In this paper, we first introduce the situation of Incomplete Factorization(IF)preconditioners. Consequently, we reduce the block tridiagonal matrix with non-singular off-diagonal blocks into a model one that has only...In this paper, we first introduce the situation of Incomplete Factorization(IF)preconditioners. Consequently, we reduce the block tridiagonal matrix with non-singular off-diagonal blocks into a model one that has only negative identity matrixfor its off-diagonals. Then we evaluate the block LU factors for the model with thehelp of M matrices. The analyses show that the evaluation is exact in some sense.For the matrices which have equal diagonal blocks and have only negative identityoff-diagonal blocks, the tendency of the factors are also focused on. Moreover,we construct a type of preconditioners with these evaluations and analyze thecondition number of the preconditioned matrices. For the model problem, we givethe evaluation and practical condition number, which shows that the evaluation isexact to some extent. At last, we implement four of these preconditioners and testthem for the model problem. The results show that our method is effective and theanalyses imply that they will be more efficient than others in parallel computing.展开更多
We study preconditioning techniques used in conjunction with the conjugate gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation ...We study preconditioning techniques used in conjunction with the conjugate gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation of electron interaction of correlated materials. Existing preconditioning techniques are not designed to be adaptive to varying numerical properties of the multi-length-scale systems. In this paper, we propose a hybrid incomplete Cholesky （HIC） preconditioner and demonstrate its adaptivity to the multi-length-scale systems. In addition, we propose an extension of the compressed sparse column with row access （CSCR） sparse matrix storage format to efficiently accommodate the data access pattem to compute the HIC preconditioner. We show that for moderately correlated materials, the HIC preconditioner achieves the optimal linear scaling of the simulation. The development of a linear-scaling preconditioner for strongly correlated materials remains an open topic.展开更多
In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems.In earlier works,the local Schu...In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems.In earlier works,the local Schur complements were computed exactly using a sparse direct solver.The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling huge problems.In this work we investigate the use of sparse approximation of the dense local Schur complements.These approximations are computed using a partial incomplete LU factorization.Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems;preliminary experiments on linear systems arising from structural mechanics are also reported.展开更多
The restrictively preconditioned conjugate gradient （RPCG） method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we...The restrictively preconditioned conjugate gradient （RPCG） method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the restrictively preconditioned conjugate gradient on normal residual （RPCGNR）, is more robust and effective than either the known RPCG method or the standard conjugate gradient on normal residual （CGNR） method when being used for solving the large sparse saddle point problems.展开更多
文摘In this paper, we first introduce the situation of Incomplete Factorization(IF)preconditioners. Consequently, we reduce the block tridiagonal matrix with non-singular off-diagonal blocks into a model one that has only negative identity matrixfor its off-diagonals. Then we evaluate the block LU factors for the model with thehelp of M matrices. The analyses show that the evaluation is exact in some sense.For the matrices which have equal diagonal blocks and have only negative identityoff-diagonal blocks, the tendency of the factors are also focused on. Moreover,we construct a type of preconditioners with these evaluations and analyze thecondition number of the preconditioned matrices. For the model problem, we givethe evaluation and practical condition number, which shows that the evaluation isexact to some extent. At last, we implement four of these preconditioners and testthem for the model problem. The results show that our method is effective and theanalyses imply that they will be more efficient than others in parallel computing.
基金supported in part by the US National Science Foundation grant 0611548in part by the US Department of Energy grant DE-FC02-06ER25793
文摘We study preconditioning techniques used in conjunction with the conjugate gradient method for solving multi-length-scale symmetric positive definite linear systems originating from the quantum Monte Carlo simulation of electron interaction of correlated materials. Existing preconditioning techniques are not designed to be adaptive to varying numerical properties of the multi-length-scale systems. In this paper, we propose a hybrid incomplete Cholesky （HIC） preconditioner and demonstrate its adaptivity to the multi-length-scale systems. In addition, we propose an extension of the compressed sparse column with row access （CSCR） sparse matrix storage format to efficiently accommodate the data access pattem to compute the HIC preconditioner. We show that for moderately correlated materials, the HIC preconditioner achieves the optimal linear scaling of the simulation. The development of a linear-scaling preconditioner for strongly correlated materials remains an open topic.
基金developed in the framework of the associated team PhyLeas(Study of parallel hybrid sparse linear solvers) funded by INRIA where the partners are INRIA,T.U.Brunswick and University of Minnesotasupported by the US Department of Energy under grant DE-FG-08ER25841 and by the Minnesota Supercomputer Institute.
文摘In this paper we study the computational performance of variants of an algebraic additive Schwarz preconditioner for the Schur complement for the solution of large sparse linear systems.In earlier works,the local Schur complements were computed exactly using a sparse direct solver.The robustness of the preconditioner comes at the price of this memory and time intensive computation that is the main bottleneck of the approach for tackling huge problems.In this work we investigate the use of sparse approximation of the dense local Schur complements.These approximations are computed using a partial incomplete LU factorization.Such a numerical calculation is the core of the multi-level incomplete factorization such as the one implemented in pARMS. The numerical and computing performance of the new numerical scheme is illustrated on a set of large 3D convection-diffusion problems;preliminary experiments on linear systems arising from structural mechanics are also reported.
基金supported by the National Basic Research Program (No.2005CB321702)the China NNSF Outstanding Young Scientist Foundation (No.10525102)the National Natural Science Foundation (No.10471146),P.R.China
文摘The restrictively preconditioned conjugate gradient （RPCG） method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the restrictively preconditioned conjugate gradient on normal residual （RPCGNR）, is more robust and effective than either the known RPCG method or the standard conjugate gradient on normal residual （CGNR） method when being used for solving the large sparse saddle point problems.
