A parallel graph partitioning algorithm for a message-passing multiprocessor
International Journal of Parallel Programming
Partitioning sparse matrices with eigenvectors of graphs
SIAM Journal on Matrix Analysis and Applications
Parallel algorithms for sparse linear systems
SIAM Review
On computing certain elements of the inverse of a sparse matrix
Communications of the ACM
Some Fast Algorithms for Sequentially Semiseparable Representations
SIAM Journal on Matrix Analysis and Applications
A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations
SIAM Journal on Matrix Analysis and Applications
A Fast Solver for HSS Representations via Sparse Matrices
SIAM Journal on Matrix Analysis and Applications
Block tridiagonal matrix inversion and fast transmission calculations
Journal of Computational Physics
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
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
ACM Transactions on Mathematical Software (TOMS)
Hi-index | 31.45 |
The FIND algorithm is a fast algorithm designed to calculate certain entries of the inverse of a sparse matrix. Such calculation is critical in many applications, e.g., quantum transport in nano-devices. We extended the algorithm to other matrix inverse related calculations. Those are required for example to calculate the less-than Green's function and the current density through the device. For a 2D device discretized as an N"xxN"y mesh, the best known algorithms have a running time of O(N"x^3N"y), whereas FIND only requires O(N"x^2N"y). Even though this complexity has been reduced by an order of magnitude, the matrix inverse calculation is still the most time consuming part in the simulation of transport problems. We could not reduce the order of complexity, but we were able to significantly reduce the constant factor involved in the computation cost. By exploiting the sparsity and symmetry, the size of the problem beyond which FIND is faster than other methods typically decreases from a 130x130 2D mesh down to a 40x40 mesh. These improvements make the optimized FIND algorithm even more competitive for real-life applications.