Support Graph Preconditioning

1. Support Graph Preconditioning +: New analytic tools, some new preconditioners +: Can use existing direct-methods software -: Current theory and techniques limited • Define a preconditioner B for matrix A • Explicitly compute the factorization B = LU • Choose nonzero structure of B to make factoring cheap (using combinatorial tools from direct methods) • Prove bounds on condition number using both algebraic and combinatorial tools

2. 3 7 1 3 7 1 6 8 6 8 4 10 4 10 9 2 9 2 5 5 Graphs and Sparse Matrices: Cholesky factorization Fill:new nonzeros in factor Symmetric Gaussian elimination: for j = 1 to n add edges between j’s higher-numbered neighbors G+(A)[chordal] G(A)

3. Spanning Tree Preconditioner [Vaidya] • A is symmetric positive definite with negative off-diagonal nzs • B is a maximum-weight spanning tree for A (with diagonal modified to preserve row sums) • factor B in O(n) space and O(n) time • applying the preconditioner costs O(n) time per iteration G(A) G(B)

4. Spanning Tree Preconditioner [Vaidya] • support each edge of A by a path in B • dilation(A edge) = length of supporting path in B • congestion(B edge) = # of supported A edges • p = max congestion, q = max dilation • condition number κ(B-1A) bounded by p·q (at most O(n2)) G(A) G(B)

5. Spanning Tree Preconditioner [Vaidya] • can improve congestion and dilation by adding a few strategically chosen edges to B • cost of factor+solve is O(n1.75), or O(n1.2) if A is planar • in experiments by Chen & Toledo, often better than drop-tolerance MIC for 2D problems, but not for 3D. G(A) G(B)

6. Support Graphs [after Gremban/Miller/Zagha] Intuition from resistive networks:How much must you amplify B to provide the conductivity of A? • The support of B for A is σ(A, B) = min{τ : xT(tB– A)x  0 for all x, all t  τ} • In the SPD case, σ(A, B) = max{λ : Ax = λBx} = λmax(A, B) • Theorem:If A, B are SPD then κ(B-1A) = σ(A, B) · σ(B, A)

7. Splitting and Congestion/Dilation Lemmas • Split A = A1+ A2 + ··· + Ak and B = B1+ B2 + ··· + Bk • Ai and Bi are positive semidefinite • Typically they correspond to pieces of the graphs of A and B (edge, path, small subgraph) • Lemma: σ(A, B)  maxi {σ(Ai , Bi)} • Lemma: σ(edge, path)  (worst weight ratio) · (path length) • In the MST case: • Ai is an edge and Bi is a path, to give σ(A, B)  p·q • Bi is an edge and Ai is the same edge, to give σ(B, A)  1

8. Algebraic framework • The support of B for A is σ(A, B) = min{τ : xT(tB– A)x  0 for all x, all t  τ} • In the SPD case, σ(A, B) = max{λ : Ax = λBx} = λmax(A, B) • If A, B are SPD then κ(B-1A) = σ(A, B) · σ(B, A) • [Boman/Hendrickson] If V·W=U, then σ(U·UT, V·VT)  ||W||22

9. Algebraic framework [Boman/Hendrickson] Lemma: If V·W=U, then σ(U·UT, V·VT)  ||W||22 Proof: • take t  ||W||22 = λmax(W·WT) = max y0 { yTW·WTy / yTy } • then yT (tI - W·WT) y  0 for all y • letting y = VTx gives xT (tV·VT - U·UT) x  0 for all x • recall σ(A, B) = min{τ : xT(tB– A)x  0 for all x, all t  τ} • thus σ(U·UT, V·VT)  ||W||22

10. [ ] a2 +b2-a2 -b2 -a2 a2 +c2 -c2-b2 -c2 b2 +c2 [ ] a2 +b2-a2 -b2 -a2 a2 -b2 b2 B =VVT A =UUT ] [ ] ] [ [ 1-c/a1 c/b/b ab-a c-b ab-a c-b-c = x U V W -a2 -c2 -a2 -b2 -b2 σ(A, B)  ||W||22 ||W||x ||W||1 = (max row sum) x (max col sum) (max congestion) x (max dilation)

11. B has positive (dotted) edges that cancel fill B has same row sums as A Strategy: Use the negative edges of B to support both the negative edges of A and the positive edges of B. -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 B A .5 .5 .5 .5 .5 .5 .5 .5 .5 A = 2D model Poisson problem B = MIC preconditioner for A Support-graph analysis of modified incomplete Cholesky

12. Every dotted (positive) edge in B is supported by two paths in B Each solid edge of B supports one or two dotted edges Tune fractions to support each dotted edge exactly 1/(2n – 2) of each solid edge is left over to support an edge of A Supporting positive edges of B

13. Analysis of MIC: Summary • Each edge of A is supported by the leftover 1/(2n – 2) fraction of the same edge of B. • Therefore σ(A, B)  2n – 2 • Easy to show σ(B, A)  1 • For this 2D model problem, condition number is O(n1/2) • Similar argument in 3D gives condition number O(n1/3) or O(n2/3) (depending on boundary conditions)

14. Open problems I • Other subgraph constructions for better bounds on||W||22? • For example [Boman], ||W||22 ||W||F2= sum(wij2) = sum of (weighted) dilations, and [Alon, Karp, Peleg, West]show there exists a spanning tree with average weighted dilation exp(O((log n loglog n)1/2)) = o(n ); this gives condition number O(n1+) and solution time O(n1.5+), compared to Vaidya O(n1.75) with augmented spanning tree • Is there a construction that minimizes ||W||22directly?

15. Open problems II • Make spanning tree methods more effective in 3D? • Vaidya gives O(n1.75) in general, O(n1.2) in 2D • Issue: 2D uses bounded excluded minors, not just separators • Spanning tree methods for more general matrices? • All SPD matrices? ([Boman, Chen, Hendrickson, Toledo]: different matroid for all diagonally dominant SPD matrices) • Finite element problems? ([Boman]: Element-by-element preconditioner for bilinear quadrilateral elements) • Analyze a multilevel method in general?

16. n1/2 n1/3 Complexity of linear solvers Time to solve model problem (Poisson’s equation) on regular mesh

17. n1/2 n1/3 Complexity of direct methods Time and space to solve any problem on any well-shaped finite element mesh