For sparse storage and quick access to projection matrix based on vector type, this paper proposes a method to solve the problems of the repetitive computation of projection coefficient, the large space occupation and...For sparse storage and quick access to projection matrix based on vector type, this paper proposes a method to solve the problems of the repetitive computation of projection coefficient, the large space occupation and low retrieval efficiency of projection matrix in iterative reconstruction algorithms, which calculates only once the projection coefficient and stores the data sparsely in binary format based on the variable size of library vector type. In the iterative reconstruction process, these binary files are accessed iteratively and the vector type is used to quickly obtain projection coefficients of each ray. The results of the experiments show that the method reduces the memory space occupation of the projection matrix and the computation of projection coefficient in iterative process, and accelerates the reconstruction speed.展开更多
In solving application problems, many largesscale nonlinear systems of equations result in sparse Jacobian matrices. Such nonlinear systems are called sparse nonlinear systems. The irregularity of the locations of non...In solving application problems, many largesscale nonlinear systems of equations result in sparse Jacobian matrices. Such nonlinear systems are called sparse nonlinear systems. The irregularity of the locations of nonzero elements of a general sparse matrix makes it very difficult to generally map sparse matrix computations to multiprocessors for parallel processing in a well balanced manner. To overcome this difficulty, we define a new storage scheme for general sparse matrices in this paper. With the new storage scheme, we develop parallel algorithms to solve large-scale general sparse systems of equations by interval Newton/Generalized bisection methods which reliably find all numerical solutions within a given domain.In Section 1, we provide an introduction to the addressed problem and the interval Newton's methods. In Section 2, some currently used storage schemes for sparse sys-terns are reviewed. In Section 3, new index schemes to store general sparse matrices are reported. In Section 4, we present a parallel algorithm to evaluate a general sparse Jarobian matrix. In Section 5, we present a parallel algorithm to solve the correspond-ing interval linear 8ystem by the all-row preconditioned scheme. Conclusions and future work are discussed in Section 6.展开更多
The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nonadiagonal sparse matrix is examined.The systems have arisen from the discretization of Poisson&...The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nonadiagonal sparse matrix is examined.The systems have arisen from the discretization of Poisson's equation using the 4th and 6th-order compact schemes.Four matrix-vector multiplication techniques based on four sparse matrix storage schemes are considered in the algorithm of the Krylov subspace methods and their effects are explored.The convergence history,error reduction,iteration-resolution relation and CPU-time are addressed.The efficacy of various methods is evaluated against a benchmark scenario in which the conventional second-order central difference scheme is employed to discretize Poisson's equation.The Krylov subspace methods,paired with four distinct matrix-vector multiplication strategies across three discretization approaches,are tested and implemented within an incompressible fluid flow solver to solve the elliptic segment of the equations.The resulting solution process CPU-time surface gives a new vision regarding speeding up a CFD code with proper selection of discretization stencil and matrixvector multiplication technique.展开更多
基金National Natural Science Foundation of China(No.6171177)
文摘For sparse storage and quick access to projection matrix based on vector type, this paper proposes a method to solve the problems of the repetitive computation of projection coefficient, the large space occupation and low retrieval efficiency of projection matrix in iterative reconstruction algorithms, which calculates only once the projection coefficient and stores the data sparsely in binary format based on the variable size of library vector type. In the iterative reconstruction process, these binary files are accessed iteratively and the vector type is used to quickly obtain projection coefficients of each ray. The results of the experiments show that the method reduces the memory space occupation of the projection matrix and the computation of projection coefficient in iterative process, and accelerates the reconstruction speed.
文摘In solving application problems, many largesscale nonlinear systems of equations result in sparse Jacobian matrices. Such nonlinear systems are called sparse nonlinear systems. The irregularity of the locations of nonzero elements of a general sparse matrix makes it very difficult to generally map sparse matrix computations to multiprocessors for parallel processing in a well balanced manner. To overcome this difficulty, we define a new storage scheme for general sparse matrices in this paper. With the new storage scheme, we develop parallel algorithms to solve large-scale general sparse systems of equations by interval Newton/Generalized bisection methods which reliably find all numerical solutions within a given domain.In Section 1, we provide an introduction to the addressed problem and the interval Newton's methods. In Section 2, some currently used storage schemes for sparse sys-terns are reviewed. In Section 3, new index schemes to store general sparse matrices are reported. In Section 4, we present a parallel algorithm to evaluate a general sparse Jarobian matrix. In Section 5, we present a parallel algorithm to solve the correspond-ing interval linear 8ystem by the all-row preconditioned scheme. Conclusions and future work are discussed in Section 6.
文摘The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nonadiagonal sparse matrix is examined.The systems have arisen from the discretization of Poisson's equation using the 4th and 6th-order compact schemes.Four matrix-vector multiplication techniques based on four sparse matrix storage schemes are considered in the algorithm of the Krylov subspace methods and their effects are explored.The convergence history,error reduction,iteration-resolution relation and CPU-time are addressed.The efficacy of various methods is evaluated against a benchmark scenario in which the conventional second-order central difference scheme is employed to discretize Poisson's equation.The Krylov subspace methods,paired with four distinct matrix-vector multiplication strategies across three discretization approaches,are tested and implemented within an incompressible fluid flow solver to solve the elliptic segment of the equations.The resulting solution process CPU-time surface gives a new vision regarding speeding up a CFD code with proper selection of discretization stencil and matrixvector multiplication technique.
