We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem(GEP).The layered piezo device is studied for resonance frequencies using different meshes,spar...We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem(GEP).The layered piezo device is studied for resonance frequencies using different meshes,sparse matrix representations,and numerical methods in COMSOL Multiphysics and ACELAN-COMPOS packages.Specifically,the matrix-vector and matrix-matrix product implementation for large sparse matrices is discussed.The shift-and-invert Lanczos method is used to solve the partial symmetric GEP numerically.Different solvers are compared in terms of the efficiency.The results of numerical experiments are presented.展开更多
Minimization algorithms are singular components in four-dimensional variational data assimilation(4DVar).In this paper,the convergence and application of the conjugate gradient algorithm(CGA),which is based on the Lan...Minimization algorithms are singular components in four-dimensional variational data assimilation(4DVar).In this paper,the convergence and application of the conjugate gradient algorithm(CGA),which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework,are investigated in the 4DVar minimization.First,the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is studied.The results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization,while the orthogonalized Lanczos algorithm converges stably.Second,the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System(GRAPES).The CGA is 40%more computationally efficient than the quasi-Newton method,although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton method.Thus,the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system.展开更多
基金funded by a grant of the Russian Science Foundation N 22-21-00318,https://rscf.ru/project/22-21-00318/at Southern Federal University.
文摘We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem(GEP).The layered piezo device is studied for resonance frequencies using different meshes,sparse matrix representations,and numerical methods in COMSOL Multiphysics and ACELAN-COMPOS packages.Specifically,the matrix-vector and matrix-matrix product implementation for large sparse matrices is discussed.The shift-and-invert Lanczos method is used to solve the partial symmetric GEP numerically.Different solvers are compared in terms of the efficiency.The results of numerical experiments are presented.
基金Supported by the China Meteorological Administration Special Public Welfare Research Fund(GYHY201506003)
文摘Minimization algorithms are singular components in four-dimensional variational data assimilation(4DVar).In this paper,the convergence and application of the conjugate gradient algorithm(CGA),which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework,are investigated in the 4DVar minimization.First,the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is studied.The results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization,while the orthogonalized Lanczos algorithm converges stably.Second,the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System(GRAPES).The CGA is 40%more computationally efficient than the quasi-Newton method,although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton method.Thus,the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system.
