The downward continuation of potential fields is a process of calculating their values in a lower plane based on those of a certain plane.This technology is not only a data processing method for resource exploration b...The downward continuation of potential fields is a process of calculating their values in a lower plane based on those of a certain plane.This technology is not only a data processing method for resource exploration but also plays an extremely important role in military applications.However,the downward continuation of potential fields is a typical linear inverse problem that is ill-posed.Generalized minimal residuals(GMRES)is an eff ective solution to ill-posed inverse problems,but it is unstable under the condition wherein the GMRES is directly applied in the calculation process.Moreover,the long-term behavior of its iterative computation is a disordered,divergent result.Therefore,to obtain stable solutions,GMRES is applied to solve the normal equations of the downward continuation of potential fields;it is also used to prequalify for occasional interruptions in the operation process by adding the damping coefficient,thus strengthening the stability conditions of the equations of residual minimization.Finally,the stable downward continuation of the potential fields method is proposed.As indicated by the theoretical data and the measured testing data,the method proposed in this paper has the advantages of high-precision and excellent stability.Furthermore,compared with the Tikhonov iteration method,the proposed method avoids the need to choose regularization parameters.展开更多
This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster aroun...This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.展开更多
The observer-based robust fault detection filter design and optimization for networked control systems (NOSs) with uncer- tain time-varying delays are addressed. The NCSs with uncertain time-varying delays are model...The observer-based robust fault detection filter design and optimization for networked control systems (NOSs) with uncer- tain time-varying delays are addressed. The NCSs with uncertain time-varying delays are modeled as parameter-uncertain systems by the matrix theory. Based on the model, an observer-based residual generator is constructed and the sufficient condition for the existence of the desired fault detection filter is derived in terms of the linear matrix inequality. Furthermore, a time domain opti- mization approach is proposed to improve the performance of the fault detection system. To prevent the false alarms, a new thresh- old function is established, and the solution of the optimization problem is given by using the singular value decomposition (SVD) of the matrix. A numerical example is provided to illustrate the effectiveness of the proposed approach.展开更多
Initiated three decades ago,integrated design of controllers and fault detectors has continuously attracted research attention.The recent development of the unified control and detection framework with an observer-bas...Initiated three decades ago,integrated design of controllers and fault detectors has continuously attracted research attention.The recent development of the unified control and detection framework with an observer-based residual generator in its core gives a more general form of the previous works.Its applications to residual centred modelling of uncertain control systems,fault detection in feedback control systems with uncertainties,fault-tolerant control(FTC)as well as control performance degradation monitoring,detection and recovery are introduced.In conclusion,some future perspectives are proposed.展开更多
In this contribution, robust fault detection problems for discrete time-delay systems with l2-norm bounded unknown inputs are studied. The basic idea of our study is first to introduce a state-memoryless observer-base...In this contribution, robust fault detection problems for discrete time-delay systems with l2-norm bounded unknown inputs are studied. The basic idea of our study is first to introduce a state-memoryless observer-based fault detection filter (FDF) as the residual generator and then to formulate such a FDF design problem as an Hen optimization problem in the sense of increasing the sensitivity of residual to the faults, while simultaneously enhancing the robustness of residual to unknown input as well as plant input. The main results consist of the formulation of such a residual generation optimization problem, solvability conditions and the derivation of an analytic solution. The residual evaluation problem is also considered, which includes the determination of residual evaluation function and threshold. A numerical example is used to demonstrate the proposed fault detection scheme.展开更多
In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to d...In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to directly identify parameters of the observer-based residual generator based on a numerically reliable data equation obtained by filtering and sampling the input and output signals.展开更多
A Fast Multipole Method (FMM) is developed as a numerical approach to the reduction of the computational cost and requirement memory capacity for a large in solving large-scale problems. In this paper it is applied to...A Fast Multipole Method (FMM) is developed as a numerical approach to the reduction of the computational cost and requirement memory capacity for a large in solving large-scale problems. In this paper it is applied to the boundary integral equation method (BIEM) for current diffraction from arbitrary 3D bodies. The boundary integral equation is discretized by higher order elements, the FMM is applied to avoid the matrix/vector product, and the resulting algebraic equation is solved by the Generalized Conjugate Residual method (GCR). Numerical examination shows that the FMM is more efficient than the direct evaluation method in computational cost and storage of computers.展开更多
Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompress...Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompressible viscous fluid flow. The spatial discretization of the formulations is based on the multi-linear interpolating functions by using the finite element method for both the fluid and solid structures. The generalized trapezoidal rule is used to obtain apparently non-symmetric linear equations in an incremental form for the variables of the flow and vibration. The nonlinear convective term and time factors are contained in the non-symmetric coefficient matrix of the equations. The generalized minimum residual (GMRES) method is used to solve the incremental equations. A new stable algorithm of GMRES-Hughes-Newmark is developed to deal with the flow-induced vibration with dynamical fluid-structure interaction in complex geometries. Good agreement between the simulations and laboratory measurements of the pressure and blade vibration accelerations in a hydro turbine passage was obtained, indicating that the GiViRES-Hughes-Newmark algorithm presented in this paper is suitable for dealing with the flow-induced vibration of structures under small deformation.展开更多
This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized...This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner.The profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings.The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.展开更多
The linear weighted regression model is one of the models studied in many articles in recent years. Some further problems, such as disturbation, influence measure and estimate efficiency, have been discussed in this p...The linear weighted regression model is one of the models studied in many articles in recent years. Some further problems, such as disturbation, influence measure and estimate efficiency, have been discussed in this paper on the basis of the regression diagnosties. The partial conclusions of this paper are the extension of the familiar concepts in the regression diagnosties theory[2' 3,7] because they are representative of this kind of model.展开更多
Krylov subspace methods are widely used for solving sparse linear algebraic equations,but they rely heavily on preconditioners,and it is difficult to find an effective preconditioner that is efficient and stable for a...Krylov subspace methods are widely used for solving sparse linear algebraic equations,but they rely heavily on preconditioners,and it is difficult to find an effective preconditioner that is efficient and stable for all problems.In this paper,a novel projection strategy including the orthogonal and the oblique projection is proposed to improve the preconditioner,which can enhance the efficiency and stability of iteration.The proposed strategy can be considered as a minimization process,where the orthogonal projection minimizes the energy norm of error and the oblique projection minimizes the 2-norm of the residual,meanwhile they can be regarded as approaches to correct the approximation by solving low-rank inverse of the matrices.The strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable one.The paper discusses in detail the projection strategy for sparse approximate inverse(SPAI)preconditioner applied to flexible GMRES and conducts the numerical test for problems from different applications.The results show that the proposed projection strategy can significantly improve the solving process,especially for some non-converging and slowly convergence systems.展开更多
This paper presents a new mathematical model for the highly nonlinear problem of frictional con- tact. A programming model, multipole boundary element method (BEM), was developed for 3-D elastic con- tact with frict...This paper presents a new mathematical model for the highly nonlinear problem of frictional con- tact. A programming model, multipole boundary element method (BEM), was developed for 3-D elastic con- tact with friction to replace the Monte Carlo method. A numerical example shows that the optimization pro- gramming model for the point-to-surface contact with friction and the fast optimization generalized minimal residual algorithm (GMRES(m)) significantly improve the analysis of such problems relative to the conven- tional BEM.展开更多
In this paper, an unstructured, collocated finite volume method for solvingthe Navier-Stokes equations was developed by virtue of auxiliary points. The derivatives weredetermined by the Gauss theorem. The proposed met...In this paper, an unstructured, collocated finite volume method for solvingthe Navier-Stokes equations was developed by virtue of auxiliary points. The derivatives weredetermined by the Gauss theorem. The proposed method could provide control volumes with arbitrarygeometry and preserve the second-order accuracy even if highly distorted grids are used. Althougharbitrary number of cell faces can be used, the hybrid quadrilateral/triangular grids are moredesirable for the simplicity of implementation and applications to engineering problems. Thepressure-velocity coupling was treated using a SIMPLE-like algorithm. The Generalized MinimumResidual (GMRES) method with the Incomplete LU (ILU) preconditioner was used to solve linearequations. Four test cases were studied for validating the proposed method. In using this method,grid quality is not important. Thus, engineers can pay mostly attention to physical mechanism ofproblems. Turbulence models can be simply integrated and the method can be straightforwardlyextended to treat three-dimensional problems.展开更多
Despite efficient parallelism in the solution of physical parameterization in the Global/Regional Assimilation and Prediction System(GRAPES),the Helmholtz equation in the dynamic core,with the increase of resolution,c...Despite efficient parallelism in the solution of physical parameterization in the Global/Regional Assimilation and Prediction System(GRAPES),the Helmholtz equation in the dynamic core,with the increase of resolution,can hardly achieve sufficient parallelism in the solving process due to a large amount of communication and irregular access.In this paper,optimizing the Helmholtz equation solution for better performance and higher efficiency has been an urgent task.An optimization scheme for the parallel solution of the Helmholtz equation is proposed in this paper.Specifically,the geometrical multigrid optimization strategy is designed by taking advantage of the data anisotropy of grid points near the pole and the isotropy of those near memory equator in the Helmholtz equation,and the Incomplete LU(ILU)decomposition preconditioner is adopted to speed up the convergence of the improved Generalized Conjugate Residual(GCR),which effectively reduces the number of iterations and the computation time.The overall solving performance of the Helmholtz equation is improved by thread-level parallelism,vectorization,and reuse of data in the cache.The experimental results show that the proposed optimization scheme can effectively eliminate the bottleneck of the Helmholtz equation as regards the solving speed.Considering the test results on a 10-node two-way server,the solution of the Helmholtz equation,compared with the original serial version,is accelerated by 100,with one-third of iterations reduced.展开更多
The purpose of this paper is to derive the generalized conjugate residual(GCR)algorithm for finding the least squares solution on a class of Sylvester matrix equations.We prove that if the system is inconsistent,the l...The purpose of this paper is to derive the generalized conjugate residual(GCR)algorithm for finding the least squares solution on a class of Sylvester matrix equations.We prove that if the system is inconsistent,the least squares solution can be obtained within finite iterative steps in the absence of round-off errors.Furthermore,we provide a method for choosing the initial matrix to obtain the minimum norm least squares solution of the problem.Finally,we give some numerical examples to illustrate the performance of GCR algorithm.展开更多
We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically...We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.展开更多
The velocity field in the Wu River at Chongqing was simulated using the shallow water equation implemented on clustered workstations. The parallel computing technique was used to increase the comput- ing power. The sh...The velocity field in the Wu River at Chongqing was simulated using the shallow water equation implemented on clustered workstations. The parallel computing technique was used to increase the comput- ing power. The shallow water equation was discretized to a linear system of equations with a direct parallel generalized minimum residual algorithm (GMRES) used to solve the linear system. Unlike other parallel GMRES methods, the direct GMRES method does not alter the sequential algorithm, but bases the paral- lelization on basic operations such as the matrix-vector product. The computed results agree well with ob- served results. The parallel computing technique significantly increases the solution speed for this large- scale problem.展开更多
基金This research is supported by the National Key Research and Development Program of China under Grant No.2018YFC1505401the Key Research and Development Projects of the Sichuan Science and Technology Department under Grant Nos.2019YFG0460,2020YFG0303,and 2021YJ0031+1 种基金the Technology Research and Development Program of China Railway Group Limited under Grant No.CZ01-Key Point-05the Fundamental Research Funds for the Central Universities under Grant No.2682021GF019.
文摘The downward continuation of potential fields is a process of calculating their values in a lower plane based on those of a certain plane.This technology is not only a data processing method for resource exploration but also plays an extremely important role in military applications.However,the downward continuation of potential fields is a typical linear inverse problem that is ill-posed.Generalized minimal residuals(GMRES)is an eff ective solution to ill-posed inverse problems,but it is unstable under the condition wherein the GMRES is directly applied in the calculation process.Moreover,the long-term behavior of its iterative computation is a disordered,divergent result.Therefore,to obtain stable solutions,GMRES is applied to solve the normal equations of the downward continuation of potential fields;it is also used to prequalify for occasional interruptions in the operation process by adding the damping coefficient,thus strengthening the stability conditions of the equations of residual minimization.Finally,the stable downward continuation of the potential fields method is proposed.As indicated by the theoretical data and the measured testing data,the method proposed in this paper has the advantages of high-precision and excellent stability.Furthermore,compared with the Tikhonov iteration method,the proposed method avoids the need to choose regularization parameters.
基金the National Natural Science Foundation of China under Grant Nos.61273311 and 61803247.
文摘This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.
基金supported by the National Natural Science Foundation of China(6107402761273083)
文摘The observer-based robust fault detection filter design and optimization for networked control systems (NOSs) with uncer- tain time-varying delays are addressed. The NCSs with uncertain time-varying delays are modeled as parameter-uncertain systems by the matrix theory. Based on the model, an observer-based residual generator is constructed and the sufficient condition for the existence of the desired fault detection filter is derived in terms of the linear matrix inequality. Furthermore, a time domain opti- mization approach is proposed to improve the performance of the fault detection system. To prevent the false alarms, a new thresh- old function is established, and the solution of the optimization problem is given by using the singular value decomposition (SVD) of the matrix. A numerical example is provided to illustrate the effectiveness of the proposed approach.
基金This work was supported by the National Natural Science Foundation of China(62020106003,62073029)the Beijing Natural Science Foundation(4202045)the Fundamental Research Funds for the Central Universities(FRF-TP-20-012A3).
文摘Initiated three decades ago,integrated design of controllers and fault detectors has continuously attracted research attention.The recent development of the unified control and detection framework with an observer-based residual generator in its core gives a more general form of the previous works.Its applications to residual centred modelling of uncertain control systems,fault detection in feedback control systems with uncertainties,fault-tolerant control(FTC)as well as control performance degradation monitoring,detection and recovery are introduced.In conclusion,some future perspectives are proposed.
基金This project was supported by the Shandong Natural Science Foundation (Y2002G05 Y2001G01).
文摘In this contribution, robust fault detection problems for discrete time-delay systems with l2-norm bounded unknown inputs are studied. The basic idea of our study is first to introduce a state-memoryless observer-based fault detection filter (FDF) as the residual generator and then to formulate such a FDF design problem as an Hen optimization problem in the sense of increasing the sensitivity of residual to the faults, while simultaneously enhancing the robustness of residual to unknown input as well as plant input. The main results consist of the formulation of such a residual generation optimization problem, solvability conditions and the derivation of an analytic solution. The residual evaluation problem is also considered, which includes the determination of residual evaluation function and threshold. A numerical example is used to demonstrate the proposed fault detection scheme.
基金This work was supported was supported in part by the European Union under grant NeCST.
文摘In this paper, a model-free approach is presented to design an observer-based fault detection system of linear continuoustime systems based on input and output data in the time domain. The core of the approach is to directly identify parameters of the observer-based residual generator based on a numerically reliable data equation obtained by filtering and sampling the input and output signals.
文摘A Fast Multipole Method (FMM) is developed as a numerical approach to the reduction of the computational cost and requirement memory capacity for a large in solving large-scale problems. In this paper it is applied to the boundary integral equation method (BIEM) for current diffraction from arbitrary 3D bodies. The boundary integral equation is discretized by higher order elements, the FMM is applied to avoid the matrix/vector product, and the resulting algebraic equation is solved by the Generalized Conjugate Residual method (GCR). Numerical examination shows that the FMM is more efficient than the direct evaluation method in computational cost and storage of computers.
基金supported by the National Natural Science Foundation of China (No. 50839003)the Natural Science Foundation of Yunnan Province (No. 2008GA027)
文摘Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompressible viscous fluid flow. The spatial discretization of the formulations is based on the multi-linear interpolating functions by using the finite element method for both the fluid and solid structures. The generalized trapezoidal rule is used to obtain apparently non-symmetric linear equations in an incremental form for the variables of the flow and vibration. The nonlinear convective term and time factors are contained in the non-symmetric coefficient matrix of the equations. The generalized minimum residual (GMRES) method is used to solve the incremental equations. A new stable algorithm of GMRES-Hughes-Newmark is developed to deal with the flow-induced vibration with dynamical fluid-structure interaction in complex geometries. Good agreement between the simulations and laboratory measurements of the pressure and blade vibration accelerations in a hydro turbine passage was obtained, indicating that the GiViRES-Hughes-Newmark algorithm presented in this paper is suitable for dealing with the flow-induced vibration of structures under small deformation.
基金financial support from the Indian National Science Academy,New Delhi,IndiaBiluru Gurubasava Mahaswamiji Institute of Technology for the encouragement and support。
文摘This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner.The profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings.The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.
文摘The linear weighted regression model is one of the models studied in many articles in recent years. Some further problems, such as disturbation, influence measure and estimate efficiency, have been discussed in this paper on the basis of the regression diagnosties. The partial conclusions of this paper are the extension of the familiar concepts in the regression diagnosties theory[2' 3,7] because they are representative of this kind of model.
基金supported by the National Key R&D Program of China(Grant No.2021YFB2401700)the National Natural Science Foundation of China(Grant No.11672362).
文摘Krylov subspace methods are widely used for solving sparse linear algebraic equations,but they rely heavily on preconditioners,and it is difficult to find an effective preconditioner that is efficient and stable for all problems.In this paper,a novel projection strategy including the orthogonal and the oblique projection is proposed to improve the preconditioner,which can enhance the efficiency and stability of iteration.The proposed strategy can be considered as a minimization process,where the orthogonal projection minimizes the energy norm of error and the oblique projection minimizes the 2-norm of the residual,meanwhile they can be regarded as approaches to correct the approximation by solving low-rank inverse of the matrices.The strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable one.The paper discusses in detail the projection strategy for sparse approximate inverse(SPAI)preconditioner applied to flexible GMRES and conducts the numerical test for problems from different applications.The results show that the proposed projection strategy can significantly improve the solving process,especially for some non-converging and slowly convergence systems.
基金Supported by the National Natural Science Foundation of China(No. 50075075)
文摘This paper presents a new mathematical model for the highly nonlinear problem of frictional con- tact. A programming model, multipole boundary element method (BEM), was developed for 3-D elastic con- tact with friction to replace the Monte Carlo method. A numerical example shows that the optimization pro- gramming model for the point-to-surface contact with friction and the fast optimization generalized minimal residual algorithm (GMRES(m)) significantly improve the analysis of such problems relative to the conven- tional BEM.
文摘In this paper, an unstructured, collocated finite volume method for solvingthe Navier-Stokes equations was developed by virtue of auxiliary points. The derivatives weredetermined by the Gauss theorem. The proposed method could provide control volumes with arbitrarygeometry and preserve the second-order accuracy even if highly distorted grids are used. Althougharbitrary number of cell faces can be used, the hybrid quadrilateral/triangular grids are moredesirable for the simplicity of implementation and applications to engineering problems. Thepressure-velocity coupling was treated using a SIMPLE-like algorithm. The Generalized MinimumResidual (GMRES) method with the Incomplete LU (ILU) preconditioner was used to solve linearequations. Four test cases were studied for validating the proposed method. In using this method,grid quality is not important. Thus, engineers can pay mostly attention to physical mechanism ofproblems. Turbulence models can be simply integrated and the method can be straightforwardlyextended to treat three-dimensional problems.
基金partially supported by the Open Project of State Key Laboratory of Plateau Ecology and Agricuture,Qinghai University(No.2020-ZZ-03)the Qinghai Province High-End Innovative Thousand Talents Program Leading Talents+1 种基金the National Natural Science Foundation of China(Nos.61762074 and 61962051)the National Natural Science Foundation of Qinghai Province(No.2019-ZJ-7034)。
文摘Despite efficient parallelism in the solution of physical parameterization in the Global/Regional Assimilation and Prediction System(GRAPES),the Helmholtz equation in the dynamic core,with the increase of resolution,can hardly achieve sufficient parallelism in the solving process due to a large amount of communication and irregular access.In this paper,optimizing the Helmholtz equation solution for better performance and higher efficiency has been an urgent task.An optimization scheme for the parallel solution of the Helmholtz equation is proposed in this paper.Specifically,the geometrical multigrid optimization strategy is designed by taking advantage of the data anisotropy of grid points near the pole and the isotropy of those near memory equator in the Helmholtz equation,and the Incomplete LU(ILU)decomposition preconditioner is adopted to speed up the convergence of the improved Generalized Conjugate Residual(GCR),which effectively reduces the number of iterations and the computation time.The overall solving performance of the Helmholtz equation is improved by thread-level parallelism,vectorization,and reuse of data in the cache.The experimental results show that the proposed optimization scheme can effectively eliminate the bottleneck of the Helmholtz equation as regards the solving speed.Considering the test results on a 10-node two-way server,the solution of the Helmholtz equation,compared with the original serial version,is accelerated by 100,with one-third of iterations reduced.
基金Supported by Fujian Natural ScienceFoundation(Grant No.2016J01005)Strategic Priority Research Program of the Chinese Academy of Sciences(Grant No.XDB18010202).
文摘The purpose of this paper is to derive the generalized conjugate residual(GCR)algorithm for finding the least squares solution on a class of Sylvester matrix equations.We prove that if the system is inconsistent,the least squares solution can be obtained within finite iterative steps in the absence of round-off errors.Furthermore,we provide a method for choosing the initial matrix to obtain the minimum norm least squares solution of the problem.Finally,we give some numerical examples to illustrate the performance of GCR algorithm.
基金Z.Mao was supported by the Fundamental Research Funds for the Central Universities(Grant 20720210037)G.E.Karniadakis was supported by the MURI/ARO on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applications(Grant W911NF-15-1-0562)X.Chen was supported by the Fujian Provincial Natural Science Foundation of China(Grants 2022J01338,2020J01703).
文摘We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDEs)by extending the algorithm in[14]for classical ordinary differential equations(ODEs).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(GMRES).We first derive the correction matrix of the SDC method for FDEs and analyze the convergence region of the SDC method.We then present several numerical examples for stiff and non-stiff FDEs including fractional linear and nonlinear ODEs as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff problems.Furthermore,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational cost.Moreover,for long-time integration of FDEs,using the geometric mesh leads to great computational savings as the total number of degrees of freedom required is relatively small.
基金Supported by the National Natural Science Foundation of China (Nos. 50379022 and 59979013)
文摘The velocity field in the Wu River at Chongqing was simulated using the shallow water equation implemented on clustered workstations. The parallel computing technique was used to increase the comput- ing power. The shallow water equation was discretized to a linear system of equations with a direct parallel generalized minimum residual algorithm (GMRES) used to solve the linear system. Unlike other parallel GMRES methods, the direct GMRES method does not alter the sequential algorithm, but bases the paral- lelization on basic operations such as the matrix-vector product. The computed results agree well with ob- served results. The parallel computing technique significantly increases the solution speed for this large- scale problem.