Spectrum SlicingSLEPc

Evaluation

Strategies for Spectrum Slicing Based onRestarted Lanczos Methods

Carmen Campos and Jose E. RomanUniversitat Politecnica de Valencia, Spain

SC2011

Spectrum SlicingSLEPc

Evaluation

Goal

Context: symmetric-definite generalized eigenvalue problem

Ax = λBx B ≥ 0

Eigenvalues are real: λ1 ≤ λ2 ≤ . . . ≤ λnVery large, sparse matrices → iterative solvers, parallel computing,small part of the spectrum

Computational interval

I In many applications: structures, electromagnetism, etc.

I All eigenvalues in a given interval [a, b] (a or b must be finite)

I Do not miss eigenvalues (could be 1000’s)

I Determine multiplicity correctly (could be as high as 400)

Spectrum SlicingSLEPc

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Outline

1 Spectrum SlicingRelated workProposed variants

2 SLEPcOverview of SLEPcImplementation of Spectrum Slicing

3 Evaluation

Spectrum SlicingSLEPc

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Spectral Transformation

The spectral transformation [Ericsson & Ruhe 1980] enablesLanczos methods to compute interior eigenvalues

Ax = λBx =⇒ (A− σB)−1Bx = θx

I Trivial mapping of eigenvalues: θ = (λ− σ)−1

I Eigenvectors are not modified

I Very fast convergence close to σ

Things to consider:

I Implicit inverse (A− σB)−1 via linear solves

I Direct linear solver for robustness

I Less effective for eigenvalues far away from σ

Spectrum SlicingSLEPc

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Spectrum SlicingIndefinite (block-)triangular factorization:

A− σB = LDLT

By Sylvester’s law of inertia, we get as a byproduct the number ofeigenvalues on the left of σ

ν(A− σB) = ν(D)

Spectrum slicing

I Multi-shift approach that sweeps all the intervalI Compute eigenvalues by chunksI Use inertia to validate sub-intervals

a b

σ1 σ2 σ3

Spectrum SlicingSLEPc

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Spectrum Slicing: Grimes et al. ApproachGrimes et al. [1994] proposed an “industrial strength” scheme

I Block Lanczos, with blocksize depending on multiplicity

I B-orthogonalization, partial and external selective reorthog.

I Create an initial trust interval, extend it until finished

I Unrestarted Lanczos, tracking eigenvalue convergence

Choice of new shift:

I Asumes there are as many eigenvalues around σi and σi+1

I Uses non-converged Ritz approximations if available

I Sometimes need to fill-in gaps

Deflation with sentinel mechanism:

I Deflation against (at least) one vector from previous shift

I Goal: orthogonality in clusters, suppress eigenvectors mostlikely to reappear

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Grimes et al.: Potential Pitfalls

Possible problems of Grimes et al. approach:

I Exploits a priori knowledge of multiplicity

I Assumes all multiplicities are of same rank

I Block size cannot be arbitrarily large, difficulties with highmultiplicities

I Irregular spectra produce bad choice of shifts, with big gaps

I Wasteful work: many repeated eigenvalues are discarded

Using an unrestarted block Lanczos has strong implications onheuristics

Spectrum SlicingSLEPc

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New Context: Restarted Lanczos

Now we have restarted Lanczos methods

I Thick-restart Lanczos [Wu et al. 1999]

I Equivalent to implicit-restart Lanczos, or symm. Krylov-Schur

New assumptions:

I Lanczos convergence is not a problem; also multiple eigs.

I Orthogonalization is relatively cheap and scales very well

I Performance of factorization degrades with n

I Triangular solves are not scalable in parallel

Goal: spectrum slicing technique that can be robust enough forirregular spectra with high multiplicity, scalable to 100’s processors

Strategy: avoid new shifts by orthogonalizing more

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Proposed Method (1)

Main idea: At each shift σi request fixed number of eigenvalues(nev), with a limited number of restarts (maxit)

Selection of new shift σi+1:

I Cannot rely on approximate Ritz values

I Separation of eigenvalues computed at σi is not reliable

I Use average eigenvalue separation in [σi−1, σi]

Backtrack: if number of eigenvalues computed in [σi−1, σi] doesnot match inertia, create a new shift somewhere inbetween

I All eigenvectors available in [σi−1, σi] are deflated

I Guarantees orthogonality of eigenvectors of multiples/clusters

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Proposed Method (2)

Deflation

I Avoid reappearance of already computed eigenvalues

I Also allow missing multiples to ariseI Two options (flag defl):

1. At σi+1, deflate all eigenvectors available in [σi, σi+1]2. Minimal deflation, with sentinels similar to Grimes

If possible, avoid backtracking

I A new factorization to compute a few eigenvalues is wasteful

I Parameter compl: try to complete interval if missingeigenvalues less or equal than compl

Spectrum SlicingSLEPc

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Proposed Method (3)

With backtracking

I nev=10, maxit=10, with deflation

a b

σ1 σ2σ3 σ4

Avoiding backtracking

I nev=10, maxit=10, with deflation, compl=5

a b

σ1 σ2 σ3

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SLEPc: Scalable Library for Eigenvalue Problem Computations

A general library for solving large-scale sparse eigenproblems onparallel computers

I For standard and generalized eigenproblems

I For real and complex arithmetic

I For Hermitian or non-Hermitian problems

I Also support for SVD and QEP

Ax = λx Ax = λBx Avi = σiui (λ2M+λC+K)x = 0

Developed at U. Politecnica de Valencia since 2000

http://www.grycap.upv.es/slepc

Current version: 3.1 (released Aug 2010)

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PETSc/SLEPc Numerical Components

PETSc

Vectors

Standard CUSP

Index Sets

Indices Block Stride Other

Matrices

CompressedSparse Row

BlockCSR

SymmetricBlock CSR

Dense CUSP Other

Preconditioners

AdditiveSchwarz

BlockJacobi

Jacobi ILU ICC LU Other

Krylov Subspace Methods

GMRES CG CGS Bi-CGStab TFQMR Richardson Chebychev Other

Nonlinear Systems

LineSearch

TrustRegion Other

Time Steppers

EulerBackward

Euler

PseudoTime Step Other

SLEPc

SVD Solvers

CrossProduct

CyclicMatrix

LanczosThick R.Lanczos

Quadratic

Linear-ization

Q-Arnoldi

Eigensolvers

Krylov-Schur Arnoldi Lanczos GD JD Other

Spectral Transformation

Shift Shift-and-invert Cayley Fold Preconditioner

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m-step Lanczos Method

Computes Vm and TmI M = (A− σB)−1B

I Vm is a basis of the Krylov space Km(M, v1), VTmBVm = I

I Tm = V ∗mBMVm provides Ritz approximations, (θi, Vmyi)

for j = 1, 2, . . . ,mw =Mvjt1:j,j = V ∗

j Bw

w = w − Vjt1:j,jtj+1,j = ‖w‖Bvj+1 = w/tj+1,j

end

Orthogonalization:

I Full B-orthogonalization

I Do not bother about partialreorthog.

I SLEPc uses iterated CGSbut MGS also available

Use MUMPS for (A− σB)−1 = L−TD−1L−1, get inertia info

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Symmetric Krylov-Schur

A restarting mechanism that filters out unwanted eigenvectors

1. Build Lanczos factorization of order m

2. Diagonalize projected matrix

3. Check convergence, sort

4. Truncate to a factorization of order p

5. Extend to a factorization of order m

6. If not finished, go to step 2

Vm

v m+1

Sm

b∗m+1Vp

v m+1

Sp

b∗pVp

v m+1

Sp

b∗p

For spectrum slicing, the basis expansion needs to orthogonalizealso against an arbitrary set of vectorsRestarts until nev converged eigenvalues (or subinterval complete)

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Computing Platform

IBM BladeCenter cluster with Myrinet interconnect

I 256 JS20 nodes

I Two 64-bit PowerPC 970+ @ 2.2 GHz processors

I 4 GB memory per node (1 TB total)

Tests with up to 128 MPI processes (2 per node)

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Test Case: Aircraft Fuselage

Simplified but realisticmodel: cylinder with skin,frames, and stringers

Parametric, “scalable”

First vibration mode (5.34 Hz)

Spectrum SlicingSLEPc

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Test Case: Matrix Properties

Analysis of frequency range[0–60] Hz

1 million dof’s

I Dimension: 1,036,698

I Nonzeros: ∼29 million

I Eigenvalues in interval: 1989

2 million dof’s

I Dimension: 2,141,646

I Nonzeros: ∼59 million

I Eigenvalues in interval: 2039

Maximum multiplicity: 2 B is singular

Spectrum SlicingSLEPc

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Evaluation: Solver Parameters1 million, 16 processors

nev maxit defl comp Shifts Rest Its Time80 10 1 - 22 36 5,109 11,07480 10 1 40 17 35 4,696 9,97880 10 0 - 33 48 7,021 14,251

120 10 1 - 14 23 4,874 10,770120 10 1 50 13 27 4,565 10,125120 10 0 - 21 29 6,417 13,052

Unrest., 300 vecs, no defl. 22 - 6,622 13,305

2 million, 32 processorsnev maxit defl comp Shifts Rest Its Time80 5 1 40 20 40 5,200 14,83380 5 0 - 30 47 6,740 18,235

120 5 1 50 12 23 4,500 13,389120 5 0 - 24 33 7,290 19,659

Unrest., 300 vecs, no defl. 24 - 7,224 18,605

Spectrum SlicingSLEPc

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Evaluation: Parallel Performance

1 million, nev=120

p Shft Rest Its Time Num. Sym. Tri. Orth.8 12 21 4,318 16,399 3,153 594 5,270 6,218

16 13 27 4,565 10,125 1,739 595 4,277 3,36432 10 21 4,101 5,500 519 653 2,839 1,42864 12 22 4,338 4,064 375 654 2,482 580

128 12 20 4,155 3,394 273 596 2,265 245

2 million, nev=120

p Shft Rest Its Time Num. Sym. Tri. Orth.32 13 24 4,873 14,429 2,012 1,691 6,536 3,90264 12 22 4,472 9,120 922 1,697 4,886 1,263

128 12 23 4,501 7,817 707 1,692 4,709 668

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Conclusion

We have developed a robust spectrum slicing method, can copewith high multiplicities

I Based on previous work by Grimes et al.

I Main focus on restarted Lanczos methods

I Our heuristics tend to favour scalability

Evaluation

I 4 times faster than plain shift-and-invert

I Compared to Grimes et al., 30-40% gain

I Reasonable scalability up to 128 processors

Future work

I Improve scalability by splitting in subcommunicators (similarto [Zhang et al. 2007])

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Thanks!

Information on SLEPc

http://www.grycap.upv.es/slepc