Three PRP-type direct search methods for unconstrained optimization are presented. The methods adopt three kinds of recently developed descent conjugate gradient methods and the idea of frame-based direct search metho...Three PRP-type direct search methods for unconstrained optimization are presented. The methods adopt three kinds of recently developed descent conjugate gradient methods and the idea of frame-based direct search method. Global convergence is shown for continuously differentiable functions. Data profile and performance profile are adopted to analyze the numerical experiments and the results show that the proposed methods are effective.展开更多
Solution to impedance distribution in electrical impedance tomography （EIT） is an ill-posed nonlinear inverse problem. It is especially difficult to reconstruct an EIT image in the center area of a measured object. ...Solution to impedance distribution in electrical impedance tomography （EIT） is an ill-posed nonlinear inverse problem. It is especially difficult to reconstruct an EIT image in the center area of a measured object. Tikhonov regularization with some prior information is a sound regnlarization method for static electrical impedance tomography under the condition that some true impedance distribution information is known a priori. This paper presents a direct search method （DSM） as pretreatment of image reconstruction through which one not only can construct a regularization matrix which may locate in areas of impedance change, but also can obtain an initial impedance distribution more similar to the true impedance distribution, as well as better current modes which can better distinguish the initial distribution and the true distribution. Simulation results indicate that, by using DSM, resolution in the center area of the measured object can be improved significantly.展开更多
In this paper, an alternating direction nonmonotone approximate Newton algorithm (ADNAN) based on nonmonotone line search is developed for solving inverse problems. It is shown that ADNAN converges to a solution of th...In this paper, an alternating direction nonmonotone approximate Newton algorithm (ADNAN) based on nonmonotone line search is developed for solving inverse problems. It is shown that ADNAN converges to a solution of the inverse problems and numerical results provide the effectiveness of the proposed algorithm.展开更多
The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The stee...The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional LMMs.In this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search rules.The corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent directions.As a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also verified.Consequently,it extends the framework of traditional LMMs.In addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the NWP-LMM.Finally,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms.In practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.展开更多
In this paper, a new superlinearly convergent algorithm for nonlinearly constrained optimization problems is presented. The search directions are directly computed by a few formulas, and neither quadratic programming ...In this paper, a new superlinearly convergent algorithm for nonlinearly constrained optimization problems is presented. The search directions are directly computed by a few formulas, and neither quadratic programming nor linear equation need to be sovled. Under mild assumptions, the new algorithm is shown to possess global and superlinear convergence.展开更多
文摘Three PRP-type direct search methods for unconstrained optimization are presented. The methods adopt three kinds of recently developed descent conjugate gradient methods and the idea of frame-based direct search method. Global convergence is shown for continuously differentiable functions. Data profile and performance profile are adopted to analyze the numerical experiments and the results show that the proposed methods are effective.
文摘Solution to impedance distribution in electrical impedance tomography （EIT） is an ill-posed nonlinear inverse problem. It is especially difficult to reconstruct an EIT image in the center area of a measured object. Tikhonov regularization with some prior information is a sound regnlarization method for static electrical impedance tomography under the condition that some true impedance distribution information is known a priori. This paper presents a direct search method （DSM） as pretreatment of image reconstruction through which one not only can construct a regularization matrix which may locate in areas of impedance change, but also can obtain an initial impedance distribution more similar to the true impedance distribution, as well as better current modes which can better distinguish the initial distribution and the true distribution. Simulation results indicate that, by using DSM, resolution in the center area of the measured object can be improved significantly.
文摘In this paper, an alternating direction nonmonotone approximate Newton algorithm (ADNAN) based on nonmonotone line search is developed for solving inverse problems. It is shown that ADNAN converges to a solution of the inverse problems and numerical results provide the effectiveness of the proposed algorithm.
基金supported by National Natural Science Foundation of China(Grant Nos.12171148 and 11771138)the Construct Program of the Key Discipline in Hunan Province.Wei Liu was supported by National Natural Science Foundation of China(Grant Nos.12101252 and 11971007)+2 种基金supported by National Natural Science Foundation of China(Grant No.11901185)National Key Research and Development Program of China(Grant No.2021YFA1001300)the Fundamental Research Funds for the Central Universities(Grant No.531118010207).
文摘The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional LMMs.In this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search rules.The corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent directions.As a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also verified.Consequently,it extends the framework of traditional LMMs.In addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the NWP-LMM.Finally,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms.In practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.
文摘In this paper, a new superlinearly convergent algorithm for nonlinearly constrained optimization problems is presented. The search directions are directly computed by a few formulas, and neither quadratic programming nor linear equation need to be sovled. Under mild assumptions, the new algorithm is shown to possess global and superlinear convergence.
