Direct methods for sparse matrices
Direct methods for sparse matrices
Communications of the ACM
The influence of relaxed supernode partitions on the multifrontal method
ACM Transactions on Mathematical Software (TOMS)
The role of elimination trees in sparse factorization
SIAM Journal on Matrix Analysis and Applications
A mapping algorithm for parallel sparse Cholesky factorization
SIAM Journal on Scientific Computing
An efficient block-oriented approach to parallel sparse Cholesky factorization
SIAM Journal on Scientific Computing
A parallel algorithm for multilevel graph partitioning and sparse matrix ordering
Journal of Parallel and Distributed Computing
The Multifrontal Solution of Indefinite Sparse Symmetric Linear
ACM Transactions on Mathematical Software (TOMS)
On computing certain elements of the inverse of a sparse matrix
Communications of the ACM
Computer Solution of Large Sparse Positive Definite
Computer Solution of Large Sparse Positive Definite
A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling
SIAM Journal on Matrix Analysis and Applications
Zoltan Data Management Service for Parallel Dynamic Applications
Computing in Science and Engineering
A new data-mapping scheme for latency-tolerant distributed sparse triangular solution
Proceedings of the 2002 ACM/IEEE conference on Supercomputing
SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems
ACM Transactions on Mathematical Software (TOMS)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
PT-Scotch: A tool for efficient parallel graph ordering
Parallel Computing
Computing entries of the inverse of a sparse matrix using the FIND algorithm
Journal of Computational Physics
A hybrid method for the parallel computation of Green's functions
Journal of Computational Physics
Performance instrumentation and measurement for terascale systems
ICCS'03 Proceedings of the 2003 international conference on Computational science
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
ACM Transactions on Mathematical Software (TOMS)
Journal of Computational Physics
A two-dimensional domain decomposition technique for the simulation of quantum-scale devices
Journal of Computational Physics
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An efficient parallel algorithm is presented for computing selected components of $A^{-1}$ where $A$ is a structured symmetric sparse matrix. Calculations of this type are useful for several applications, including electronic structure analysis of materials in which the diagonal elements of the Green's functions are needed. The algorithm proposed here is a direct method based on a block $LDL^T$ factorization. The selected elements of $A^{-1}$ we compute lie in the nonzero positions of $L+L^T$. We use the elimination tree associated with the block $LDL^T$ factorization to organize the parallel algorithm, and reduce the synchronization overhead by passing the data level by level along this tree using the technique of local buffers and relative indices. We demonstrate the efficiency of our parallel implementation by applying it to a discretized two dimensional Hamiltonian matrix. We analyze the performance of the parallel algorithm by examining its load balance and communication overhead, and show that our parallel implementation exhibits an excellent weak scaling on a large-scale high performance distributed-memory parallel machine.