Large-scale multi-objective optimization problems(LSMOPs)pose challenges to existing optimizers since a set of well-converged and diverse solutions should be found in huge search spaces.While evolutionary algorithms a...Large-scale multi-objective optimization problems(LSMOPs)pose challenges to existing optimizers since a set of well-converged and diverse solutions should be found in huge search spaces.While evolutionary algorithms are good at solving small-scale multi-objective optimization problems,they are criticized for low efficiency in converging to the optimums of LSMOPs.By contrast,mathematical programming methods offer fast convergence speed on large-scale single-objective optimization problems,but they have difficulties in finding diverse solutions for LSMOPs.Currently,how to integrate evolutionary algorithms with mathematical programming methods to solve LSMOPs remains unexplored.In this paper,a hybrid algorithm is tailored for LSMOPs by coupling differential evolution and a conjugate gradient method.On the one hand,conjugate gradients and differential evolution are used to update different decision variables of a set of solutions,where the former drives the solutions to quickly converge towards the Pareto front and the latter promotes the diversity of the solutions to cover the whole Pareto front.On the other hand,objective decomposition strategy of evolutionary multi-objective optimization is used to differentiate the conjugate gradients of solutions,and the line search strategy of mathematical programming is used to ensure the higher quality of each offspring than its parent.In comparison with state-of-the-art evolutionary algorithms,mathematical programming methods,and hybrid algorithms,the proposed algorithm exhibits better convergence and diversity performance on a variety of benchmark and real-world LSMOPs.展开更多
Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering (CAE) and computer aided manufacturing ...Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering (CAE) and computer aided manufacturing (CAM). This paper presents a high-efficiency improved symmetric successive over-relaxation (ISSOR) preconditioned conjugate gradient (PCG) method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.展开更多
In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient proje...In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.展开更多
In this paper, a modified Polak-Ribière-Polyak conjugate gradient projection method is proposed for solving large scale nonlinear convex constrained monotone equations based on the projection method of Solodov an...In this paper, a modified Polak-Ribière-Polyak conjugate gradient projection method is proposed for solving large scale nonlinear convex constrained monotone equations based on the projection method of Solodov and Svaiter. The obtained method has low-complexity property and converges globally. Furthermore, this method has also been extended to solve the sparse signal reconstruction in compressive sensing. Numerical experiments illustrate the efficiency of the given method and show that such non-monotone method is suitable for some large scale problems.展开更多
A hybridization of the three–term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Polak and Ribi`ere, and Polyak is suggested. Based on an eigenvalue analysi...A hybridization of the three–term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Polak and Ribi`ere, and Polyak is suggested. Based on an eigenvalue analysis, it is shown that search directions of the proposed method satisfy the sufficient descent condition, independent of the line search and the objective function convexity. Global convergence of the method is established under an Armijo–type line search condition. Numerical experiments show practical efficiency of the proposed method.展开更多
The fast convergence without initial value dependence is the key to solving large angle relative orientation.Therefore,a hybrid conjugate gradient algorithm is proposed in this paper.The concrete process is:①stochast...The fast convergence without initial value dependence is the key to solving large angle relative orientation.Therefore,a hybrid conjugate gradient algorithm is proposed in this paper.The concrete process is:①stochastic hill climbing(SHC)algorithm is used to make a random disturbance to the given initial value of the relative orientation element,and the new value to guarantee the optimization direction is generated.②In local optimization,a super-linear convergent conjugate gradient method is used to replace the steepest descent method in relative orientation to improve its convergence rate.③The global convergence condition is that the calculation error is less than the prescribed limit error.The comparison experiment shows that the method proposed in this paper is independent of the initial value,and has higher accuracy and fewer iterations.展开更多
Two new formulaes of the main parameter βk of the conjugate gradient method are presented, which respectively can be seen as the modifications of method HS and PRP. In comparison with classic conjugate gradient metho...Two new formulaes of the main parameter βk of the conjugate gradient method are presented, which respectively can be seen as the modifications of method HS and PRP. In comparison with classic conjugate gradient methods, the new methods take both available gradient and function value information. Furthermore, their modifications are proposed. These methods are shown to be global convergent under some assumptions. Numerical results are also reported.展开更多
We extend a results presented by Y.F. Hu and C.Storey (1991) [1] on the global convergence result for conjugate gradient methods with different choices for the parameter β k . In this note, the condit...We extend a results presented by Y.F. Hu and C.Storey (1991) [1] on the global convergence result for conjugate gradient methods with different choices for the parameter β k . In this note, the conditions given on β k are milder than that used by Y.F. Hu and C. Storey.展开更多
It is well known that Newton and quasi-Newton algorithms are effective to small and medium scale smooth problems because they take full use of corresponding gradient function’s information but fail to solve nonsmooth...It is well known that Newton and quasi-Newton algorithms are effective to small and medium scale smooth problems because they take full use of corresponding gradient function’s information but fail to solve nonsmooth problems.The perfect algorithm stems from concept of‘bundle’successfully addresses both smooth and nonsmooth complex problems,but it is regrettable that it is merely effective to small and medium optimization models since it needs to store and update relevant information of parameter’s bundle.The conjugate gradient algorithm is effective both large-scale smooth and nonsmooth optimization model since its simplicity that utilizes objective function’s information and the technique of Moreau-Yosida regularization.Thus,a modified three-term conjugate gradient algorithm was proposed,and it has a sufficiently descent property and a trust region character.At the same time,it possesses the global convergence under mild assumptions and numerical test proves it is efficient than similar optimization algorithms.展开更多
In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the ite...In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.展开更多
Y Liu and C Storey(1992)proposed the famous LS conjugate gradient method which has good numerical results.However,the LS method has very weak convergence under the Wolfe-type line search.In this paper,we give a new de...Y Liu and C Storey(1992)proposed the famous LS conjugate gradient method which has good numerical results.However,the LS method has very weak convergence under the Wolfe-type line search.In this paper,we give a new descent gradient method based on the LS method.It can guarantee the sufficient descent property at each iteration and the global convergence under the strong Wolfe line search.Finally,we also present extensive preliminary numerical experiments to show the efficiency of the proposed method by comparing with the famous PRP^+method.展开更多
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
This paper discusses the global convergence of a class of nonmonotone conjugate gra- dient methods(NM methods) for nonconvex object functions.This class of methods includes the nonmonotone counterpart of modified Po...This paper discusses the global convergence of a class of nonmonotone conjugate gra- dient methods(NM methods) for nonconvex object functions.This class of methods includes the nonmonotone counterpart of modified Polak- Ribière method and modified Hestenes- Stiefel method as special cases展开更多
Recently, Gilbert and Nocedal([3]) investigated global convergence of conjugate gradient methods related to Polak-Ribiere formular, they restricted beta(k) to non-negative value. [5] discussed the same problem as that...Recently, Gilbert and Nocedal([3]) investigated global convergence of conjugate gradient methods related to Polak-Ribiere formular, they restricted beta(k) to non-negative value. [5] discussed the same problem as that in [3] and relaxed beta(k) to be negative with the objective function being convex. This paper allows beta(k) to be selected in a wider range than [5]. Especially, the global convergence of the corresponding algorithm without sufficient decrease condition is proved.展开更多
When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the comput...When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the computational efficiency and to save storage, the Conjugate Gradient (CG) method is presented. The CG is an effective method for solving a large system of linear equations and belongs to the method of iteration with rapid convergence and high precision. An example is given and calculated results are compared to validate the proposed methods.展开更多
The gradient element of the aperture gradient map is utilized directly to generate the aperture shape without modulation.This process can be likened to choosing the direction of negative gradient descent for the gener...The gradient element of the aperture gradient map is utilized directly to generate the aperture shape without modulation.This process can be likened to choosing the direction of negative gradient descent for the generic aperture shape optimiza-tion.The negative gradient descent direction is more suitable under local conditions and has a slow convergence rate.To overcome these limitations,this study introduced conjugate gradients into aperture shape optimization based on gradient modulation.First,the aperture gradient map of the current beam was obtained for the proposed aperture shape optimiza-tion method,and the gradients of the aperture gradient map were modulated using conjugate gradients to form a modulated gradient map.The aperture shape was generated based on the modulated gradient map.The proposed optimization method does not change the optimal solution of the original optimization problem,but changes the iterative search direction when generating the aperture shape.The performance of the proposed method was verified using cases of head and neck cancer,and prostate cancer.The optimization results indicate that the proposed optimization method better protects the organs at risk and rapidly reduces the objective function value by ensuring a similar dose distribution to the planning target volume.Compared to the contrasting methods,the normal tissue complication probability obtained by the proposed optimization method decreased by up to 4.61%,and the optimization time of the proposed method decreased by 5.26%on average for ten cancer cases.The effectiveness and acceleration of the proposed method were verified through comparative experiments.According to the comparative experiments,the results indicate that the proposed optimization method is more suitable for clinical applications.It is feasible for the aperture shape optimization involving the proposed method.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
The purpose of these investigations is to find the numerical outcomes of the fractional kind of biological system based on Leptospirosis by exploiting the strength of artificial neural networks aided by scale conjugat...The purpose of these investigations is to find the numerical outcomes of the fractional kind of biological system based on Leptospirosis by exploiting the strength of artificial neural networks aided by scale conjugate gradient,called ANNs-SCG.The fractional derivatives have been applied to get more reliable performances of the system.The mathematical form of the biological Leptospirosis system is divided into five categories,and the numerical performances of each model class will be provided by using the ANNs-SCG.The exactness of the ANNs-SCG is performed using the comparison of the reference and obtained results.The reference solutions have been obtained by using theAdams numerical scheme.For these investigations,the data selection is performed at 82%for training,while the statics for both testing and authentication is selected as 9%.The procedures based on the recurrence,mean square error,error histograms,regression,state transitions,and correlation will be accomplished to validate the fitness,accuracy,and reliability of the ANNs-SCG scheme.展开更多
The purpose of this study is to present the numerical performancesand interpretations of the SEIR nonlinear system based on the Zika virusspreading by using the stochastic neural networks based intelligent computingso...The purpose of this study is to present the numerical performancesand interpretations of the SEIR nonlinear system based on the Zika virusspreading by using the stochastic neural networks based intelligent computingsolver. The epidemic form of the nonlinear system represents the four dynamicsof the patients, susceptible patients S(y), exposed patients hospitalized inhospital E(y), infected patients I(y), and recovered patients R(y), i.e., SEIRmodel. The computing numerical outcomes and performances of the systemare examined by using the artificial neural networks (ANNs) and the scaledconjugate gradient (SCG) for the training of the networks, i.e., ANNs-SCG.The correctness of the ANNs-SCG scheme is observed by comparing theproposed and reference solutions for three cases of the SEIR model to solvethe nonlinear system based on the Zika virus spreading dynamics throughthe knacks of ANNs-SCG procedure based on exhaustive experimentations.The outcomes of the ANNs-SCG algorithm are found consistently in goodagreement with standard numerical solutions with negligible errors. Moreover,the procedure’s constancy, dependability, and exactness are perceived by usingthe values of state transitions, error histogram measures, correlation, andregression analysis.展开更多
基金supported in part by the National Key Research and Development Program of China(2018AAA0100100)the National Natural Science Foundation of China(61906001,62136008,U21A20512)+1 种基金the Key Program of Natural Science Project of Educational Commission of Anhui Province(KJ2020A0036)Alexander von Humboldt Professorship for Artificial Intelligence Funded by the Federal Ministry of Education and Research,Germany。
文摘Large-scale multi-objective optimization problems(LSMOPs)pose challenges to existing optimizers since a set of well-converged and diverse solutions should be found in huge search spaces.While evolutionary algorithms are good at solving small-scale multi-objective optimization problems,they are criticized for low efficiency in converging to the optimums of LSMOPs.By contrast,mathematical programming methods offer fast convergence speed on large-scale single-objective optimization problems,but they have difficulties in finding diverse solutions for LSMOPs.Currently,how to integrate evolutionary algorithms with mathematical programming methods to solve LSMOPs remains unexplored.In this paper,a hybrid algorithm is tailored for LSMOPs by coupling differential evolution and a conjugate gradient method.On the one hand,conjugate gradients and differential evolution are used to update different decision variables of a set of solutions,where the former drives the solutions to quickly converge towards the Pareto front and the latter promotes the diversity of the solutions to cover the whole Pareto front.On the other hand,objective decomposition strategy of evolutionary multi-objective optimization is used to differentiate the conjugate gradients of solutions,and the line search strategy of mathematical programming is used to ensure the higher quality of each offspring than its parent.In comparison with state-of-the-art evolutionary algorithms,mathematical programming methods,and hybrid algorithms,the proposed algorithm exhibits better convergence and diversity performance on a variety of benchmark and real-world LSMOPs.
基金Project supported by the National Natural Science Foundation of China(Nos.5130926141030747+3 种基金41102181and 51121005)the National Basic Research Program of China(973 Program)(No.2011CB013503)the Young Teachers’ Initial Funding Scheme of Sun Yat-sen University(No.39000-1188140)
文摘Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering (CAE) and computer aided manufacturing (CAM). This paper presents a high-efficiency improved symmetric successive over-relaxation (ISSOR) preconditioned conjugate gradient (PCG) method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.
文摘In this paper, a class of the stochastic generalized linear complementarity problems with finitely many elements is proposed for the first time. Based on the Fischer-Burmeister function, a new conjugate gradient projection method is given for solving the stochastic generalized linear complementarity problems. The global convergence of the conjugate gradient projection method is proved and the related numerical results are also reported.
文摘In this paper, a modified Polak-Ribière-Polyak conjugate gradient projection method is proposed for solving large scale nonlinear convex constrained monotone equations based on the projection method of Solodov and Svaiter. The obtained method has low-complexity property and converges globally. Furthermore, this method has also been extended to solve the sparse signal reconstruction in compressive sensing. Numerical experiments illustrate the efficiency of the given method and show that such non-monotone method is suitable for some large scale problems.
基金Supported by Research Council of Semnan University
文摘A hybridization of the three–term conjugate gradient method proposed by Zhang et al. and the nonlinear conjugate gradient method proposed by Polak and Ribi`ere, and Polyak is suggested. Based on an eigenvalue analysis, it is shown that search directions of the proposed method satisfy the sufficient descent condition, independent of the line search and the objective function convexity. Global convergence of the method is established under an Armijo–type line search condition. Numerical experiments show practical efficiency of the proposed method.
基金National Natural Science Foundation of China(Nos.4156108241161061)。
文摘The fast convergence without initial value dependence is the key to solving large angle relative orientation.Therefore,a hybrid conjugate gradient algorithm is proposed in this paper.The concrete process is:①stochastic hill climbing(SHC)algorithm is used to make a random disturbance to the given initial value of the relative orientation element,and the new value to guarantee the optimization direction is generated.②In local optimization,a super-linear convergent conjugate gradient method is used to replace the steepest descent method in relative orientation to improve its convergence rate.③The global convergence condition is that the calculation error is less than the prescribed limit error.The comparison experiment shows that the method proposed in this paper is independent of the initial value,and has higher accuracy and fewer iterations.
基金supported by the Teaching and Research Award Program for the Outstanding Young Teachers in Higher Education Institutesof Ministry of Educationthe Natural Science Foundation of Inner Mongolia Autonomous Region (2010BS0108)SPH-IMU (Z20090135)
文摘Two new formulaes of the main parameter βk of the conjugate gradient method are presented, which respectively can be seen as the modifications of method HS and PRP. In comparison with classic conjugate gradient methods, the new methods take both available gradient and function value information. Furthermore, their modifications are proposed. These methods are shown to be global convergent under some assumptions. Numerical results are also reported.
文摘We extend a results presented by Y.F. Hu and C.Storey (1991) [1] on the global convergence result for conjugate gradient methods with different choices for the parameter β k . In this note, the conditions given on β k are milder than that used by Y.F. Hu and C. Storey.
基金This work is supported by the National Natural Science Foundation of China(Grant No.11661009)the Guangxi Science Fund for Distinguished Young Scholars(No.2015GXNSFGA139001)+1 种基金the Guangxi Natural Science Key Fund(No.2017GXNSFDA198046)Innovation Project of Guangxi Graduate Education(No.YCSW2018046).
文摘It is well known that Newton and quasi-Newton algorithms are effective to small and medium scale smooth problems because they take full use of corresponding gradient function’s information but fail to solve nonsmooth problems.The perfect algorithm stems from concept of‘bundle’successfully addresses both smooth and nonsmooth complex problems,but it is regrettable that it is merely effective to small and medium optimization models since it needs to store and update relevant information of parameter’s bundle.The conjugate gradient algorithm is effective both large-scale smooth and nonsmooth optimization model since its simplicity that utilizes objective function’s information and the technique of Moreau-Yosida regularization.Thus,a modified three-term conjugate gradient algorithm was proposed,and it has a sufficiently descent property and a trust region character.At the same time,it possesses the global convergence under mild assumptions and numerical test proves it is efficient than similar optimization algorithms.
文摘In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method (PCCG). The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix. The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix. This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems, and simultaneously contrasted with other methods. The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations, ft is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.
基金Supported by The Youth Project Foundation of Chongqing Three Gorges University(13QN17)Supported by the Fund of Scientific Research in Southeast University(the Support Project of Fundamental Research)
文摘Y Liu and C Storey(1992)proposed the famous LS conjugate gradient method which has good numerical results.However,the LS method has very weak convergence under the Wolfe-type line search.In this paper,we give a new descent gradient method based on the LS method.It can guarantee the sufficient descent property at each iteration and the global convergence under the strong Wolfe line search.Finally,we also present extensive preliminary numerical experiments to show the efficiency of the proposed method by comparing with the famous PRP^+method.
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Supported by the National Natural Science Foundation of China(1 0 1 6 1 0 0 2 ) and Guangxi Natural Sci-ence Foundation (0 1 3 5 0 0 4 )
文摘This paper discusses the global convergence of a class of nonmonotone conjugate gra- dient methods(NM methods) for nonconvex object functions.This class of methods includes the nonmonotone counterpart of modified Polak- Ribière method and modified Hestenes- Stiefel method as special cases
文摘Recently, Gilbert and Nocedal([3]) investigated global convergence of conjugate gradient methods related to Polak-Ribiere formular, they restricted beta(k) to non-negative value. [5] discussed the same problem as that in [3] and relaxed beta(k) to be negative with the objective function being convex. This paper allows beta(k) to be selected in a wider range than [5]. Especially, the global convergence of the corresponding algorithm without sufficient decrease condition is proved.
文摘When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the computational efficiency and to save storage, the Conjugate Gradient (CG) method is presented. The CG is an effective method for solving a large system of linear equations and belongs to the method of iteration with rapid convergence and high precision. An example is given and calculated results are compared to validate the proposed methods.
基金supported by the Natural Science Foundation of Shanxi Province(No.20210302124403)the Research Project Supported by Shanxi Scholarship Council of China(No.2021-111)the Science and Technology Innovation Project of Colleges and Universities in Shanxi Province(No.2022L353).
文摘The gradient element of the aperture gradient map is utilized directly to generate the aperture shape without modulation.This process can be likened to choosing the direction of negative gradient descent for the generic aperture shape optimiza-tion.The negative gradient descent direction is more suitable under local conditions and has a slow convergence rate.To overcome these limitations,this study introduced conjugate gradients into aperture shape optimization based on gradient modulation.First,the aperture gradient map of the current beam was obtained for the proposed aperture shape optimiza-tion method,and the gradients of the aperture gradient map were modulated using conjugate gradients to form a modulated gradient map.The aperture shape was generated based on the modulated gradient map.The proposed optimization method does not change the optimal solution of the original optimization problem,but changes the iterative search direction when generating the aperture shape.The performance of the proposed method was verified using cases of head and neck cancer,and prostate cancer.The optimization results indicate that the proposed optimization method better protects the organs at risk and rapidly reduces the objective function value by ensuring a similar dose distribution to the planning target volume.Compared to the contrasting methods,the normal tissue complication probability obtained by the proposed optimization method decreased by up to 4.61%,and the optimization time of the proposed method decreased by 5.26%on average for ten cancer cases.The effectiveness and acceleration of the proposed method were verified through comparative experiments.According to the comparative experiments,the results indicate that the proposed optimization method is more suitable for clinical applications.It is feasible for the aperture shape optimization involving the proposed method.
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.
基金National Research Council of Thailand(NRCT)and Khon Kaen University:N42A650291.
文摘The purpose of these investigations is to find the numerical outcomes of the fractional kind of biological system based on Leptospirosis by exploiting the strength of artificial neural networks aided by scale conjugate gradient,called ANNs-SCG.The fractional derivatives have been applied to get more reliable performances of the system.The mathematical form of the biological Leptospirosis system is divided into five categories,and the numerical performances of each model class will be provided by using the ANNs-SCG.The exactness of the ANNs-SCG is performed using the comparison of the reference and obtained results.The reference solutions have been obtained by using theAdams numerical scheme.For these investigations,the data selection is performed at 82%for training,while the statics for both testing and authentication is selected as 9%.The procedures based on the recurrence,mean square error,error histograms,regression,state transitions,and correlation will be accomplished to validate the fitness,accuracy,and reliability of the ANNs-SCG scheme.
基金support from the NSRF via the program anagement Unit for Human Resources&Institutional Development,Research and Innovation[Grant number B05F640183]Chiang Mai University.Watcharaporn Cholamjiak would like to thank National Research Council of Thailand (N42A650334)Thailand Science Research and Innovation,the University of Phayao (Grant No.FF66-UoE).
文摘The purpose of this study is to present the numerical performancesand interpretations of the SEIR nonlinear system based on the Zika virusspreading by using the stochastic neural networks based intelligent computingsolver. The epidemic form of the nonlinear system represents the four dynamicsof the patients, susceptible patients S(y), exposed patients hospitalized inhospital E(y), infected patients I(y), and recovered patients R(y), i.e., SEIRmodel. The computing numerical outcomes and performances of the systemare examined by using the artificial neural networks (ANNs) and the scaledconjugate gradient (SCG) for the training of the networks, i.e., ANNs-SCG.The correctness of the ANNs-SCG scheme is observed by comparing theproposed and reference solutions for three cases of the SEIR model to solvethe nonlinear system based on the Zika virus spreading dynamics throughthe knacks of ANNs-SCG procedure based on exhaustive experimentations.The outcomes of the ANNs-SCG algorithm are found consistently in goodagreement with standard numerical solutions with negligible errors. Moreover,the procedure’s constancy, dependability, and exactness are perceived by usingthe values of state transitions, error histogram measures, correlation, andregression analysis.