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A combined unifrontal/multifrontal method for unsymmetric sparse matrices

Timothy A. Davis, Iain S. Duff
1999 ACM Transactions on Mathematical Software  
The (uni-)frontal method avoids this extra work by factorizing the matrix with a single frontal matrix. Rows and columns are added to the frontal matrix, and pivot rows and columns are removed.  ...  In the multifrontal method, work on a frontal matrix can be suspended, the frontal matrix can be stored for later reuse, and a new frontal matrix can be generated.  ...  -j ) + max(i -j ) } , 5 {ai j #o aji#O where it is assumed the diagonal is nonzero so all terms in the summation are non-negative.  ... 
doi:10.1145/305658.287640 fatcat:5efpfopl4nam3dvvvxdydsvnam

A survey of direct methods for sparse linear systems

Timothy A. Davis, Sivasankaran Rajamanickam, Wissam M. Sid-Lakhdar
2016 Acta Numerica  
sparse matrix problems.  ...  They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations.  ...  QR factorization is an effective alternative. Non-multifrontal sparse QR methods are discussed in Section 7; here we present the multifrontal QR method.  ... 
doi:10.1017/s0962492916000076 fatcat:u4dqyjkjqnelll5e3ywm7lqkca

IMF: An Incomplete Multifrontal $LU$-Factorization for Element-Structured Sparse Linear Systems

Nick Vannieuwenhoven, Karl Meerbergen
2013 SIAM Journal on Scientific Computing  
We propose an incomplete multifrontal LU -factorization (IMF) that extends supernodal multifrontal methods to incomplete factorizations.  ...  We propose an incomplete multifrontal LU -factorization (IMF) that extends supernodal multifrontal methods to incomplete factorizations.  ...  Furthermore, the ratio of the non-zeros in the block diagonal matrix to the total non-zeros in the preconditioner increases.  ... 
doi:10.1137/100818996 fatcat:xutjmao4fnanropwxkfttjbogu

A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations

Nicholas I. M. Gould, Jennifer A. Scott, Yifan Hu
2007 ACM Transactions on Mathematical Software  
A factorization phase that uses the pivot sequence to factorize the matrix (some codes scale the matrix prior to the factorization). 4.  ...  The factorization phase implements a modified multifrontal algorithm.  ... 
doi:10.1145/1236463.1236465 fatcat:zcqti6k6u5anbfht3sfwb2jibu

AN EFFICIENT APPROACH FOR MULTIFRONTAL ALGORITHM TO SOLVE NON-POSITIVE-DEFINITE FINITE ELEMENT EQUATIONS IN ELECTROMAGNETIC PROBLEMS

Jin Tian, Zhi-Qing Lv, Xiao-Wei Shi, Le Xu, Feng Wei
2009 Electromagnetic Waves  
Based on the method, multifrontal (MF) algorithm is applied in non-positive-definite FEM computation. Numerical results show that the hybrid ECM/MF algorithm is stable and effective.  ...  The method can be used to decompose sparse symmetric non-positive-definite finite element (FEM) matrices.  ...  One of the significant advancements in direct methods for a sparse matrix solution is the development of the multifrontal (MF) algorithm [15] [16] [17] [18] .  ... 
doi:10.2528/pier09070207 fatcat:lwu3nfokpjcsjespmbpp2iuawi

Multifrontal QR Factorization for Multicore Architectures over Runtime Systems [chapter]

Emmanuel Agullo, Alfredo Buttari, Abdou Guermouche, Florent Lopez
2013 Lecture Notes in Computer Science  
This paper evaluates the usability of runtime systems for complex applications, namely, sparse matrix multifrontal factorizations which constitute extremely irregular workloads, with tasks of different  ...  The multifrontal method, introduced by Duff and Reid [12] as a method for the factorization of sparse, symmetric linear systems, can be adapted to the QR factorization of a sparse matrix thanks to the  ...  fact that the R factor of a matrix A and the Cholesky factor of the normal equation matrix A T A share the same structure.  ... 
doi:10.1007/978-3-642-40047-6_53 fatcat:nuekjepqxbae7ocsklphedcyre

Logarithmic barriers for sparse matrix cones

Martin S. Andersen, Joachim Dahl, Lieven Vandenberghe
2013 Optimization Methods and Software  
The algorithms are based on the multifrontal method for sparse Cholesky factorization.  ...  Linearized Cholesky factorization and matrix completion Let L(t), D(t) be the matrices in the factorization X(t) = X + tY = L(t)D(t)L(t) T and let U i (t) be the ith update matrix in the multifrontal factorization  ...  In the context of multifrontal factorizations, the grouping into supernodes has the advantage that only one frontal matrix is required per supernode.  ... 
doi:10.1080/10556788.2012.684353 fatcat:k2apjixbnbeedj4d4ts7hkgfoi

Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers [article]

Björn Engquist, Lexing Ying
2010 arXiv   pre-print
Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case  ...  The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain.  ...  In practice, instead of generating the sweeping factorization of the original matrix A, we choose to generate the factorization for the matrix A α associated with the modified Helmholtz equation ∆u(x)  ... 
arXiv:1007.4291v2 fatcat:fvknv3b6inayfbkf3hkiokjkxa

A fast direct solver for elliptic problems on general meshes in 2D

Phillip G. Schmitz, Lexing Ying
2012 Journal of Computational Physics  
We follow the approach in Xia et al. (2009) on combining the multifrontal method with hierarchical matrices.  ...  However, the efficiency of the sparse Cholesky decomposition depends on choosing a reordering to reduce fill-in of non-zeros in the factors.  ...  Moreover, most of matrix operations such as matrix-vector product, matrix addition, matrix multiplication, matrix inversion, and some matrix factorizations, can be carried out in the hierarchical matrix  ... 
doi:10.1016/j.jcp.2011.10.013 fatcat:dy766mrwrrfbbft2qigpddp7hi

A parallel formulation of interior point algorithms

George Karypis, Anshul Gupta, Vipin Kumar
1994 Supercomputing, Proceedings  
In our parallel interior point algorithm, we use our recently developed parallel multifrontal algorithm that has the smallest communication overhead over all parallel algorithms for Cholesky factorization  ...  A key component of the interior point algorithm is the solution of a sparse system of linear equations using Cholesky factorization.  ...  Multifrontal Method Let M be an m m symmetric positive de nite matrix and L be its Cholesky factor.  ... 
doi:10.1145/602805.602808 fatcat:23vhjjrnfrfe7lx4khfmtnu5ja

A parallel formulation of interior point algorithms

George Karypis, Anshul Gupta, Vipin Kumar
1994 Supercomputing, Proceedings  
In our parallel interior point algorithm, we use our recently developed parallel multifrontal algorithm that has the smallest communication overhead over all parallel algorithms for Cholesky factorization  ...  A key component of the interior point algorithm is the solution of a sparse system of linear equations using Cholesky factorization.  ...  Multifrontal Method Let M be an m m symmetric positive de nite matrix and L be its Cholesky factor.  ... 
doi:10.1145/602770.602808 fatcat:joxeq6lb7fh45jefcwyhdskrua

Fast and Accurate Simulation of Multithreaded Sparse Linear Algebra Solvers

Luka Stanisic, Emmanuel Agullo, Alfredo Buttari, Abdou Guermouche, Arnaud Legrand, Florent Lopez, Brice Videau
2015 2015 IEEE 21st International Conference on Parallel and Distributed Systems (ICPADS)  
the QR factorization of a sparse matrix thanks to the fact that the R factor of a matrix A and the Cholesky factor of the normal equation matrix A T A share the same structure under the hypothesis that  ...  In particular, non trivial results can be obtained such as the fact that the TF17 matrix benefits much more from using several nodes than the sls matrix. B.  ... 
doi:10.1109/icpads.2015.67 dblp:conf/icpads/StanisicABGLLV15 fatcat:mscn54jpxzgfpmtyjcmelqn5j4

Parallel computation of pseudospectra of large sparse matrices

Dany Mezher, Bernard Philippe
2002 Parallel Computing  
Computing the smallest singular value Given a large sparse matrix A 2 C nÂn , we consider the matrix B 2 C 2nÂ2n B ¼ 0 A À1 A ÀH 0 : The eigenvalues of B are the singular values of A À1 and their negatives  ...  As a trade-off, it converges only linearly with the convergence factor 1=2.  ... 
doi:10.1016/s0167-8191(01)00136-3 fatcat:3y3q7z4sfzbwhdwxyseqm5wmna

A sweeping preconditioner for Yee's finite difference approximation of time-harmonic Maxwell's equations

Paul Tsuji, Lexing Ying
2012 Frontiers of Mathematics in China  
. , n do Let G m be as defined in (3.10) and H m be the matrix resulting from the finite difference discretization of (3.9). Construct the multifrontal factorization of H m .  ...  using the multifrontal factorization of H F . 5. for m = b + 1, . . . , n do u m = T m u m .  ... 
doi:10.1007/s11464-012-0191-8 fatcat:y6o563b2tfc5tezszn7tzcu3h4

Design of a Multicore Sparse Cholesky Factorization Using DAGs

J. D. Hogg, J. K. Reid, J. A. Scott
2010 SIAM Journal on Scientific Computing  
PaStiX [26, 27] is non-multifrontal C/Pthreads/MPI code that is primarily designed for positive-definite systems.  ...  SuiteSparseQR [13] is a recent C++ sparse QR factorization package based on the multifrontal algorithm.  ...  When a factorize block or solve block task completes, we decrement the block's count to flag this event with a negative value.  ... 
doi:10.1137/090757216 fatcat:meflt2mkxfapzghp6fyg6ptmqa
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