A class of geometric asynchronous parallel algorithms for solving large-scale discrete PDE eigenvalues has been studied by the author (Sun in Sci China Math 41(8): 701–725, 2011;Sun in Math Numer Sin 34(1): 1–24, 20...A class of geometric asynchronous parallel algorithms for solving large-scale discrete PDE eigenvalues has been studied by the author (Sun in Sci China Math 41(8): 701–725, 2011;Sun in Math Numer Sin 34(1): 1–24, 2012;Sun in J Numer Methods Comput Appl 42(2): 104–125, 2021;Sun in Math Numer Sin 44(4): 433–465, 2022;Sun in Sci China Math 53(6): 859–894, 2023;Sun et al. in Chin Ann Math Ser B 44(5): 735–752, 2023). Different from traditional preconditioning algorithm with the discrete matrix directly, our geometric pre-processing algorithm (GPA) algorithm is based on so-called intrinsic geometric invariance, i.e., commutativity between the stiff matrix A and the grid mesh matrix G:AG=GA, Thus, the large-scale system solvers can be replaced with a much smaller block-solver as a pretreatment. In this paper, we study a sole PDE and assume G satisfies a periodic condition G^(m)=I,m<<dim(G). Four special cases have been studied in this paper: two-point ODE eigen-problem, Laplace eigen-problems over L-shaped region, square ring, and 3D hexahedron. Two conclusions that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” and “commutativity of grid mesh matrix and mass matrix is the essential condition for the GPA algorithm” have been obtained.展开更多
A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditi...A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditioning,the authors apply the intrinsic geometric invariance,the Grid matrix G and the discrete PDE mass matrix B,stiff matrix A satisfies commutative operator BG=GB and AG=GA,where G satisfies G^(m)=I,m<<dim(G).A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.In this paper,the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle,square,tetrahedron,cube,and so on.They give the general form of pre-processing matrix,theory and numerical test of GPA.The conclusion that“the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron”is obtained through research,and it is further found that“commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.展开更多
基金supported by the Basic Research Plan on High Performance Computing of Institute of Software(No.ISCAS-PYFX-202302).
文摘A class of geometric asynchronous parallel algorithms for solving large-scale discrete PDE eigenvalues has been studied by the author (Sun in Sci China Math 41(8): 701–725, 2011;Sun in Math Numer Sin 34(1): 1–24, 2012;Sun in J Numer Methods Comput Appl 42(2): 104–125, 2021;Sun in Math Numer Sin 44(4): 433–465, 2022;Sun in Sci China Math 53(6): 859–894, 2023;Sun et al. in Chin Ann Math Ser B 44(5): 735–752, 2023). Different from traditional preconditioning algorithm with the discrete matrix directly, our geometric pre-processing algorithm (GPA) algorithm is based on so-called intrinsic geometric invariance, i.e., commutativity between the stiff matrix A and the grid mesh matrix G:AG=GA, Thus, the large-scale system solvers can be replaced with a much smaller block-solver as a pretreatment. In this paper, we study a sole PDE and assume G satisfies a periodic condition G^(m)=I,m<<dim(G). Four special cases have been studied in this paper: two-point ODE eigen-problem, Laplace eigen-problems over L-shaped region, square ring, and 3D hexahedron. Two conclusions that “the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron” and “commutativity of grid mesh matrix and mass matrix is the essential condition for the GPA algorithm” have been obtained.
基金supported by the Basic Research Plan on High Performance Computing of Institute of Software(No.ISCAS-PYFX-202302)the National Key R&D Program of China(No.2020YFB1709502)the Advanced Space Propulsion Laboratory of BICE and Beijing Engineering Research Center of Efficient and Green Aerospace Propulsion Technology(No.Lab ASP-2019-03)。
文摘A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditioning,the authors apply the intrinsic geometric invariance,the Grid matrix G and the discrete PDE mass matrix B,stiff matrix A satisfies commutative operator BG=GB and AG=GA,where G satisfies G^(m)=I,m<<dim(G).A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.In this paper,the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle,square,tetrahedron,cube,and so on.They give the general form of pre-processing matrix,theory and numerical test of GPA.The conclusion that“the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron”is obtained through research,and it is further found that“commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.
