Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems

1
Finding Rightmost Eigenvalues of Large, Sparse,
Nonsymmetric Parameterized Eigenvalue Problems

• Minghao Wu
• AMSC Program
• mwu_at_math.umd.edu
• Advisor
• Dr. Howard Elman
• Department of Computer Science
• elman_at_cs.umd.edu

2
Motivation

• To determine the stability of the linearized
  system of the form

• The steady state solution x is
• - stable, if all the eigenvalues of Ax ?Bx
  have
• negative real parts
• - unstable, otherwise.

3
Problem Statement

• To find the rightmost eigenvalues of

• where matrices A and B are
• Real N-by-N
• Large
• Sparse
• Nonsymmetric
• Depend on one or several parameters

4
Method

• Eigensolver Arnoldi Algorithm
• - Iterative method
• - Based on Krylov subspace

- Computes Arnoldi decomposition
where Uk uk1 an orthonormal basis of
Hk k-by-k upper Hessenberg
matrix, k N ßk scalar, ek k-by-1 vector 0 0
0 1
5
Arnoldi Algorithm (continue)

• Eigenvalues of Hk approximate eigenvalues of A
• Premultiply previous equation by
  transpose(Uk)

Let (?,z) be an eigenpair of Hk, then
Residual
As k increases, Residual will decrease. When
k N, Residual 0.
6
Matrix Transformation

• Motivation
• - Arnoldi Algorithm cannot solve generalized
  eigenvalue problem
• - It converges to well separated extremal
  eigenvalues, not
• rightmost eigenvalues
• Shift Invert Transformation

7
Matlab Code of Arnoldi Method

• Arnoldi Algorithm (with Shift Invert matrix
• transformation) routine
• v,X,U,HSI_Arnoldi(A,B,k,sigma)
• Input
• A, B matrix A and B in Ax ?Bx
• k number of eigenpairs wanted
• sigma the shift s in shift invert matrix
  transformation
• Output
• v a vector of k computed eigenvalues
• X k eigenvectors associated with the
  eigenvalues
• U the Krylov basis Uk1
• H the upper Hessenberg matrix Hk

8
Test Problem

• Olmstead Model (see Olmstead et al (1986))

with boundary conditions
This model represents the flow of a layer of
viscoelastic fluid heated from below. u the
speed of the fluid S related to viscoelastic
forces b,c scalars, R scalar, Rayleigh number
9
Test Problem (continue)

• Discretize the model with finite differences
• - grid size h 1/(N/2)
• - (4) can be written as dy/dt f(y) with

• Evaluate the Jacobian matrix A df/dy at
  steady state
• solution y
• - N 1000, b 2, c 0.1, R 0.6
• - y 0
• - A df/dy(y) is a nonsymmetric sparse
  matrix with bandwidth 6

10
Test Problem (continue)

• Computational Result
• Rightmost eigenvalues ?1,2 0 0.4472i
• Residual Axi - ?i xi 8.4504e-012, i1,2
• The result agrees with the literature.

11
Implicitly Restarted Arnoldi (IRA)

• Motivation
• - Large k is not practical
• Example

• When B is singular, Arnoldi algorithm may give
  rise to spurious
• eigenvalues

12
IRA (continue)

• Basic idea of Implicitly Restarted Arnoldi
• Filter out the unwanted eigendirections from
  the starting vector by using the most recent
  spectrum information and a clever filtering
  technique
• IRA steps
• 1. Compute m eigenpairs (kltmN) by Arnoldi
  method
• with starting vector u1
• 2. Filter out the m-k unwanted
  eigendirections from u1
• (Key Technique shifted QR algorithm)
• 3. Restart the process with filtered starting
  vector till
• the k eigenvalues of interest converge

13
Test Problem
K 200-by-200 matrix, full rank C 200-by-100
matrix, full rank M 200-by-200 matrix, full
rank. Eigenvalue problem with this kind of block
structure appears in the stability analysis of
steady state solution of Navier Stokes
equations for incompressible flow.
14
Test Problem (continue)

• Use Matlab function rand to generate K, C, M
• -2.7377 Re(?) 49.9129
• Find out 10 rightmost eigenvalues
• Use the IRA code written by Fei Xue

15
Test Problem (continue)
Computational Result (shift s 60)
16
Future Work (AMSC 664)

• Solve the third test problem
• Implement iterative solvers for linear systems