Sparse Rank-Revealing LU Factorization
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1 Sparse Rank-Revealing LU Factorization Householder Symposium XV on Numerical Linear Algebra Peebles, Scotland, June 2002 Michael O Sullivan and Michael Saunders Dept of Engineering Science Dept of Management Sci & Eng University of Auckland Stanford University Auckland, New Zealand Stanford, CA michael.osullivan@auckland.ac.nz saunders@stanford.edu Sparse Rank-Revealing LU Factorization p.1/30
2 GOALS Sparse Rank-Revealing LU Factorization p.2/30
3 Goals Given a sparse matrix A (m n, usually m n), Maintain sparsity Determine if A is ill-conditioned? Determine which columns to delete? (or replace) P 1 AP 2 = LU, U = Sparse Rank-Revealing LU Factorization p.3/30
4 Motivation Dynamic Programming (Mike O Sullivan s thesis) P substochastic A = P I, at most one singularity Optimization (MINOS and SNOPT) Basis Repair I A = B, square basis, perhaps ill-conditioned Basis Repair II A = (B S) T, look for better B Sparse Rank-Revealing LU Factorization p.4/30
5 LU with Threshold Pivoting Lmax = stability tolerance = 10 or 3.99 or (bound on L ij ) At each stage of Gaussian elimination: A A lu T α j = biggest element in col j β i = biggest element in row i Amax = biggest element in A (= max α j = max β i ) Sparse Rank-Revealing LU Factorization p.5/30
6 RANK-REVEALING FACTORS Sparse Rank-Revealing LU Factorization p.6/30
7 Rank-Revealing Factors A = XDY T = Demmel et al. (1999) X, Y full column rank, well conditioned D diagonal SVD QR with column interchanges LU with Rook Pivoting LU with Complete Pivoting Sparse Rank-Revealing LU Factorization p.7/30
8 Sparse Rank-Revealing Factors QR multifrontal Pierce and Lewis (1997) TPP (Threshold Partial Pivoting) a pq Lmax α q Not RR TRP (Threshold Rook Pivoting) Gupta (2001) WSMP LUSOL a pq Lmax α q and β p TCP (Threshold Complete Pivoting) a pq Lmax Amax LUSOL where Lmax (e.g. 10 or 3.99 or ) bounds L ij Sparse Rank-Revealing LU Factorization p.8/30
9 LUSOL Sparse Rank-Revealing LU Factorization p.9/30
10 LUSOL A = LU + updates, L well-conditioned Gill, Murray, Saunders and Wright (1987) Revised , Markowitz strategy for sparse pivots (cf. MA28, Y12M, LA05, MOPS, MA48) TPP (Threshold Partial Pivoting) Search only a few sparse cols and rows Zlatev 1981 Store α j at top of col j Suhl & Suhl 1990 TRP (Threshold Rook Pivoting) TCP (Threshold Complete Pivoting) New New Sparse Rank-Revealing LU Factorization p.10/30
11 Stability Tolerance Lmax P 1 AP 2 = LDU, unit diags on L, U Threshold pivoting bounds the off-diags of L and perhaps U: TPP TRP TCP } } L ij Lmax 10.0 L ij, U ij Lmax 5.0 TRP, TCP are more Rank-Revealing with low Lmax: cond(l), cond(u) < (1 + Lmax) n Sparse Rank-Revealing LU Factorization p.11/30
12 Elimination Step Allowable pivots with Lmax = 3.0 TPP can pivot on 1 TRP can pivot on 4 TCP must pivot on (Markowitz) Just a few columns and rows change Sparse Rank-Revealing LU Factorization p.12/30
13 Maintaining α j α j New α j??? A ij stored column-wise easy to update modified α j Sparse Rank-Revealing LU Factorization p.13/30
14 Implementing TRP Sparse Rank-Revealing LU Factorization p.14/30
15 Threshold Rook Pivoting Same Markowitz search strategy as TPP: cols of length 1, rows of length 1, cols of length 2, rows of length 2,... but have to search longer α j are stored at top of each column β i stored in separate array (new) A ij stored column-wise β i more expensive than α j Could store A ij row-wise faster but more storage Sparse Rank-Revealing LU Factorization p.15/30
16 Implementing TCP Sparse Rank-Revealing LU Factorization p.16/30
17 Partial Pivoting vs Complete Pivoting Dense Computing L and U O(n 3 ) PP Finding α 1 O(n) CP Finding Amax O(n 3 ) Not so bad! Sparse Computing L and U O(nnz(L + U)) TPP Finding all α j O(nnz(L + U)) TCP Finding Amax O(n 2 )? Too much Sparse Rank-Revealing LU Factorization p.17/30
18 Finding Amax from α j Naive Method α j = biggest element in column j Amax = biggest element in A jmax = column containing Amax Find Amax by searching all α j O(n 2 ) Warning from Iain Theorem Need to search all α j only if Amax decreases Sparse Rank-Revealing LU Factorization p.18/30
19 Proof If col jmax is not modified: α j Amax α j Amax = If col jmax is modified: α j Amax α j Must search Amax + 5 +? for Amax Sparse Rank-Revealing LU Factorization p.19/30
20 Better: Store α j in a Heap Thanks to John Gilbert Amax Ha(k) α j Hj(k) j Hk(j) location of j in heap Sparse Rank-Revealing LU Factorization p.20/30
21 Calls to Heap Functions build heap from all α j for k = 1 : min(m, n) Choose pivot, do elimination Find α j for modified cols delete entry for pivot column for l = 2 : lenpivrow change entry for each modified column end end Remarkably little work Sparse Rank-Revealing LU Factorization p.21/30
22 NUMERICAL RESULTS Sparse Rank-Revealing LU Factorization p.22/30
23 Markowitz, then Dense CP Modified Zlatev strategy Markowitz1 until 30% dense: Search at least 5 cols, 4 rows 2 Markowitz2 until 50% dense: Search at least 5 cols, 0 rows 3 Dense Complete Pivoting Sparse Rank-Revealing LU Factorization p.23/30
24 Problem memplus from Harwell-Boeing collection A = , nonzeros, Scaled Lmax nnz(l+u) Time TRP RR RR RR RR RR TCP RR RR RR PP (SuperLU, colamd) Sparse Rank-Revealing LU Factorization p.24/30
25 Problem BRATU2D from CUTE optimization collection A = , nonzeros Permuted triangle 64 singularities P 1 AP 2 = Marching pattern from PDE TRP and TCP must have Lmax < 4.0 Sparse Rank-Revealing LU Factorization p.25/30
26 TRP Profile CUTE Problem BRATU2D A = , nonzeros TRP, Lmax= 1.26, LU = nonzeros Update β i for modified rows 57.7% Markowitz Find stable pivot 31.4% Elimination The algebra 4.0% Dense CP % Update α j for modified cols 1.6% Sparse Rank-Revealing LU Factorization p.26/30
27 TCP Profile Harwell-Boeing Problem memplus A = , nonzeros TCP, Lmax = 10.0, LU = nonzeros Markowitz Find stable pivot 65.0% Dense CP % Elimination The algebra 7.4% Update α j for modified cols 4.7% change Heap update 0.1% (!) Sparse Rank-Revealing LU Factorization p.27/30
28 CONCLUSIONS Sparse Rank-Revealing LU Factorization p.28/30
29 Conclusions TPP: Work-horse, usually reliable RRLUs can be rather dense. Scaling essential TRP: Markowitz search costs more TCP: Markowitz search costs MUCH more but Heap allows Amax to be maintained cheaply TRP, TCP are usually Rank-Revealing with Lmax 4.0 (but sometimes needs 2.5) SNOPT optimization code on CUTE test problems: TRP reveals same rank as TCP almost always (much more cheaply) Sparse Rank-Revealing LU Factorization p.29/30
30 HEAPS of THANKS Iain Duff John Gilbert R. Sedgewick Cormen, Leiserson and Rivest Les Foster Philip Gill The CUTE and Harwell-Boeing Teams Michael Friedlander Sparse Rank-Revealing LU Factorization p.30/30
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