Direct Algorithms for Sparse Schur Complements and Inverses

Transcription

1 Direct Algorithms for Sparse Schur Complements and Inverses Dr. Ryan Chilton MyraMath

2 Outline Examine some less common sparse direct algorithms: Partial linear solution. Schur complements. Sampling the inverse operator. Apply them as frontends for low-rank skeletonization: Cross approximation. Range estimation. Ritz projection. Motivations: fast direct solvers for FE-BI s and FE-DDM s.

3 Refresher: Factor A=LL T Reorder: Left(0), Right(1), Separator(2). A 01 = A 10 = all zero! Right looking. Factor A 00 /A 11, schur downdate A 22, factor. FEM mesh: Reordered matrix: Algorithm steps: A 00 0 A 02 Factor A 22 0 A 11 A 12 Schur Downdate A 22 A 20 A 21 A 22 Solve A 20 Solve A 21 Separator induces these zeroes. They can t fill-in! Factor A 00 Factor A 11 Left (0) Separator (2) Right (1) Note A 00 and A 11 also sparse, apply idea recursively. Leads to a tree of operations, eliminating from bottom up.

4 Selected profiling data. Example problem under study: I x J x K brick (N = IJK) N= 80 3 = 512K N=100 3 = 1M N=128 3 = 2.1M GEMM 12sec 42s 161s 35s 3D=O(n 1.87 ) 2D=O(n 1.53 ) 1D=O(n 1.08 ) 105sec 367s 1559s 405s Intel E x8=16 Xeon at 2.4GHz, MKL 48 3 Discrete graph laplacian (7-point): well understood spectrum. Structured grid: easy to reorder using nested dissection.

5 Partial solution x=r it A -1 R j b In plain english: only b(j) nonzero, only x(i) is needed. = Many engineering QoI s use only boundary-valued b and x. Lx=b L T x=b Solve Partial Solve j j i i O(n 4/3 ) time, like x=a -1 b. Only O(n 2/3 ) space per RHS, not O(n).

6 Schur complement S=B T A -1 B Concept: form saddle system of A and B, then quit early. Arise from FE-BI hybrids, eg scattering from apertures.

7 Sampling the inverse Z(i,j), Z=A -1 Closely related to Schur complement, Z(i,j) = R it A -1 R j Arise in FETI/DDM, iterate/exchange fields at boundaries. Scatter, solve, gather. Scatter, solve, gather. Tabulating Z(i,j) opens up reuse/preconditioning options.

8 Cross Approximating Z(i,j) [1/2] Alternately sample row/column with largest error modulus. log 10 (Z-UV T ) Estimated Error Actual Error SVD(Z) Key idea: partialsolve() can efficiently extract rows/columns: c = Z([i],j) = solver.partialsolve([i],j,x=1.0,'left') r = Z(i,[j]) = solver.partialsolve(i,[j],x=1.0,'right')

9 Cross Approximating Z(i,j) [2/2] Beats solver.inverse() at large N, especially at low rank/tol. 8 digits 6 digits 4 digits But in parallel the gap narrows, BLAS3 vs BLAS1 effects.

10 (error) Range estimation of Z(i,j) [1/2] Apply action of Z to random vectors X, form image Y=ZX. If Z has rapidly decaying σ s, Y probably spans range(z). // Find Q = span(z) X = rand(z.cols,k) Y = Z.apply(X) [Q,R,π] = QR(Y,0) k=4 k=8 x SVD(Z) SVD(UV T ) Pass 1, Pass 2.. // Build k-svd from Q W = Z.apply(Q) [U,Ʃ,V] = svd(w,0) Z (Q U) Ʃ (V) k=16 k=32 Key idea: partialsolve() can efficiently apply Y=Z(i,j) X: Y = Z([i],[j])*X = solver.partialsolve([i],[j],x,'left')

11 Range estimation of Z(i,j) [2/2] All the same problem instances as before (sizes,shapes). 8 digits 6 digits 4 digits Availability of all forcing data up front leads to speedup. Can be faster than parallel solver.inverse(), even at modest N.

12 Ritz Projection of Z(i,j) [1/3] What about approximating more than just one block? (B)lock (L)ow (R)ank (H)eirarchical Matrix Optimization(BLR)/amortization(H) opportunities do exist.

13 All of exterior, partitioned into (leaf) groups. G3 G2 G1 Y(3,0) = colspan Z(3,0) G0 Y(0,3) = colspan Z(0,3) = rowspan Z(3,0) Ritz Projection of Z(i,j) [2/3] First pass: find row/column spans using fat partialsolve() k k k k R = Y(3,0) T Z(3,0) Y(0,3) k k = T [schur] R = solver.schur(y 30,Y 03 ) [U,Ʃ,V] = svd(r03) Z 30 (Y 30 U) Ʃ (V Y 03 ) X Y Second pass: Ritz projection using solver.schur(), k-svd

14 Ritz Projection of Z(i,j) [3/3] Fill an H-matrix representation of Z restricted to boundary. 1385sec Factor Form Y [partialsolve] Form B [schur,qr,svd] Form Z [inverse] Algorithm quickly furnishes all (admissible) blocks. Can form H-matrix of S=B T A -1 B with a few minor changes.

15 Wrapping Up Examined several uncommon sparse direct algorithms: Partial linear solution: x=r it A -1 R j b (sparse b, sifted x) Schur complements: B T A -1 B, B T A -1 C, all sparse Sampling the inverse operator: Z(i,j) = R i A -1 R j Used them as frontends for low-rank/skeletonization: Cross approximation: partialsolve() can extract row/column Range estimation: partialsolve() can apply Z(i,j) quickly Ritz projection: schur()+partialsolve(), amortization over blocks Essential tools for FEBI/DDM methods (sparsity+lowrank).

16 Contact: MyraMath: sparse factor/solve/schur/inverse/partialsolve. MyraKL: BLAS/LAPACK API for MyraMath, or use MKL. Free software (GPL), or dual license