Linear Least Squares(LLS) problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics...Linear Least Squares(LLS) problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm(SAIA) is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.展开更多
For the (2+1)-dimensional Broer–Kaup–Kupershmidt(BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the ...For the (2+1)-dimensional Broer–Kaup–Kupershmidt(BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion(CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.展开更多
This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)...This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)method,by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy.To end that,a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process.A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required.The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection,and a penalty function is also employed to remove the orthogonal constraints.According to the extreme principle,a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function.A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations.The two examples of one-dimensional heat transfer equation and nonlinear Burgers’equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references,and the dominant characteristics of the dynamics are well captured in case of few bases used only.展开更多
基金supported by Open Fund of Engineering Laboratory of Spatial Information Technology of Highway Geological Disaster Early Warning in Hunan Province(Changsha University of Science&Technology,kfj150602)Hunan Province Science and Technology Program Funded Projects,China(2015NK3035)+1 种基金the Land and Resources Department Scientific Research Project of Hunan Province,China(2013-27)the Education Department Scientific Research Project of Hunan Province,China(13C1011)
文摘Linear Least Squares(LLS) problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm(SAIA) is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.
基金Project supported by the Zhejiang Provincial Natural Science Foundation of China(Grant No.LQ13A010014)the National Natural Science Foundation of China(Grant Nos.11326164,11401528,11435005,and 11375090)
文摘For the (2+1)-dimensional Broer–Kaup–Kupershmidt(BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion(CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.
基金supported by Natural Science Foundation of China under Great Nos.11072053 and 11372068,and the National Basic Research Program of China under Grant No.2014CB74410.
文摘This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)method,by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy.To end that,a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process.A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required.The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection,and a penalty function is also employed to remove the orthogonal constraints.According to the extreme principle,a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function.A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations.The two examples of one-dimensional heat transfer equation and nonlinear Burgers’equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references,and the dominant characteristics of the dynamics are well captured in case of few bases used only.