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• Spectrum Slicing SLEPc

Evaluation

Strategies for Spectrum Slicing Based on Restarted Lanczos Methods

Carmen Campos and Jose E. Roman Universitat Politècnica de València, Spain

SC2011

• Spectrum Slicing SLEPc

Evaluation

Goal

Context: symmetric-definite generalized eigenvalue problem

Ax = λBx B ≥ 0

Eigenvalues are real: λ1 ≤ λ2 ≤ . . . ≤ λn Very 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 Slicing SLEPc

Evaluation

Outline

1 Spectrum Slicing Related work Proposed variants

2 SLEPc Overview of SLEPc Implementation of Spectrum Slicing

3 Evaluation

• Spectrum Slicing SLEPc

Evaluation

Spectral Transformation

The spectral transformation [Ericsson & Ruhe 1980] enables Lanczos 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 Slicing SLEPc

Evaluation

Spectrum Slicing Indefinite (block-)triangular factorization:

A− σB = LDLT

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

ν(A− σB) = ν(D)

Spectrum slicing

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

a b

σ1 σ2 σ3

• Spectrum Slicing SLEPc

Evaluation

Spectrum Slicing: Grimes et al. Approach Grimes 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 most likely to reappear

• Spectrum Slicing SLEPc

Evaluation

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 high multiplicities

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 on heuristics

• Spectrum Slicing SLEPc

Evaluation

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 for irregular spectra with high multiplicity, scalable to 100’s processors

Strategy: avoid new shifts by orthogonalizing more

• Spectrum Slicing SLEPc

Evaluation

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] does not 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

• Spectrum Slicing SLEPc

Evaluation

Proposed Method (2)

Deflation

I Avoid reappearance of already computed eigenvalues

I Also allow missing multiples to arise I 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 missing eigenvalues less or equal than compl

• Spectrum Slicing SLEPc

Evaluation

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

• Spectrum Slicing SLEPc

Evaluation

SLEPc: Scalable Library for Eigenvalue Problem Computations

A general library for solving large-scale sparse eigenproblems on parallel 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 (λ 2M+λC+K)x = 0

Developed at U. Politècnica de València since 2000

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

Current version: 3.1 (released Aug 2010)

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

• Spectrum Slicing SLEPc

Evaluation

PETSc/SLEPc Numerical Components

PETSc

Vectors

Standard CUSP

Index Sets

Indices Block Stride Other

Matrices

Compressed Sparse Row

Block CSR

Symmetric Block CSR

Dense CUSP Other

Preconditioners

Block Jacobi

Jacobi ILU ICC LU Other

Krylov Subspace Methods

GMRES CG CGS Bi-CGStab TFQMR Richardson Chebychev Other

Nonlinear Systems

Line Search

Trust Region Other

Time Steppers

Euler Backward

Euler

Pseudo Time Step Other

SLEPc

SVD Solvers

Cross Product

Cyclic Matrix

Lanczos Thick R. Lanczos

Linear- ization

Q-Arnoldi

Eigensolvers

Krylov-Schur Arnoldi Lanczos GD JD Other

Spectral Transformation

Shift Shift-and-invert Cayley Fold Preconditioner

• Spectrum Slicing SLEPc

Evaluation

m-step Lanczos Method

Computes Vm and Tm I M = (A− σB)−1B I Vm is a basis of the Krylov space Km(M, v1), V TmBVm = I I Tm = V

∗ mBMVm provides Ritz approximations, (θ̃i, Vmyi)

for j = 1, 2, . . . ,m w =Mvj t1:j,j = V

∗ j Bw

w = w − Vjt1:j,j tj+1,j = ‖w‖B vj+1 = w/tj+1,j

end

Orthogonalization:

I Full B-orthogonalization

I Do not bother about partial reorthog.

I SLEPc uses iterated CGS but MGS also available

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

• Spectrum Slicing SLEPc

Evaluation

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+1Ṽp

v m + 1

S̃p

b̃∗p Ṽp

v m + 1

S̃p

b̃∗p

For spectrum slicing, the basis expansion needs to orthogonalize also against an arbitrary set of vectors Restarts until nev converged eigenvalues (or subinterval complete)

• Spectrum Slicing SLEPc

Evaluation

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)

• Spectrum Slicing SLEPc

Evaluation

Test Case: Aircraft Fuselage

Simplified but realistic model: cylinder with skin, frames, and stringers

Parametric, “scalable”

First vibration mode (5.34 Hz)

• Spectrum Slicing SLEPc

Evaluation

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