It is essential to ac quire sound speed profiles(SSPs)in high-precision spatiotemporal resolution for undersea acoustic activities.However,conventional observation methods cannot obtain high-resolution SSPs.Besides,S ...It is essential to ac quire sound speed profiles(SSPs)in high-precision spatiotemporal resolution for undersea acoustic activities.However,conventional observation methods cannot obtain high-resolution SSPs.Besides,S SPs are complex and changeable in time and space,especially in coastal areas.We proposed a new space-time multigrid three-dimensional variational method with weak constraint term(referred to as STC-MG3DVar)to construct high-precision spatiotemporal resolution SSPs in coastal areas,in which sound velocity is defined as the analytical variable,and the Chen-Millero sound velocity empirical formula is introduced as a weak constraint term into the cost function of the STC-MG3DVar.The spatiotemporal correlation of sound velocity observations is taken into account in the STC-MG3DVar method,and the multi-scale information of sound velocity observations from long waves to short waves can be successively extracted.The weak constraint term can optimize sound velocity by the physical relationship between sound velocity and temperature-salinity to obtain more reasonable and accurate SSPs.To verify the accuracy of the STC-MG3DVar,SSPs observations and CTD observations(temperature observations,salinity observations)are obtained from field experiments in the northern coastal area of the Shandong Peninsula.The average root mean square error(RMSE)of the STC-MG3DVar-constructed SSPs is 0.132 m/s,and the STC-MG3DVar method can improve the SSPs construction accuracy over the space-time multigrid 3DVar without weak constraint term(ST-MG3DVar)by 10.14%and over the spatial multigrid 3DVar with weak constraint term(SC-MG3DVar)by 44.19%.With the advantage of the constraint term and the spatiotemporal correlation information,the proposed STC-MG3DVar method works better than the ST-MG3DVar and the SCMG3DVar in constructing high-precision spatiotemporal re solution SSPs.展开更多
In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonl...In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.展开更多
To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and invest...To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.展开更多
In order to solve the so-called "bull-eye" problem caused by using a simple bilinear interpolation as an observational mapping operator in the cost function in the multigrid three-dimensional variational (3DVAR) d...In order to solve the so-called "bull-eye" problem caused by using a simple bilinear interpolation as an observational mapping operator in the cost function in the multigrid three-dimensional variational (3DVAR) data assimilation scheme, a smoothing term, equivalent to a penalty term, is introduced into the cost function to serve as a means of troubleshooting. A theoretical analysis is first performed to figure out what on earth results in the issue of "bull-eye", and then the meaning of such smoothing term is elucidated and the uniqueness of solution of the multigrid 3DVAR with the smoothing term added is discussed through the theoretical deduction for one-dimensional (1D) case, and two idealized data assimilation experiments (one- and two-dimensional (2D) cases). By exploring the relationship between the smoothing term and the recursive filter theoretically and practically, it is revealed why satisfied analysis results can be achieved by using such proposed solution for the issue of the multigrid 3DVAR.展开更多
Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid...Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, V-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.展开更多
In this paper,standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full elli...In this paper,standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full elliptic regularity,so they can be used to tackle more general elliptic problems.Numerical experiments are reported to support our theory.展开更多
We propose an efficient and robust algorithm to solve the steady Euler equa- tions on unstructured grids.The new algorithm is a Newton-iteration method in which each iteration step is a linear multigrid method using b...We propose an efficient and robust algorithm to solve the steady Euler equa- tions on unstructured grids.The new algorithm is a Newton-iteration method in which each iteration step is a linear multigrid method using block lower-upper symmetric Gauss-Seidel(LU-SGS)iteration as its smoother To regularize the Jacobian matrix of Newton-iteration,we adopted a local residual dependent regularization as the replace- ment of the standard time-stepping relaxation technique based on the local CFL number The proposed method can be extended to high order approximations and three spatial dimensions in a nature way.The solver was tested on a sequence of benchmark prob- lems on both quasi-uniform and local adaptive meshes.The numerical results illustrated the efficiency and robustness of our algorithm.展开更多
Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method (EXCMG) is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial val...Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method (EXCMG) is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.展开更多
A system of linear time-dependent hyperbolic partial differential equations in the form of the time-domain Maxwell′s equations is numerically solved using ageometric multigrid method.The multilevel method is an adapt...A system of linear time-dependent hyperbolic partial differential equations in the form of the time-domain Maxwell′s equations is numerically solved using ageometric multigrid method.The multilevel method is an adaptation of Ni′s cell-vertex based multigrid technique,originally proposed for accelerating steady state convergence of nonlinear time-dependent Euler equations of gas dynamics.We discuss issues pertaining to the application of the geometric multigrid method to a system of equations where the major issue is of accurately propagating linear waves over large distances leading to major constraints on the required grid resolution in terms of points-perwavelength.展开更多
In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain a...In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform.We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps.The variable V-cycle multigrid preconditioner are also available.展开更多
A new forecasting system-the System of Multigrid Nonlinear Least-squares Four-dimensional Variational(NLS-4DVar)Data Assimilation for Numerical Weather Prediction(SNAP)-was established by building upon the multigrid N...A new forecasting system-the System of Multigrid Nonlinear Least-squares Four-dimensional Variational(NLS-4DVar)Data Assimilation for Numerical Weather Prediction(SNAP)-was established by building upon the multigrid NLS-4DVar data assimilation scheme,the operational Gridpoint Statistical Interpolation(GSI)−based data-processing and observation operators,and the widely used Weather Research and Forecasting numerical model.Drawing upon lessons learned from the superiority of the operational GSI analysis system,for its various observation operators and the ability to assimilate multiple-source observations,SNAP adopts GSI-based data-processing and observation operator modules to compute the observation innovations.The multigrid NLS-4DVar assimilation framework is used for the analysis,which can adequately correct errors from large to small scales and accelerate iteration solutions.The analysis variables are model state variables,rather than the control variables adopted in the conventional 4DVar system.Currently,we have achieved the assimilation of conventional observations,and we will continue to improve the assimilation of radar and satellite observations in the future.SNAP was evaluated by case evaluation experiments and one-week cycling assimilation experiments.In the case evaluation experiments,two six-hour time windows were established for assimilation experiments and precipitation forecasts were verified against hourly precipitation observations from more than 2400 national observation sites.This showed that SNAP can absorb observations and improve the initial field,thereby improving the precipitation forecast.In the one-week cycling assimilation experiments,six-hourly assimilation cycles were run in one week.SNAP produced slightly lower forecast RMSEs than the GSI 4DEnVar(Four-dimensional Ensemble Variational)as a whole and the threat scores of precipitation forecasts initialized from the analysis of SNAP were higher than those obtained from the analysis of GSI 4DEnVar.展开更多
In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the fu...In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of fow equation in original full-order space,and the discrete empirical interpolation method(DEIM)is applied to speed up the solution of Peng-Robinson equation of state in reduced-order subspace.The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reducedorder subspace,which not only preserves good prediction accuracy of full-order model,but also gains dramatic computational acceleration by multigrid and DEIM.Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method.In addition,the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng-Robinson equation of state is demonstrated and carried out in detail.Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors.The general acceleration is up to 50-60 times faster than that of standard semi-implicit method in two-dimensional simulations,but the average relative errors of numerical results between these two methods only have the order of magnitude 10^(−4)-10^(−6)%.展开更多
In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic mul...In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.展开更多
This paper proposes a new method for data assimilation of the surface radial current observed by High Frequency ground wave radar and optimization of the bottom friction coefficient.In this method,the shallow water wa...This paper proposes a new method for data assimilation of the surface radial current observed by High Frequency ground wave radar and optimization of the bottom friction coefficient.In this method,the shallow water wave equation is introduced into the cost function of the multigrid three-dimensional variation data assimilation method as the weak constraint term,the surface current and the bottom friction coefficient are defined as the analytical variables,and the high spatiotemporal resolution surface radial flow observed by the high-frequency ground wave radar is used to optimize the surface current and bottom friction coefficient.This method can effectively consider the spatiotemporal correlation of radar data and extract multiscale information from surface radial flow data from long waves to short waves.Introducing the shallow water wave equation into the cost function as a weak constraint condition can adjust both the momentum and mass fields simultaneously to obtain more reasonable analysis information.The optimized bottom friction coefficient is introduced into the regional ocean numerical model to carry out numerical experiments.The test results show that the bottom friction coefficient obtained by this method can effectively improve the accuracy of the numerical simulation of sea surface height in the offshore area and reduce the simulation error.展开更多
基金Supported by the National Natural Science Foundation of China(No.41876014)the Open Project of Tianjin Key Laboratory of Oceanic Meteorology(No.2020TKLOMYB04)。
文摘It is essential to ac quire sound speed profiles(SSPs)in high-precision spatiotemporal resolution for undersea acoustic activities.However,conventional observation methods cannot obtain high-resolution SSPs.Besides,S SPs are complex and changeable in time and space,especially in coastal areas.We proposed a new space-time multigrid three-dimensional variational method with weak constraint term(referred to as STC-MG3DVar)to construct high-precision spatiotemporal resolution SSPs in coastal areas,in which sound velocity is defined as the analytical variable,and the Chen-Millero sound velocity empirical formula is introduced as a weak constraint term into the cost function of the STC-MG3DVar.The spatiotemporal correlation of sound velocity observations is taken into account in the STC-MG3DVar method,and the multi-scale information of sound velocity observations from long waves to short waves can be successively extracted.The weak constraint term can optimize sound velocity by the physical relationship between sound velocity and temperature-salinity to obtain more reasonable and accurate SSPs.To verify the accuracy of the STC-MG3DVar,SSPs observations and CTD observations(temperature observations,salinity observations)are obtained from field experiments in the northern coastal area of the Shandong Peninsula.The average root mean square error(RMSE)of the STC-MG3DVar-constructed SSPs is 0.132 m/s,and the STC-MG3DVar method can improve the SSPs construction accuracy over the space-time multigrid 3DVar without weak constraint term(ST-MG3DVar)by 10.14%and over the spatial multigrid 3DVar with weak constraint term(SC-MG3DVar)by 44.19%.With the advantage of the constraint term and the spatiotemporal correlation information,the proposed STC-MG3DVar method works better than the ST-MG3DVar and the SCMG3DVar in constructing high-precision spatiotemporal re solution SSPs.
文摘In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.
基金supported by the National Natural Science Foundation of the People’s Republic of China“The research of finite element methods for eigenvalue problems in inverse scattering”(12261024)。
文摘To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.
基金The National Basic Research Program of China under contract No. 2013CB430304the National High-Tech R&D Program of China under contract No. 2013AA09A505the National Natural Science Foundation of China under contract Nos 41030854,40906015,40906016,41106005 and 41176003
文摘In order to solve the so-called "bull-eye" problem caused by using a simple bilinear interpolation as an observational mapping operator in the cost function in the multigrid three-dimensional variational (3DVAR) data assimilation scheme, a smoothing term, equivalent to a penalty term, is introduced into the cost function to serve as a means of troubleshooting. A theoretical analysis is first performed to figure out what on earth results in the issue of "bull-eye", and then the meaning of such smoothing term is elucidated and the uniqueness of solution of the multigrid 3DVAR with the smoothing term added is discussed through the theoretical deduction for one-dimensional (1D) case, and two idealized data assimilation experiments (one- and two-dimensional (2D) cases). By exploring the relationship between the smoothing term and the recursive filter theoretically and practically, it is revealed why satisfied analysis results can be achieved by using such proposed solution for the issue of the multigrid 3DVAR.
基金Projects(2006AA06Z105, 2007AA06Z134) supported by the National High-Tech Research and Development Program of ChinaProjects(2007, 2008) supported by China Scholarship Council (CSC)
文摘Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, V-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.
基金supported by the National Basic Research Program of China under the grant 2005CB321701the National Science Foundation(NSF) of China(10731060)111 project(B08018)
文摘In this paper,standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full elliptic regularity,so they can be used to tackle more general elliptic problems.Numerical experiments are reported to support our theory.
文摘We propose an efficient and robust algorithm to solve the steady Euler equa- tions on unstructured grids.The new algorithm is a Newton-iteration method in which each iteration step is a linear multigrid method using block lower-upper symmetric Gauss-Seidel(LU-SGS)iteration as its smoother To regularize the Jacobian matrix of Newton-iteration,we adopted a local residual dependent regularization as the replace- ment of the standard time-stepping relaxation technique based on the local CFL number The proposed method can be extended to high order approximations and three spatial dimensions in a nature way.The solver was tested on a sequence of benchmark prob- lems on both quasi-uniform and local adaptive meshes.The numerical results illustrated the efficiency and robustness of our algorithm.
基金Supported by National Natural Science Foundation of China (10771063)the Doctor Programme of the National Education Committee (20050542006)
文摘Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method (EXCMG) is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.
文摘A system of linear time-dependent hyperbolic partial differential equations in the form of the time-domain Maxwell′s equations is numerically solved using ageometric multigrid method.The multilevel method is an adaptation of Ni′s cell-vertex based multigrid technique,originally proposed for accelerating steady state convergence of nonlinear time-dependent Euler equations of gas dynamics.We discuss issues pertaining to the application of the geometric multigrid method to a system of equations where the major issue is of accurately propagating linear waves over large distances leading to major constraints on the required grid resolution in terms of points-perwavelength.
基金This research was supported by the National Natural Science Foundation of China under grant 10071015
文摘In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform.We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps.The variable V-cycle multigrid preconditioner are also available.
基金the National Key Research and Development Program of China(Grant No.2016YFA0600203)the National Natural Science Foundation of China(Grant No.41575100)+1 种基金the Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZDY-SSW-DQC012)the CMA Special Public Welfare Research Fund(Grant No.GYHY201506002).
文摘A new forecasting system-the System of Multigrid Nonlinear Least-squares Four-dimensional Variational(NLS-4DVar)Data Assimilation for Numerical Weather Prediction(SNAP)-was established by building upon the multigrid NLS-4DVar data assimilation scheme,the operational Gridpoint Statistical Interpolation(GSI)−based data-processing and observation operators,and the widely used Weather Research and Forecasting numerical model.Drawing upon lessons learned from the superiority of the operational GSI analysis system,for its various observation operators and the ability to assimilate multiple-source observations,SNAP adopts GSI-based data-processing and observation operator modules to compute the observation innovations.The multigrid NLS-4DVar assimilation framework is used for the analysis,which can adequately correct errors from large to small scales and accelerate iteration solutions.The analysis variables are model state variables,rather than the control variables adopted in the conventional 4DVar system.Currently,we have achieved the assimilation of conventional observations,and we will continue to improve the assimilation of radar and satellite observations in the future.SNAP was evaluated by case evaluation experiments and one-week cycling assimilation experiments.In the case evaluation experiments,two six-hour time windows were established for assimilation experiments and precipitation forecasts were verified against hourly precipitation observations from more than 2400 national observation sites.This showed that SNAP can absorb observations and improve the initial field,thereby improving the precipitation forecast.In the one-week cycling assimilation experiments,six-hourly assimilation cycles were run in one week.SNAP produced slightly lower forecast RMSEs than the GSI 4DEnVar(Four-dimensional Ensemble Variational)as a whole and the threat scores of precipitation forecasts initialized from the analysis of SNAP were higher than those obtained from the analysis of GSI 4DEnVar.
基金This study is supported by the National Natural Science Foundation of China(Nos.51904031,51936001)the Beijing Natural Science Foundation(No.3204038)the Jointly Projects of Beijing Natural Science Foundation and Beijing Municipal Education Commission(No.KZ201810017023).
文摘In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of fow equation in original full-order space,and the discrete empirical interpolation method(DEIM)is applied to speed up the solution of Peng-Robinson equation of state in reduced-order subspace.The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reducedorder subspace,which not only preserves good prediction accuracy of full-order model,but also gains dramatic computational acceleration by multigrid and DEIM.Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method.In addition,the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng-Robinson equation of state is demonstrated and carried out in detail.Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors.The general acceleration is up to 50-60 times faster than that of standard semi-implicit method in two-dimensional simulations,but the average relative errors of numerical results between these two methods only have the order of magnitude 10^(−4)-10^(−6)%.
文摘In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.
基金supported by the National Natural Science Foundation of China (Nos. 41506039, 41776004, 41775100 and 41606039)the National Key Research and Development Program of China (No. 2016YFC1401800)+1 种基金the Fundamental Research Funds for the Central Universities (No. 2016B12514)the National Programme on Global Change and Air-Sea Interaction of China (No. GASI-IPO VAI-04)
文摘This paper proposes a new method for data assimilation of the surface radial current observed by High Frequency ground wave radar and optimization of the bottom friction coefficient.In this method,the shallow water wave equation is introduced into the cost function of the multigrid three-dimensional variation data assimilation method as the weak constraint term,the surface current and the bottom friction coefficient are defined as the analytical variables,and the high spatiotemporal resolution surface radial flow observed by the high-frequency ground wave radar is used to optimize the surface current and bottom friction coefficient.This method can effectively consider the spatiotemporal correlation of radar data and extract multiscale information from surface radial flow data from long waves to short waves.Introducing the shallow water wave equation into the cost function as a weak constraint condition can adjust both the momentum and mass fields simultaneously to obtain more reasonable analysis information.The optimized bottom friction coefficient is introduced into the regional ocean numerical model to carry out numerical experiments.The test results show that the bottom friction coefficient obtained by this method can effectively improve the accuracy of the numerical simulation of sea surface height in the offshore area and reduce the simulation error.