonstationary fuzzy neural network (NFNN) model by combining NFISs and neural networks (NNs). The proposed model synthesizes the logical inference and language expression abilities of NFIS with the learning mechanism of NNs. Hence, it makes NNs more translatable and also achieves self-learning of fuzzy rules. Besides, a fuzzy c-means network (FCMnet) clustering and a modified conjugate gradient method with constrained Armijo-type rule (MCGA) are proposed to initialize and train NFNN, respectively. The proposed FCMnet first combines the fuzzy c-means (FCM) algorithm with NNs. Thus, it not only improves the performance of FCM, but also makes NNs more interpretable. For MCGA, we construct a specific conjugate coefficient to ensure the sufficient descent property and propose a constrained Armijo-type rule to search a suitable learning rate in each iteration. By adopting these two techniques, MCGA achieves a fast convergence speed. In addition, both weak and strong convergence results are proven rigorously. Experiments on 14 datasets are carried out to illustrate the competitive performance of our models.

GNSS Signal Compression Acquisition Algorithm Based on Sensing Matrix Optimization

Due to the sparsity of GNSS signal in the correlation domain, compressed sensing theory is considered to be a promising technology for GNSS signal acquisition. However, the detection probability of the traditional compression acquisition algorithm is low under low signal-to-noise ratio (SNR) conditions. This paper proposes a GNSS compression acquisition algorithm based on sensing matrix optimization. The Frobenius norm of the difference between Gram matrix and an approximate equiangular tight frame (ETF) matrix is taken as the objective function, and the modified conjugate gradient method is adopted to reduce the mutual coherence between the measurement matrix and the sparse basis. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the detection probability compared with the existing compression acquisition algorithms under the same SNR.

A Modified Conjugate Residual Method and Nearest Kronecker Product Preconditioner for the Generalized Coupled Sylvester Tensor Equations

This paper is devoted to proposing a modified conjugate residual method for solving the generalized coupled Sylvester tensor equations. To further improve its convergence rate, we derive a preconditioned modified conjugate residual method based on the Kronecker product approximations for solving the tensor equations. A theoretical analysis shows that the proposed method converges to an exact solution for any initial tensor at most finite steps in the absence round-off errors. Compared with a modified conjugate gradient method, the obtained numerical results illustrate that our methods perform much better in terms of the number of iteration steps and computing time.

Two modified conjugate gradient methods for unconstrained optimization with applications in image restoration problems

Two modified conjugate gradient methods for unconstrained optimization with applications in image restoration problems

Gradient-wise search strategy for blind image deblurring

Blind image deblurring is a long-standing challenging problem to improve the sharpness of an image as a prerequisite step. Many iterative methods are widely used for the deblurring image, but care must be taken to ensure that the methods have fast convergence and accuracy solutions. To address these problems, we propose a gradient-wise step size search strategy for iterative methods to achieve robustness and accelerate the deblurring process. We further modify the conjugate gradient method with the proposed strategy to solve the bling image deblurring problem. The gradient-wise step size aims to update gradient for each pixel individually, instead of updating step size by the fixed factor. The modified conjugate gradient method improves the convergence performance computation speed with a gradient-wise step size. Experimental results show that our method effectively estimates the sharp image for both motion blur images and defocused images. The results of synthetic datasets and natural images are better than what is achieved with other state-of-the-art blind image deblurring methods.

Double iterative algorithm for solving different constrained solutions of multivariate quadratic matrix equations

<abstract><p>Constrained solutions are required in some practical problems. This paper studies the problem of different constrained solutions for a class of multivariate quadratic matrix equations, which has rarely been studied before. First, the quadratic matrix equations are transformed into linear matrix equations by Newton's method. Second, the different constrained solutions of the linear matrix equations are solved by the modified conjugate gradient method, and then the different constrained solutions of the multivariate quadratic matrix equations are obtained. Finally, the convergence of the algorithm is proved. The algorithm can be well performed on the computers, and the effectiveness of the method is verified by numerical examples.</p></abstract>

A new modified conjugate gradient method under the strong Wolfe line search for solving unconstrained optimization problems

Conjugate gradient (CG) method is well-known due to efficiency to solve the problems of unconstrained optimization because of its convergence properties and low computation cost. Nowadays, the method is widely developed to compete with existing methods in term of their efficiency. In this paper, a modification of CG method will be proposed under strong Wolfe line search. A new CG coefficient is presented based on the idea of make use some parts of the previous existing CG methods to retain the advantages. The proposed method guarantees that the sufficient descent condition holds and globally convergent under inexact line search. Numerical testing provides strong indication that the proposed method has better capability when solving unconstrained optimization compared to the other methods under inexact line search specifically strong Wolfe–Powell line search.

Modified conjugate gradient method for solving sparse recovery problem with nonconvex penalty

Sparse recovery is a strategy for effectively reconstructing a signal by obtaining sparse solutions of underdetermined linear systems. As an important feature of a signal, sparsity is often measured by the ℓ1-norm. However, the approximate solutions obtained via ℓ1-norm regularization mostly underestimate the original signal. To overcome this defect, here we employ a class of nonconvex penalty functions proposed by Selesnick and Farshchian which preserve convexity of the cost function under certain conditions. To solve the problem, we suggest a nonmonotone modification of the generalized shrinkage conjugate gradient method proposed by Esmaeili et al., based on a modified secant equation. We establish global convergence of the method with standard suppositions. Numerical tests are made to shed light on performance of the proposed method.

A modified conjugate gradient method based on the self-scaling memoryless BFGS update

A modified conjugate gradient method based on the self-scaling memoryless BFGS update

A Modified Conjugate Gradient Method with Global-convergence for Unconstrianed Optimization Probems

A Modified Conjugate Gradient Method with Global-convergence for Unconstrianed Optimization Probems

A Descent Four-Term Conjugate Gradient Method with Global Convergence Properties for Large-Scale Unconstrained Optimisation Problems

The conjugate gradient method is a useful method to solve large-scale unconstrained optimisation problems and to be used in some applications in several fields such as engineering, medical science, image restorations, neural network, and many others. The main benefit of the conjugate gradient method is not using the second derivative or its approximation, such as Newton’s method or its approximation. Moreover, the algorithm of the conjugate gradient method is simple and easy to apply. This study proposes a new modified conjugate gradient method that contains four terms depending on popular two- and three-term conjugate gradient methods. The new algorithm satisfies the descent condition. In addition, the new CG algorithm possesses the convergence property. In the numerical results part, we compare the new algorithm with famous methods such as CG-Descent. We conclude from numerical results that the new algorithm is more efficient than other popular CG methods such as CG-Descent 6.8 in terms of number of function evaluations, number of gradient evaluations, number of iterations, and CPU time.

Reconstruction of the heat transfer coefficients and heat fluxes in heat conduction problems

This paper studies the reconstruction of the spatial-dependent-heat transfer coefficient γ(x) and- heat flux q(x), using the Levenberg–Marquardt (L–M) connected with the surrogate functional method and the modified conjugate gradient method (MCGM) (problem I). In addition to studying the reconstruction of γ(x) and space–time-dependent heat flux q(x,t) using the MCGM (problem II). The numerical results obtained by the L–M and MCGM demonstrate the efficiency, accuracy, and robustness of the proposed methods. The obtained results are compared with those obtained from an inverse approach connected with COMSOL Multiphysics under the same conditions. Finally, a conclusion and some remarks are presented.