scientific computing seminar

Scientific Computing Seminar

AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues*

C. Bekas Y. SaadComp. Science & Engineering Dept.University of Minnesota, Twin Cities

* Work supported by NSF under grants NSF/ITR-0082094, NSF/ACI-0305120 and by the Minnesota Supercomputing Institute

AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

Introduction and Motivation

C. Bekas: SC Seminar

Target ProblemCompute a large number of the smallest eigenvalues of large sparse

matrices Numerous important applications, including:

• structural engineering• computational materials science (electronic structure calculations)• Signal/Image processing and Control

Shift and Invert techniques can be very successful(Grimes et al 94, MSC.NASTRAN) …

BUT quickly become impractical to use:

For very large problem sizes (…N>106) a supercomputer is needed

When we need to compute several hundreds or thousands of eigenvalues (deep in the spectrum) reorthogonalization costs dominate and become prohibitive!


Introduction and Motivation

Component Mode Synthesis (CMS) (Hurty ’60, Graig-Bampton ’68)Well known alternative. Used for many years in Structural Engineering.But it too suffers from limitations due to problem size…

AMLS, (Bennighof, Lehoucq, Kaplan and collaborators) Multilevel CMS method (solves the dimensionality problem) Automatic computation of substructures (easy application) Approximation: Truncated Congruence Transformation Builds very large projection basis without reorthogonalization Successful in computing thousands of eigenvalues in vibro-acoustic analysis (N>107) in a few hours on workstations (Kropp–Heiserer, 02)

Accuracy issues AMLS accuracy is adequate in Structural Eng. (in the order of the discretization error) , but higher accuracy is needed in other applications, (i.e. electronic structure calculations) AMLS is an one shot approach: no (iterative) refinement is done



Recent Advances

The success of AMLS in Struct. Eng. and its potential for other applications has sparked several new research initiatives.

Some include:

- Bekas and SaadPurely algebraic analysis of AMLS1) Approximation to a nonlinear (Schur) eigenvalue problem2) Approximation of the resolvent (A- I)-1 by a careful projection3) Improvements: a) 2nd order expansions, b) Krylov projections and combinations

- Yang et alAlgebraic substructuring. Careful selection of added eigenvectors for improved accuracy.

- Elssel and VossA priori error bounds for algebraic substructuring.



In this talk

We will describe

- Approximation mechanism behind AMLS. We will review our recent purely algebraic analysis of AMLS (Bekas – Saad, 04).

- Iterative application of AMLS1) Refinement of approximated eigenvalues2) Multiple shifts in AMLS3) Computation of eigenvalues in an interval

- Numerical examples


1 2

Subdivide into 2 subdomains: 1 and 2

Component Mode Synthesis: a model problem

Y

X

Consider the model problem:

on the unit square . We wish to compute smallest eigenvalues.

Component Mode Synthesis

• Solve problem on each i

• “Combine” partial solutions



CMS: Approximation procedure. Ignore coupling!

Component Mode Synthesis: Approximation

CMS: Approximation procedure (2)

• Solve B v = v • Remember that B is block diagonal• Thus: we have to solve smaller decoupled eigenproblems

• Then, CMS approximates the coupling among the subdomains by the application of a carefully selected operator on the interface unknowns



Recent CMS method: AMLS

Block Gaussian eliminator matrix: such that:

Where S = C – E> B-1 E is the Schur Complement

Equivalent Generalized Eigenproblem : UTAUu= UTU u



AMLS Approximation

AMLS: Approximation procedure. Ignore coupling!

Solve decoupled problems Form basis

FINAL APPROXIMATION:



AMLS: Multilevel application

S1

E1

E1*

B2

B1

S2

S3

Scheme applied recursively.Resulting to thousands of subdomains. Successful in computing thousands smallest eigenvalues in vibro-acoustic analysis with problem size N>107

In the following we analyze the approximation mechanism of one step of AMLS, adopting a purely algebraic setting.

This will naturally lead to improved versions of the method



AMLS: approximation to a nonlinear eigenproblem!

AMLS: Approximation procedure. DO NOT ignore coupling this time

Leads to:

Substitute equation (1) in equation (2) to give equivalent non-linear problem:




Define:

Resolvent:

Basic Property

We can show that:




Neumann Series of the Resolvent:

Truncate the series. Keep the first two terms only. Then…

Approximate problem:

Remember:

AMLS:



AMLS: The projection view-point

Initial problem:

We can show that if: Then, is eigenvalue of (1) and

Respective eigenvector




1. AMLS solves approximately (truncation)

4. Remedy: augment the space of approximants by eigenvectors of B. Why?

2. AMLS, change of basis:

3. Thus:




We examine the difference:

Expansion in terms of eigenvectors of B:

Then, this difference is:

Thus: the difference is large in eigenvectors of B with eigenvalues close to the smallest eigenvalues of A.




Therefore: augmenting the space of approximants with ‘’smallest’’eigenvectors of B is well justified. Can we bound the error?

Let: mB “smallest” eigenvectors of B

XB: restriction to the B-part (upper part) of the space of approximants. Then:



Improvements

The algebraic framework we have described leads to improvements:

Nonlinear Schur complement problem: Introduce an additional term of the truncated Neumann series Utilize corresponding 2nd order projection

Approximation of the resolvent (B- I)-1: Approximate with Krylov subspaces on B-1 and Utilize corresponding projection

Combinations of the above improvements lead to hybrid algorithms with enhanced stability and robustness



Augmenting with Krylov Subspaces

We need to approximate

Therefore it is natural to consider the Krylov subspace

Vk is an orthonormal block Krylov basis

The columns of US are eigenvectors of the nonlinear Schur complement problem

or in general the block Krylov subspace many j



Second Order Approximation

Neumann Series of the Resolvent:

AMLS approximation

Can we do better? 2nd order approximation

Quadratic Eigenvalue Problem:



Second Order Approximation: Solving the QED

Quadratic Eigenvalue Problem:



Linearization leads to equivalent generalized eigenproblem:

Second Order Approximation: Projection view point

Eigenvector of A:Better approx. by the QEP

We can add more eigenvectors of B…or add “second order” vectors:

ADD THE VECTORS

Construct basis

Solve



Second Order Approximation: Error bounds

Augmenting the space of approximants with eigenvectors of B and “second order vectors” leads to quadratic error bounds compared to adding just eigenvectors

Let: mB smallest eigenvectors of B

XB: restriction to the B-part (upper part) of the space of approximants. Then:



Improvements

The algebraic framework we have described leads to improvements:

Our previous workNonlinear Schur complement problem:

Introduce an additional term of the truncated Neumann seriesApproximation of the resolvent (B- I)-1:

Approximate with Krylov subspaces on B-1 and Utilize corresponding projection

Recent Work Framework for the iterative refinement of the AMLS approximations Utilization of many different shifts (A-k I)-1 that can be used for… …computation of eigenvalues deep in the spectrum More robust method. Currently examining connections with shift-invert Lanczos (Grimes et al) and Rational Krylov (Ruhe)



Iterative Refinement (1/3)



OR WITH SOME REARRANGEMENT…


NON-LINEAR SCHUR COMPLEMENT

AMLS APPROXIMATIONR: AMLS REMAINDER



Using Multiple Shifts in AMLS (1/3)

MATRIX: BCSSTK111st shift: 3, close to eigenvalues in the interval [2.9,3]2nd shift: 69, close to eigenvalues in the interval [68,70]

CAN WE COMBINE THESE TWO SHIFTS?




Smallest eigenvectors of

Smallest eigenvectors of

PROJECTED PROBLEM



Numerical Experiments



Numerical Experiments: Standard v.s. Krylov

Thus: Small /favors the Krylov version

Matrix: BCSSTK11(N=1473), Reorder using Nes. Disec., size of Schur complement NS=94

Bef. reord./

After reord./




Number of Schur eigenvectors: 5Number of added vectors:

AMLS: 30 eigenvectors of B (left) and 40 eigenvectors of B (right) Krylov: 30=5 x 6 (left) and 40=5 x 8 (right)

REORDERED VERSION




Number of Schur eigenvectors: 5Number of added vectors:

AMLS: 30 eigenvectors of B (left) and 40 eigenvectors of B (right) Krylov: 30=5 x 6 (left) and 40=5 x 8 (right)

NOT REORDERED VERSION



Second order method

Matrix: BCSSTK11Approximate S() u= u with a Quadratic eigenvalue problem instead of the original generalized eigenvalue problem of AMLS



Second order method

Matrix: BCSSTK11Compare 2nd order AMLS (20 added vectors) v.s. standard AMLS with increasing number of added eigenvectors of B (mB=20,40,60,100)

1 2 3 4 5 6 7 8 9 10

10-8

10-6

10-4

10-2

Eigenvalues: smallest to larger

abs.

rela

tive

erro

r

AMLS, mB = 100AMLS, mB = 60AMLS, mB = 40AMLS, mB = 202nd order AMLS, mS = 10

2nd order: 10 Schur vectors: 20 added Krylov vectors. 30 vectors in total

AMLS: 10 Schur vectors: 20, 40, 60 and 100 added eigenvectors of B



Combine AMLS with Krylov AMLS

1 2 3 4 5 6 7 8 9 1010

-6

10-4

10-2

100

102


abs.

rela

tive

erro

r

AMLS, k=20Krylov AMLS, m=2

Matrix: 5-point stencil discretization of the LaplacianUse only the 2 smallest Schur vectors. Then, for larger eigenvalues the Krylov version can fail.

REMEDY: Augment subspace with eigenvectors of B too!

VB: eigenvectors of BVK: Krylov basisUS: Schur eigenvectors



Combine AMLS with Krylov AMLS

Matrix: 5-point stencil discretization of the LaplacianUse only 2 smallest Schur vectors. Then, for larger eigenvalues the Krylov version can fail.

REMEDY: Augment subspace with eigenvectors of B too!

VB: eigenvectors of BVK: Krylov subspaceUS: Schur eigenvectors

1 2 3 4 5 6 7 8 9 1010

-6

10-5

10-4

10-3

10-2

10-1


abs.

rela

tive

erro

r

AMLS, k=30Krylov - Standard AMLS, m=2, k=20



Iterative Refinement

Matrix: 5-point stencil discretization of the Laplacian

Next shift k is 1st smallest approximate eigenvalue at step k-1

Next shift k is 5th smallest approximate eigenvalue at step k-1



Multiple Shifts

Dimension of each Qi : 40

Matrix: 5-point stencil discretization of the Laplacian

Dimension of each Qi : 60



Conclusions

AMLS is a promising alternative to Shift-Invert methods for very large problems

In this work we have presented an analysis of AMLS based on a completely algebraic framework:

• AMLS is a nonlinear (spectral) Schur complement method that utilizes…• Projection on carefully selected eigenspaces to approximate the solution

Improvements• Based on the algebraic framework we have proposed Krylov projection subspaces and second order approximations (and combinations) with significant improvements.

Iterative Refinement• Approximations can be iteratively refined, allowing for very good accuracy• Many different shifts are combined and thus we can compute eigenvalues deep in the spectrum

Current Work• Investigate strategies to compute all eigenvalues in a (large) interval [a,b]



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