We present a flexible version of GPBi-CG algorithm which allows for the use of a different preconditioner at each step of the algorithm. In particular, a result of the flexibility of the variable preconditioner is to ...We present a flexible version of GPBi-CG algorithm which allows for the use of a different preconditioner at each step of the algorithm. In particular, a result of the flexibility of the variable preconditioner is to use any iterative method. For example, the standard GPBi-CG algorithm itself can be used as a preconditioner, as can other Krylov subspace methods or splitting methods. Numerical experiments are conducted for flexible GPBi-CG for a few matrices including some nonsymmetric matrices. These experiments illustrate the convergence and robustness of the flexible iterative method.展开更多
Electrical capacitance tomography （ECT） is a promising technique for multi-phase flow measurement due to its high speed, low cost and non-intrusive sensing. Image reconstruction for ECT is an inverse problem of find...Electrical capacitance tomography （ECT） is a promising technique for multi-phase flow measurement due to its high speed, low cost and non-intrusive sensing. Image reconstruction for ECT is an inverse problem of finding the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. The conjugate gradient （CG） method is a popular image reconstruction method for ECT, in spite of its low convergence rate. In this paper, an advanced version of the CG method, the projected CG method, is used for image reconstruction of an ECT system. The solution space is projected into the Kryiov subspace and the inverse problem is solved by the CG method in a low-dimensional specific subspace. Both static and dynamic experiments were carried out for gas-solid two-phase flows. The flow regimes are identified using the reconstructed images obtained with the projected CG method. The results obtained indicate that the projected CG method improves the quality of reconstructed images and dramatically reduces computation time, as compared to the traditional sensitivity, Landweber, and CG methods. Furthermore, the projected CG method was also used to estimate the important parameters of the pneumatic conveying process, such as the volume concentration, flow velocity and mass flow rate of the solid phase. Therefore, the projected CG method is considered suitable for online gas-solid two-phase flow measurement,展开更多
When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the comput...When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the computational efficiency and to save storage, the Conjugate Gradient （CG） method is presented. The CG is an effective method for solving a large system of linear equations and belongs to the method of iteration with rapid convergence and high precision. An example is given and calculated results are compared to validate the proposed methods.展开更多
Recently Y. Saad proposed a flexible inner-outer preconditioned GMRES algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/GMRES (k), in wh...Recently Y. Saad proposed a flexible inner-outer preconditioned GMRES algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/GMRES (k), in which the preconditioner is constructed in the iteration step of CGS, by several steps of GMRES(k). Numerical experiments show that the residual of the outer iteration decreases rapidly. We also found the interesting residual behaviour of GMRES for the skewsymmetric linear system Ax = b, which gives a convergence result for restarted GMRES (k). For convenience, we discuss real systems.展开更多
The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the ...The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the external torque, a dynamic equation of three-dimensional attitude motion of the system is formulated. The attitude motion planning problem of the coupled-rigid-body spacecraft can be converted to a dis- crete nonlinear programming （NLP） problem using the Chebyshev-Gauss pseudospectral method （CGPM）. Solutions of the NLP problem can be obtained using the sequential quadratic programming （SQP） algorithm. Since the collocation points of the CGPM are Chebyshev-Gauss （CG） points, the integration of cost function can be approximated by the Clenshaw-Curtis quadrature, and the corresponding quadrature weights can be calculated efficiently using the fast Fourier transform （FFT）. To improve computational efficiency and numerical stability, the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of state and con- trol variables. Furthermore, numerical float errors of the state differential matrix and barycentric weights can be alleviated using trigonometric identity especially when the number of CG points is large. A simple yet efficient method is used to avoid sensitivity to the initial values for the SQP algorithm using a layered optimization strategy from a feasible solution to an optimal solution. Effectiveness of the proposed algorithm is perfect for attitude motion planning of a two-rigid-body spacecraft coupled by a ball-in-socket joint through numerical simulation.展开更多
文摘We present a flexible version of GPBi-CG algorithm which allows for the use of a different preconditioner at each step of the algorithm. In particular, a result of the flexibility of the variable preconditioner is to use any iterative method. For example, the standard GPBi-CG algorithm itself can be used as a preconditioner, as can other Krylov subspace methods or splitting methods. Numerical experiments are conducted for flexible GPBi-CG for a few matrices including some nonsymmetric matrices. These experiments illustrate the convergence and robustness of the flexible iterative method.
基金support from the National Natural Science Foundation of China(50937005,61001135,61201350)the Natural Science Foundation of Tianjin Municipal Science and Technology Commission(11JCYBJC06900)
文摘Electrical capacitance tomography （ECT） is a promising technique for multi-phase flow measurement due to its high speed, low cost and non-intrusive sensing. Image reconstruction for ECT is an inverse problem of finding the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. The conjugate gradient （CG） method is a popular image reconstruction method for ECT, in spite of its low convergence rate. In this paper, an advanced version of the CG method, the projected CG method, is used for image reconstruction of an ECT system. The solution space is projected into the Kryiov subspace and the inverse problem is solved by the CG method in a low-dimensional specific subspace. Both static and dynamic experiments were carried out for gas-solid two-phase flows. The flow regimes are identified using the reconstructed images obtained with the projected CG method. The results obtained indicate that the projected CG method improves the quality of reconstructed images and dramatically reduces computation time, as compared to the traditional sensitivity, Landweber, and CG methods. Furthermore, the projected CG method was also used to estimate the important parameters of the pneumatic conveying process, such as the volume concentration, flow velocity and mass flow rate of the solid phase. Therefore, the projected CG method is considered suitable for online gas-solid two-phase flow measurement,
文摘When material properties, geometry parameters and applied loads are assumed to be stochastic, the vibration equation of a system is transformed to static problem by using Newmark method. In order to improve the computational efficiency and to save storage, the Conjugate Gradient （CG） method is presented. The CG is an effective method for solving a large system of linear equations and belongs to the method of iteration with rapid convergence and high precision. An example is given and calculated results are compared to validate the proposed methods.
基金Supported by the State Major Key Project for Basic Researchesthe Doctorial Program Foundation of China
文摘Recently Y. Saad proposed a flexible inner-outer preconditioned GMRES algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/GMRES (k), in which the preconditioner is constructed in the iteration step of CGS, by several steps of GMRES(k). Numerical experiments show that the residual of the outer iteration decreases rapidly. We also found the interesting residual behaviour of GMRES for the skewsymmetric linear system Ax = b, which gives a convergence result for restarted GMRES (k). For convenience, we discuss real systems.
基金supported by the National Natural Science Foundation of China(No.11472058)
文摘The attitude optimal control problem （OCP） of a two-rigid-body space- craft with two rigid bodies coupled by a ball-in-socket joint is considered. Based on conservation of angular momentum of the system without the external torque, a dynamic equation of three-dimensional attitude motion of the system is formulated. The attitude motion planning problem of the coupled-rigid-body spacecraft can be converted to a dis- crete nonlinear programming （NLP） problem using the Chebyshev-Gauss pseudospectral method （CGPM）. Solutions of the NLP problem can be obtained using the sequential quadratic programming （SQP） algorithm. Since the collocation points of the CGPM are Chebyshev-Gauss （CG） points, the integration of cost function can be approximated by the Clenshaw-Curtis quadrature, and the corresponding quadrature weights can be calculated efficiently using the fast Fourier transform （FFT）. To improve computational efficiency and numerical stability, the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of state and con- trol variables. Furthermore, numerical float errors of the state differential matrix and barycentric weights can be alleviated using trigonometric identity especially when the number of CG points is large. A simple yet efficient method is used to avoid sensitivity to the initial values for the SQP algorithm using a layered optimization strategy from a feasible solution to an optimal solution. Effectiveness of the proposed algorithm is perfect for attitude motion planning of a two-rigid-body spacecraft coupled by a ball-in-socket joint through numerical simulation.
