Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very important.With the increasing emergence of large⁃scale problems,the subspace technology ...Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very important.With the increasing emergence of large⁃scale problems,the subspace technology has become particularly important and widely used in the field of optimization.In this study,a new CG method was put forward,which combined subspace technology and a cubic regularization model.Besides,a special scaled norm in a cubic regularization model was analyzed.Under certain conditions,some significant characteristics of the search direction were given and the convergence of the algorithm was built.Numerical comparisons show that for the 145 test functions under the CUTEr library,the proposed method is better than two classical CG methods and two new subspaces conjugate gradient methods.展开更多
A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The conv...A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.展开更多
In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explai...In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.展开更多
Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image...Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.展开更多
Natural stones used as floor and wall coverings are exposed to many different abrasive forces,so it is essential to choose suitable materials for wear resistance in terms of the life of the structure.The abrasion resi...Natural stones used as floor and wall coverings are exposed to many different abrasive forces,so it is essential to choose suitable materials for wear resistance in terms of the life of the structure.The abrasion resistance of natural stones can be determined in the laboratory by applying the B?hme abrasion resistance(BAR)test.However,the direct analysis of BAR in the laboratory has disadvantages such as wasting time and energy,experimental errors,and health impacts.To eliminate these disadvantages,the estimation of BAR using artificial neural networks(ANN)was proposed.Different natural stone samples were collected from Türkiye,and uniaxial compressive strength(UCS),flexural strength(FS),water absorption rate(WA),unit volume weight(UW),effective porosity(n),and BAR tests were carried out.The outputs of these tests were gathered and a data set,consisting of a total of 105 data,was randomly divided into two groups:testing and training.In the current study,the success of three different training algorithms of Levenberg-Marquardt(LM),Bayesian regularization(BR),and scaled conjugate gradient(SCG)were compared for BAR prediction of natural stones.Statistical criteria such as coefficient of determination(R~2),mean square error(MSE),mean square error(RMSE),and mean absolute percentage error(MAPE),which are widely used and adopted in the literature,were used to determine predictive validity.The findings of the study indicated that ANN is a valid method for estimating the BAR value.Also,the LM algorithm(R~2=0.9999,MSE=0.0001,RMSE=0.0110,and MAPE=0.0487)in training and the BR algorithm(R~2=0.9896,MSE=0.0589,RMSE=0.2427,and MAPE=1.2327)in testing showed the best prediction performance.It has been observed that the proposed method is quite practical to implement.Using the artificial neural networks method will provide an advantage in similar laborintensive experimental studies.展开更多
3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical...3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical data based on the quasi-analytical approximation （QA） and re-weighted regularized conjugate gradient method （RRCG） algorithms using Visual Fortran 6.5. Application of the QA approximation to forward modeling and Frechet derivative computations speeds up the calculation dramatically. The trial calculation for synthetic data of theoretical model showed that the program is fast and highly precise.展开更多
结构动载荷识别反问题是典型的病态问题,需要应用正则方法克服其病态特性而获得稳定的解。与直接正则化算法Tikhonov方法相比,共轭梯度最小二乘(Conjugate Gradient Least Squares,CGLS)迭代算法在载荷识别反问题的正则化过程有无须对...结构动载荷识别反问题是典型的病态问题,需要应用正则方法克服其病态特性而获得稳定的解。与直接正则化算法Tikhonov方法相比,共轭梯度最小二乘(Conjugate Gradient Least Squares,CGLS)迭代算法在载荷识别反问题的正则化过程有无须对传递矩阵求逆、无须明确正则化参数的优点。提出共轭梯度最小二乘迭代正则化算法和启发式迭代收敛终止准则,用于三自由度仿真模型和壳结构试验模型的冲击载荷识别,并与经典的Landweber迭代正则化算法和直接正则化算法Tikhonov方法比较。仿真和实验结果表明:CGLS迭代正则化算法在识别精度、收敛速度、计算效率和抗噪性方面有明显优势。展开更多
基金Sponsored by the National Natural Science Foundation of China(Grant No.11901561).
文摘Many methods have been put forward to solve unconstrained optimization problems,among which conjugate gradient method(CG)is very important.With the increasing emergence of large⁃scale problems,the subspace technology has become particularly important and widely used in the field of optimization.In this study,a new CG method was put forward,which combined subspace technology and a cubic regularization model.Besides,a special scaled norm in a cubic regularization model was analyzed.Under certain conditions,some significant characteristics of the search direction were given and the convergence of the algorithm was built.Numerical comparisons show that for the 145 test functions under the CUTEr library,the proposed method is better than two classical CG methods and two new subspaces conjugate gradient methods.
基金Subsidized by The Special Funds For Major State Basic Research Projects G1999032803.
文摘A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.
基金Supported by the National Natural Science Foundation of China(Grant No.11471253 and No.11571311)
文摘In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.
基金supported by the National Basic Research Program (No.2005CB321702)the National Outstanding Young Scientist Foundation(No. 10525102)the Specialized Research Grant for High Educational Doctoral Program(Nos. 20090211120011 and LZULL200909),Hong Kong RGC grants and HKBU FRGs
文摘Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.
文摘Natural stones used as floor and wall coverings are exposed to many different abrasive forces,so it is essential to choose suitable materials for wear resistance in terms of the life of the structure.The abrasion resistance of natural stones can be determined in the laboratory by applying the B?hme abrasion resistance(BAR)test.However,the direct analysis of BAR in the laboratory has disadvantages such as wasting time and energy,experimental errors,and health impacts.To eliminate these disadvantages,the estimation of BAR using artificial neural networks(ANN)was proposed.Different natural stone samples were collected from Türkiye,and uniaxial compressive strength(UCS),flexural strength(FS),water absorption rate(WA),unit volume weight(UW),effective porosity(n),and BAR tests were carried out.The outputs of these tests were gathered and a data set,consisting of a total of 105 data,was randomly divided into two groups:testing and training.In the current study,the success of three different training algorithms of Levenberg-Marquardt(LM),Bayesian regularization(BR),and scaled conjugate gradient(SCG)were compared for BAR prediction of natural stones.Statistical criteria such as coefficient of determination(R~2),mean square error(MSE),mean square error(RMSE),and mean absolute percentage error(MAPE),which are widely used and adopted in the literature,were used to determine predictive validity.The findings of the study indicated that ANN is a valid method for estimating the BAR value.Also,the LM algorithm(R~2=0.9999,MSE=0.0001,RMSE=0.0110,and MAPE=0.0487)in training and the BR algorithm(R~2=0.9896,MSE=0.0589,RMSE=0.2427,and MAPE=1.2327)in testing showed the best prediction performance.It has been observed that the proposed method is quite practical to implement.Using the artificial neural networks method will provide an advantage in similar laborintensive experimental studies.
文摘3D inversion of borehole-surface electrical data for complex geo-electrical models is still a challenging problem in geophysical exploration. We have developed a program for 3D inversion to borehole-surface electrical data based on the quasi-analytical approximation （QA） and re-weighted regularized conjugate gradient method （RRCG） algorithms using Visual Fortran 6.5. Application of the QA approximation to forward modeling and Frechet derivative computations speeds up the calculation dramatically. The trial calculation for synthetic data of theoretical model showed that the program is fast and highly precise.
文摘结构动载荷识别反问题是典型的病态问题,需要应用正则方法克服其病态特性而获得稳定的解。与直接正则化算法Tikhonov方法相比,共轭梯度最小二乘(Conjugate Gradient Least Squares,CGLS)迭代算法在载荷识别反问题的正则化过程有无须对传递矩阵求逆、无须明确正则化参数的优点。提出共轭梯度最小二乘迭代正则化算法和启发式迭代收敛终止准则,用于三自由度仿真模型和壳结构试验模型的冲击载荷识别,并与经典的Landweber迭代正则化算法和直接正则化算法Tikhonov方法比较。仿真和实验结果表明:CGLS迭代正则化算法在识别精度、收敛速度、计算效率和抗噪性方面有明显优势。
