We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonline...We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.展开更多
A box model of the interhemispheric thermohaline circulation （THC） in atmosphere-ocean for global climate is considered. By using the generalized variational iteration method, the approximate solution of a simplifie...A box model of the interhemispheric thermohaline circulation （THC） in atmosphere-ocean for global climate is considered. By using the generalized variational iteration method, the approximate solution of a simplified nonlinear model is studied. The generalized variational iteration method is an analytic method, and the obtained analytic solution can be operated sequentially. The authors also diversify qualitative and quantitative behaviors for corresponding physical quantities.展开更多
A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations.Common differential operators as well as the variational forms are defined within the ...A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations.Common differential operators as well as the variational forms are defined within the context of nonlocal operators.The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity,which is necessary for the eigenvalue analysis such as the waveguide problem.The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields.The governing equations are converted into nonlocal integral form.An hourglass energy functional is introduced for the elimination of zeroenergy modes.Finally,the proposed method is validated by testing three classical benchmark problems.展开更多
A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO mode...A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.展开更多
The quantitative analysis shows that no theoretical model for 3-d magnetoelastic bodies, in literatures to date, can commonly simulate two kinds of distinct experimental phenomena on magnetoelastic interaction of ferr...The quantitative analysis shows that no theoretical model for 3-d magnetoelastic bodies, in literatures to date, can commonly simulate two kinds of distinct experimental phenomena on magnetoelastic interaction of ferromagnetic structures. This makes it difficult to effectively discribe the magnetoelastic mechanical behavior of structures with complex geometry, such as shells. Therefore, it is a key step for simulating magnetoelastic mechanical characteristics of structures with complex geometry to establish a 3-d model which also can commonly characterize the two distinct experimental phenomena. A theoretical model for three dimension magnetizable elastic bodies, which is commonly suitable for the two kinds of experimental phenomena on magnetoelastic interaction of ferromagnetic plates, is presented by the variational principle for the total energy functional of the coupling system of the 3-d ferromagnetic bodies. It is found that for the case of linear isotropic magnetic materials, the magnetic forces obtained by this model include not only the body magnetic force which is the same as that got from the magnetic dipole model, but also a distribution of the magnetic traction on the surface of the magnetizable body. And the value of the traction is equal to the jumping one of the Faraday electromagnetic stress on the two sides of the surface, which does not appear in any model, such as magnetic dipole model and axiomatic model.展开更多
The scientific design and preliminary results of the data assimilation component of the Global-Regional Prediction and Assimilation System (GRAPES) recently developed in China Meteorological Administration (CMA) are p...The scientific design and preliminary results of the data assimilation component of the Global-Regional Prediction and Assimilation System (GRAPES) recently developed in China Meteorological Administration (CMA) are presented in this paper. This is a three-dimensional variational (3DVar) assimilation system set up on global and regional grid meshes favorable for direct assimilation of the space-based remote sensing data and matching the frame work of the prediction model GRAPES. The state variables are assumed to decompose balanced and unbalanced components. By introducing a simple transformation from the state variables to the control variables with a recursive or spectral filter, the convergence rate of iteration for minimization of the cost function in 3DVar is greatly accelerated. The definition of dynamical balance depends on the characteristic scale of the circulation considered. The ratio of the balanced to the unbalanced parts is controlled by the prescribed statistics of background errors. Idealized trials produce the same results as the analytic solution. The results of real data case studies show the capability of the system to improve analysis compared to the traditional schemes. Finally, further development of the system is discussed.展开更多
This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assim...This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.展开更多
The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known varia...The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made.展开更多
Based on the porous media theory and by taking into account the efects of the pore fuid viscidity, energy exchanges due to the additional thermal conduction and convection between solid and fuid phases, a mathematical...Based on the porous media theory and by taking into account the efects of the pore fuid viscidity, energy exchanges due to the additional thermal conduction and convection between solid and fuid phases, a mathematical model for the dynamic-thermo-hydro-mechanical coupling of a non-local thermal equilibrium fuid-saturated porous medium, in which the two constituents are assumed to be incompressible and immiscible, is established under the assumption of small de- formation of the solid phase, small velocity of the fuid phase and small temperature changes of the two constituents. The mathematical model of a local thermal equilibrium fuid-saturated porous medium can be obtained directly from the above one. Several Gurtin-type variational principles, especially Hu-Washizu type variational principles, for the initial boundary value problems of dy- namic and quasi-static responses are presented. It should be pointed out that these variational principles can be degenerated easily into the case of isothermal incompressible fuid-saturated elastic porous media, which have been discussed previously.展开更多
The corresponding solution for a class of disturbed KdV equation is considered using the analytic method. From the generalized variational iteration theory, the problem of solving soliton for the corresponding equatio...The corresponding solution for a class of disturbed KdV equation is considered using the analytic method. From the generalized variational iteration theory, the problem of solving soliton for the corresponding equation translates into the problem of variational iteration. And then the approximate solution of the soliton for the equation is obtained.展开更多
The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We...The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time T_M, with T_M in the interval [1, 2×3^(1/2)]/M for M > 0 and in the interval [2×2^(1/2), 2(1 +3^(1/2))]/|M | for M < 0. The exact value of TM is however unknown.We investigate in this work the determination of the minimal time T_M employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L^2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.展开更多
A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding...A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding one established by Karatzes et al. before. By applying some analysis methods different from earlier works, the sufficient and necessary conditions of the existence of optimal control have been obtained. If an optimal control exists, it is a 'transient reflection' of state process for two lines.展开更多
In the present work, the data assimilation problem in meteorology and physical oceanography is re-examined using the variational optimal control approaches in combination with regularization techniques in inverse prob...In the present work, the data assimilation problem in meteorology and physical oceanography is re-examined using the variational optimal control approaches in combination with regularization techniques in inverse problem. Here the estimations of the initial condition, boundary condition and model parameters are performed simultaneously in the framework of variational data assimilation. To overcome the difficulty of ill-posedness, especially for the model parameters distributed in space and time, an additional term is added into the cost functional as a stabilized functional. Numerical experiments show that even with noisy observations the initial conditions and model parameters are recovered to an acceptable degree of accuracy.展开更多
This paper aims at constructing an emission source inversion model using a variational processing method and adaptive nudging scheme for the Community Multiscale Air Quality Model (CMAQ) based on satellite data to inv...This paper aims at constructing an emission source inversion model using a variational processing method and adaptive nudging scheme for the Community Multiscale Air Quality Model (CMAQ) based on satellite data to investigate the applicability of high resolution OMI (Ozone Monitoring Instrument) column concentration data for air quality forecasts over the North China. The results show a reasonable consistency and good correlation between the spatial distributions of NO2 from surface and OMI satellite measurements in both winter and summer. Such OMI products may be used to implement integrated variational analysis based on observation data on the ground. With linear and variational corrections made, the spatial distribution of OMI NO2 clearly revealed more localized distributing characteristics of NO2 concentration. With such information, emission sources in the southwest and southeast of North China are found to have greater impacts on air quality in Beijing. When the retrieved emission source inventory based on high-resolution OMI NO2 data was used, the coupled Weather Research Forecasting CMAQ model (WRF-CMAQ) performed significantly better in forecasting NO2 concentration level and its tendency as reflected by the more consistencies between the NO2 concentrations from surface observation and model result. In conclusion, satellite data are particularly important for simulating NO2 concentrations on urban and street-block scale. High-resolution OMI NO2 data are applicable for inversing NOx emission source inventory, assessing the regional pollution status and pollution control strategy, and improving the model forecasting results on urban scale.展开更多
Without applying any stable element techniques in the mixed methods, two simple generalized mixed element(GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner(H–R...Without applying any stable element techniques in the mixed methods, two simple generalized mixed element(GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner(H–R) variational principle. The main features of the GME formulations are that the common C0-continuous polynomial shape functions for displacement methods are used to express both displacement and stress variables, and the coefficient matrix of these formulations is not only automatically symmetric but also invertible. Hence, the numerical results of the generalized mixed methods based on the GME formulations are stable. Displacement as well as stress results can be obtained directly from the algebraic system for finite element analysis after introducing stress and displacement boundary conditions simultaneously. Numerical examples show that displacement and stress results retain the same accuracy. The results of the noncompatible generalized mixed method proposed herein are more accurate than those of the standard noncompatible displacement method. The noncompatible generalized mixed element is less sensitive to element geometric distortions.展开更多
A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains multipole correctio...A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains multipole corrections, converges quickly and is variationally, changeable. The new potential and the ECM are applied to variationally studying the potential energies of eight electronic states of several diatomic molecular ions: the A2π state of CO+, the X2∑ g + state of Li 2 + , the X2∑ g + state of He 2 + , the 12∏u state of Na 2 + , the A2∏u state of N 2 + , the X1∑+ state of KrH+, the X2∑+ state of SiO+ and the A2π state of SO+ ion. The present results agree excellently with the experiment-based Rydberg-Klein-Rees (RKR) potentials, and are superior to the commonly used Huxley-Murrell-Sorbie (HMS) analytical potentials, and are better in some cases than some quantum mechanicalab initio potentials in the ionic asymptotic and dissociation regions.展开更多
For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process...For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.展开更多
The detailed kinematic structure of a heavy rain event that occurred in the middle reaches of the Yangtze River was investigated using dual-Doppler radar observation. A variational analysis method was developed to ob... The detailed kinematic structure of a heavy rain event that occurred in the middle reaches of the Yangtze River was investigated using dual-Doppler radar observation. A variational analysis method was developed to obtain the three-dimensional wind fields. Before the analysis, a data preprocessing procedure was carried out, in which the temporal variation with the scanning time interval and the effect of the earth curvature on the data position were taken into account. The analysis shows that a shear line in the lower and middle levels played an important role in the rainfall event. The precipitation fell mainly on the south end of the shear line where southerly flow prevailed and convergence and updraft were obvious. With the movement and decay of the shear line, the precipitation moved and decayed correspondingly.展开更多
Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.
基金supported in part by the National Key Basic Research Program of China 2015CB856000Major Program of NNSFC under Grant 91130005,DOE Grant DE-SC0009248ONR Grant N00014-13-1-0338.
文摘We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.
基金This research is supported by the National Natural Science Foundation of China under Grant Nos. 40576012 and 90111011, the State Key Development Program for Basics Research of China under Grant No. 2004CB418304, the Key Project of the Chinese Academy of Sciences under Grant No. KZCX3-SW-221 and in part by E- Institutes of Shanghai Municipal Education Commission under Grant No. E03004.
文摘A box model of the interhemispheric thermohaline circulation （THC） in atmosphere-ocean for global climate is considered. By using the generalized variational iteration method, the approximate solution of a simplified nonlinear model is studied. The generalized variational iteration method is an analytic method, and the obtained analytic solution can be operated sequentially. The authors also diversify qualitative and quantitative behaviors for corresponding physical quantities.
文摘A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations.Common differential operators as well as the variational forms are defined within the context of nonlocal operators.The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity,which is necessary for the eigenvalue analysis such as the waveguide problem.The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields.The governing equations are converted into nonlocal integral form.An hourglass energy functional is introduced for the elimination of zeroenergy modes.Finally,the proposed method is validated by testing three classical benchmark problems.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 90111011 and 10471039）, the National Key Basic Research Special Foundation of China （Grant Nos 2003CB415101-03 and 2004CB418304）, the Key Basic Research Foundation of the Chinese Academy of Sciences （Grant No KZCX3-SW-221） and in part by E-Institutes of Shanghai Municipal Education Commission （Grant No N.E03004）.
文摘A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19572031)the National Science Fundation for Outstanding Young Scientiests in Chinaa united foundation of the State Education Committee of China and National Natural
文摘The quantitative analysis shows that no theoretical model for 3-d magnetoelastic bodies, in literatures to date, can commonly simulate two kinds of distinct experimental phenomena on magnetoelastic interaction of ferromagnetic structures. This makes it difficult to effectively discribe the magnetoelastic mechanical behavior of structures with complex geometry, such as shells. Therefore, it is a key step for simulating magnetoelastic mechanical characteristics of structures with complex geometry to establish a 3-d model which also can commonly characterize the two distinct experimental phenomena. A theoretical model for three dimension magnetizable elastic bodies, which is commonly suitable for the two kinds of experimental phenomena on magnetoelastic interaction of ferromagnetic plates, is presented by the variational principle for the total energy functional of the coupling system of the 3-d ferromagnetic bodies. It is found that for the case of linear isotropic magnetic materials, the magnetic forces obtained by this model include not only the body magnetic force which is the same as that got from the magnetic dipole model, but also a distribution of the magnetic traction on the surface of the magnetizable body. And the value of the traction is equal to the jumping one of the Faraday electromagnetic stress on the two sides of the surface, which does not appear in any model, such as magnetic dipole model and axiomatic model.
基金Key Technologies Research and Development Program (Grant No. 2001BA607B and 2001BA607B02)National Natural Science Foundation of China (Grant No. 40518001)
文摘The scientific design and preliminary results of the data assimilation component of the Global-Regional Prediction and Assimilation System (GRAPES) recently developed in China Meteorological Administration (CMA) are presented in this paper. This is a three-dimensional variational (3DVar) assimilation system set up on global and regional grid meshes favorable for direct assimilation of the space-based remote sensing data and matching the frame work of the prediction model GRAPES. The state variables are assumed to decompose balanced and unbalanced components. By introducing a simple transformation from the state variables to the control variables with a recursive or spectral filter, the convergence rate of iteration for minimization of the cost function in 3DVar is greatly accelerated. The definition of dynamical balance depends on the characteristic scale of the circulation considered. The ratio of the balanced to the unbalanced parts is controlled by the prescribed statistics of background errors. Idealized trials produce the same results as the analytic solution. The results of real data case studies show the capability of the system to improve analysis compared to the traditional schemes. Finally, further development of the system is discussed.
基金sponsored by the U.S. National Science Foundation (Grant No.ATM0205599)the U.S. Offce of Navy Research under Grant N000140410471Dr. James A. Hansen was partially supported by US Offce of Naval Research (Grant No. N00014-06-1-0500)
文摘This study examines the performance of coupling the deterministic four-dimensional variational assimilation system （4DVAR） with an ensemble Kalman filter （EnKF） to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme （E4DVAR） benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence： the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.
基金Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060)Qiu Shi Science & Technology Foundation, DPFIHE, MCSEC and MCMCAS.
文摘The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made.
基金Project supported by the National Natural Science Foundation of China(No.10272070)and the Development Foun-dation of the Education Commission of Shanghai,China.
文摘Based on the porous media theory and by taking into account the efects of the pore fuid viscidity, energy exchanges due to the additional thermal conduction and convection between solid and fuid phases, a mathematical model for the dynamic-thermo-hydro-mechanical coupling of a non-local thermal equilibrium fuid-saturated porous medium, in which the two constituents are assumed to be incompressible and immiscible, is established under the assumption of small de- formation of the solid phase, small velocity of the fuid phase and small temperature changes of the two constituents. The mathematical model of a local thermal equilibrium fuid-saturated porous medium can be obtained directly from the above one. Several Gurtin-type variational principles, especially Hu-Washizu type variational principles, for the initial boundary value problems of dy- namic and quasi-static responses are presented. It should be pointed out that these variational principles can be degenerated easily into the case of isothermal incompressible fuid-saturated elastic porous media, which have been discussed previously.
基金Supported by the National Natural Science Foundation of China under Grant No. 40876010the Knowledge Innovation Project of Chinese Academy of Sciences under Grant No. KZCX2-YW-Q03-08+3 种基金the R & D Special Fund for Public Welfare Industry (meteorology) under Grant No. GYHY200806010the LASG State Key Laboratory Special Fundthe E-Institutes of Shanghai Municipal Education Commission under Grant No. E03004the Natural Science Foundation of Zhejiang Province under Grant No. Y6090164
文摘The corresponding solution for a class of disturbed KdV equation is considered using the analytic method. From the generalized variational iteration theory, the problem of solving soliton for the corresponding equation translates into the problem of variational iteration. And then the approximate solution of the soliton for the equation is obtained.
文摘The advection-diffusion equation y_t~ε-εy_(xx)~ε+ M y_x~ε= 0,(x, t) ∈(0, 1) ×(0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time T_M, with T_M in the interval [1, 2×3^(1/2)]/M for M > 0 and in the interval [2×2^(1/2), 2(1 +3^(1/2))]/|M | for M < 0. The exact value of TM is however unknown.We investigate in this work the determination of the minimal time T_M employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and ε. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter ε, of the optimality system associated to the control of minimal L^2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.
文摘A kind of discounted problems for singular diffusion control have been studied. The drift and diffusion coefficients of state process are nonlinear. The class of models has been basically extended from a corresponding one established by Karatzes et al. before. By applying some analysis methods different from earlier works, the sufficient and necessary conditions of the existence of optimal control have been obtained. If an optimal control exists, it is a 'transient reflection' of state process for two lines.
基金This work was supported by the National Natural Science Foundation of China (Grant Nos. 40075014 and 40175014)the Shanghai Science and Technology Association Foundation (Grant No. 02DJ14032).
文摘In the present work, the data assimilation problem in meteorology and physical oceanography is re-examined using the variational optimal control approaches in combination with regularization techniques in inverse problem. Here the estimations of the initial condition, boundary condition and model parameters are performed simultaneously in the framework of variational data assimilation. To overcome the difficulty of ill-posedness, especially for the model parameters distributed in space and time, an additional term is added into the cost functional as a stabilized functional. Numerical experiments show that even with noisy observations the initial conditions and model parameters are recovered to an acceptable degree of accuracy.
基金supported by the Ministry of Science and Technology of the People’s Republic of China (Grant No. G1999045700)the China Meteorological Administration Project (Grant No. CMATG2007Z04)
文摘This paper aims at constructing an emission source inversion model using a variational processing method and adaptive nudging scheme for the Community Multiscale Air Quality Model (CMAQ) based on satellite data to investigate the applicability of high resolution OMI (Ozone Monitoring Instrument) column concentration data for air quality forecasts over the North China. The results show a reasonable consistency and good correlation between the spatial distributions of NO2 from surface and OMI satellite measurements in both winter and summer. Such OMI products may be used to implement integrated variational analysis based on observation data on the ground. With linear and variational corrections made, the spatial distribution of OMI NO2 clearly revealed more localized distributing characteristics of NO2 concentration. With such information, emission sources in the southwest and southeast of North China are found to have greater impacts on air quality in Beijing. When the retrieved emission source inventory based on high-resolution OMI NO2 data was used, the coupled Weather Research Forecasting CMAQ model (WRF-CMAQ) performed significantly better in forecasting NO2 concentration level and its tendency as reflected by the more consistencies between the NO2 concentrations from surface observation and model result. In conclusion, satellite data are particularly important for simulating NO2 concentrations on urban and street-block scale. High-resolution OMI NO2 data are applicable for inversing NOx emission source inventory, assessing the regional pollution status and pollution control strategy, and improving the model forecasting results on urban scale.
基金supported by the National Natural Science Foundation of China (Grant 11502286)
文摘Without applying any stable element techniques in the mixed methods, two simple generalized mixed element(GME) formulations were derived by combining the minimum potential energy principle and Hellinger–Reissner(H–R) variational principle. The main features of the GME formulations are that the common C0-continuous polynomial shape functions for displacement methods are used to express both displacement and stress variables, and the coefficient matrix of these formulations is not only automatically symmetric but also invertible. Hence, the numerical results of the generalized mixed methods based on the GME formulations are stable. Displacement as well as stress results can be obtained directly from the algebraic system for finite element analysis after introducing stress and displacement boundary conditions simultaneously. Numerical examples show that displacement and stress results retain the same accuracy. The results of the noncompatible generalized mixed method proposed herein are more accurate than those of the standard noncompatible displacement method. The noncompatible generalized mixed element is less sensitive to element geometric distortions.
基金supported by the National Natural Science Foundation of China(Grant No.10074048)the Science Foundation of the Ministry of Education of China.
文摘A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains multipole corrections, converges quickly and is variationally, changeable. The new potential and the ECM are applied to variationally studying the potential energies of eight electronic states of several diatomic molecular ions: the A2π state of CO+, the X2∑ g + state of Li 2 + , the X2∑ g + state of He 2 + , the 12∏u state of Na 2 + , the A2∏u state of N 2 + , the X1∑+ state of KrH+, the X2∑+ state of SiO+ and the A2π state of SO+ ion. The present results agree excellently with the experiment-based Rydberg-Klein-Rees (RKR) potentials, and are superior to the commonly used Huxley-Murrell-Sorbie (HMS) analytical potentials, and are better in some cases than some quantum mechanicalab initio potentials in the ionic asymptotic and dissociation regions.
基金supported by the National Natural Science Foundations of China (Grant 11502286)
文摘For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.
基金This work was supported by the National Natural Science Foundation of China(Grant No.40175010)
文摘 The detailed kinematic structure of a heavy rain event that occurred in the middle reaches of the Yangtze River was investigated using dual-Doppler radar observation. A variational analysis method was developed to obtain the three-dimensional wind fields. Before the analysis, a data preprocessing procedure was carried out, in which the temporal variation with the scanning time interval and the effect of the earth curvature on the data position were taken into account. The analysis shows that a shear line in the lower and middle levels played an important role in the rainfall event. The precipitation fell mainly on the south end of the shear line where southerly flow prevailed and convergence and updraft were obvious. With the movement and decay of the shear line, the precipitation moved and decayed correspondingly.
基金This work was supported in part bythe National Natural Science Foundation of China (Grant No. 19631060) Mathematical Tian Yuan Foundation, Qiu Shi Science & Technology Foundation, RFDP and MCEC.
文摘Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.
