The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of t...The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of the same type: In a breakdown situation the Galerkin method is unable to calculate an approximate solution, while the least-squares method, although does not really break down, is unsucessful in reducing the norm of its residual. In this paper we first establish a unified theorem which gives a relationship between breakdowns in the two methods. We further illustrate theoretically and experimentally that if the coefficient matrix of a lienar system is of high defectiveness with the associated eigenvalues less than 1, then the restarted Galerkin and least-squares methods will be in great risks of complete breakdowns. It appears that our findings may help to understand phenomena observed practically and to derive treatments for breakdowns of this type.展开更多
Recently,the authors proposed a low-cost approach,named Optimization Approach for Linear Systems(OPALS)for solving any kind of a consistent linear system regarding the structure,characteristics,and dimension of the co...Recently,the authors proposed a low-cost approach,named Optimization Approach for Linear Systems(OPALS)for solving any kind of a consistent linear system regarding the structure,characteristics,and dimension of the coefficient matrix A.The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques.In this work,we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem(LLSP)and the Minimum Norm Linear System Problem(MNLSP)using any iterative low-cost gradient-type method,avoiding the construction of the matrices AT A or AAT,and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction.The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems.Moreover,the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems.We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.展开更多
This paper studies the discrete minus one norm least-squares methods for the stress formulation of pure displacement linear elasticity in two dimensions. The proposed least-squares functional is defined as the sum of ...This paper studies the discrete minus one norm least-squares methods for the stress formulation of pure displacement linear elasticity in two dimensions. The proposed least-squares functional is defined as the sum of the L2- and H-1-norms of the residual equations weighted appropriately. The minus one norm in the functional is replaced by the discrete minus one norm and then the discrete minus one norm least-squares methods are analyzed with various numerical results focusing on the finite element accuracy and multigrid convergence performances.展开更多
The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditio...The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditioning of the equations. Based on the singular value decomposition (SVD) of the coefficient matrix, an error based truncation algorithm is proposed in this paper. By rejection of selected small singular values, the influence of noise can be reduced. A simply-supported beam is used as a simulation example to compare the results to other methods. Illustrative numerical examples demonstrate the good efficiency and stability of the algorithm in the nondestructive identification of structural damage through modal data.展开更多
Unusually severe weather is occurring more frequently due to global climate change. Heat waves, rainstorms, snowstorms, and droughts are becoming increasingly common all over the world, threatening human lives and pro...Unusually severe weather is occurring more frequently due to global climate change. Heat waves, rainstorms, snowstorms, and droughts are becoming increasingly common all over the world, threatening human lives and property. Both temperature and precipitation are representative variables usually used to directly reflect and forecast the influences of climate change. In this study, daily data （from 1953 to 1995） and monthly data （from 1950 to 2010） of temperature and precipitation in five regions of the Amur River were examined. The significance of changes in temperature and precipitation was tested using the Mann-Kendall test method. The amplitudes were computed using the linear least-squares regression model, and the extreme temperature and precipitation were analyzed using hydrological statistical methods. The results show the following： the mean annual temperature increased significantly from 1950 to 2010 in the five regions, mainly due to the warming in spring and winter; the annual precipitation changed significantly from 1950 to 2010 only in the lower mainstream of the Amur River; the frequency of extremely low temperature events decreased from 1953 to 1995 in the mainstream of the Amur River; the frequency of high temperature events increased from 1953 to 1995 in the mainstream of the Amur River; and the frequency of extreme precipitation events did not change significantly from 1953 to 1995 in the mainstream of the Amur River. This study provides a valuable theoretical basis for settling disputes between China and Russia on sustainable development and utilization of water resources of the Amur River.展开更多
Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asympt...Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ2-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.展开更多
文摘The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of the same type: In a breakdown situation the Galerkin method is unable to calculate an approximate solution, while the least-squares method, although does not really break down, is unsucessful in reducing the norm of its residual. In this paper we first establish a unified theorem which gives a relationship between breakdowns in the two methods. We further illustrate theoretically and experimentally that if the coefficient matrix of a lienar system is of high defectiveness with the associated eigenvalues less than 1, then the restarted Galerkin and least-squares methods will be in great risks of complete breakdowns. It appears that our findings may help to understand phenomena observed practically and to derive treatments for breakdowns of this type.
文摘Recently,the authors proposed a low-cost approach,named Optimization Approach for Linear Systems(OPALS)for solving any kind of a consistent linear system regarding the structure,characteristics,and dimension of the coefficient matrix A.The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques.In this work,we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem(LLSP)and the Minimum Norm Linear System Problem(MNLSP)using any iterative low-cost gradient-type method,avoiding the construction of the matrices AT A or AAT,and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction.The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems.Moreover,the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems.We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.
基金The paper was supported by Korea Research Foundation Grant (KRF-2002-070-C00014).
文摘This paper studies the discrete minus one norm least-squares methods for the stress formulation of pure displacement linear elasticity in two dimensions. The proposed least-squares functional is defined as the sum of the L2- and H-1-norms of the residual equations weighted appropriately. The minus one norm in the functional is replaced by the discrete minus one norm and then the discrete minus one norm least-squares methods are analyzed with various numerical results focusing on the finite element accuracy and multigrid convergence performances.
文摘The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditioning of the equations. Based on the singular value decomposition (SVD) of the coefficient matrix, an error based truncation algorithm is proposed in this paper. By rejection of selected small singular values, the influence of noise can be reduced. A simply-supported beam is used as a simulation example to compare the results to other methods. Illustrative numerical examples demonstrate the good efficiency and stability of the algorithm in the nondestructive identification of structural damage through modal data.
基金supported by the Innovative Project of Scientific Research for Postgraduates in Ordinary Universities in Jiangsu Province (Grant No. CX09B_161Z)the Cultivation Project for Excellent Doctoral Dissertations in Hohai University+1 种基金the Fundamental Research Funds for the Central Universities (Grant No.2010B18714)Special Funds for Scientific Research on Public Causes of the Ministry of Water Resources of China (Grant No. 201001052)
文摘Unusually severe weather is occurring more frequently due to global climate change. Heat waves, rainstorms, snowstorms, and droughts are becoming increasingly common all over the world, threatening human lives and property. Both temperature and precipitation are representative variables usually used to directly reflect and forecast the influences of climate change. In this study, daily data （from 1953 to 1995） and monthly data （from 1950 to 2010） of temperature and precipitation in five regions of the Amur River were examined. The significance of changes in temperature and precipitation was tested using the Mann-Kendall test method. The amplitudes were computed using the linear least-squares regression model, and the extreme temperature and precipitation were analyzed using hydrological statistical methods. The results show the following： the mean annual temperature increased significantly from 1950 to 2010 in the five regions, mainly due to the warming in spring and winter; the annual precipitation changed significantly from 1950 to 2010 only in the lower mainstream of the Amur River; the frequency of extremely low temperature events decreased from 1953 to 1995 in the mainstream of the Amur River; the frequency of high temperature events increased from 1953 to 1995 in the mainstream of the Amur River; and the frequency of extreme precipitation events did not change significantly from 1953 to 1995 in the mainstream of the Amur River. This study provides a valuable theoretical basis for settling disputes between China and Russia on sustainable development and utilization of water resources of the Amur River.
基金supported by National Natural Science Foundation of China (Grant No. 10871072)Natural Science Foundation of Shanxi Province of China (Grant No. 2007011014)PhD Program Scholarship Fund of ECNU 2009
文摘Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ2-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.
