Constructing memory-minimizing schedules for multifrontal methods

We are interested in the memory usage of multifrontal methods. Starting from the algorithms introduced by Liu, we propose new schedules to allocate and process tasks that improve memory usage. This generalizes two existing factorization and memory-allocation schedules by allowing a more flexible task allocation together with a specific tree traversal. We present optimal algorithms for this new class of schedules, and demonstrate experimentally their benefit for some real-world matrices from sparse matrix collections where either the active memory or the total memory is minimized.

[1]  Robert E. Tarjan,et al.  The Space Complexity of Pebble Games on Trees , 1980, Inf. Process. Lett..

[2]  Iain S. Duff,et al.  The Multifrontal Solution of Unsymmetric Sets of Linear Equations , 1984 .

[3]  Joseph W. H. Liu.  An application of generalized tree pebbling to sparse matrix factorization , 1987 .

[4]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[5]  W. H. Liu,et al.  AN APPLICATION OF GENERALIZED TREE PEBBLING TO SPARSE MATRIX FACTORIZATION , 2022 .

[6]  Joseph W. H. Liu.  The role of elimination trees in sparse factorization , 1990 .

[7]  Patrick R. Amestoy,et al.  Multifrontal parallel distributed symmetric and unsymmetric solvers , 2000 .

[8]  Jean-Yves L'Excellent,et al.  Optimal memory minimization algorithms for the multifrontal method , 2004 .

[9]  Patrick R. Amestoy,et al.  An Approximate Minimum Degree Ordering Algorithm , 1996, SIAM J. Matrix Anal. Appl..

[10]  Gil Utard,et al.  Analysis and Improvments of the Memory Usage of a Multifrontal Solver , 2003 .

[11]  Patrick R. Amestoy,et al.  An unsymmetrized multifrontal LU factorization , 2000 .

[12]  John K. Reid,et al.  The Multifrontal Solution of Indefinite Sparse Symmetric Linear , 1983, TOMS.

[13]  Jürgen Schulze.  Towards a Tighter Coupling of Bottom-Up and Top-Down Sparse Matrix Ordering Methods , 2001 .

[14]  Iain S. Duff,et al.  On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix , 2000, SIAM J. Matrix Anal. Appl..

[15]  Patrick Amestoy,et al.  Memory Management Issues in Sparse Multifrontal Methods On Multiprocessors , 1993, Int. J. High Perform. Comput. Appl..

[16]  Gil Utard,et al.  Impact of reordering on the memory of a multifrontal solver , 2003, Parallel Comput..

[17]  Barry W. Peyton,et al.  Progress in Sparse Matrix Methods for Large Linear Systems On Vector Supercomputers , 1987 .

[18]  Iain S. Duff,et al.  MA27 -- A set of Fortran subroutines for solving sparse symmetric sets of linear equations , 1982 .

[19]  Joseph W. H. Liu,et al.  On the storage requirement in the out-of-core multifrontal method for sparse factorization , 1986, TOMS.

[20]  Iain S. Duff,et al.  The Rutherford-Boeing sparse matrix collection , 1997 .