Fast Iterative Solution of the Time-Harmonic Elastic Wave ...manuelbaumann.de/projects/mmb_defense.pdf · Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple

Fast Iterative Solution of the Time-HarmonicElastic Wave Equation at Multiple Frequencies

Manuel M. Baumann

January 10, 2018

Manuel Baumann PhD Defense Talk 1 / 13

Question you have asked me today...

Are you nervous? → Yes!

Questions you have asked me during the last years...

What is your PhD project about?

What is numerical linear algebra?

What have you been doing all day?(The German word for this is: rumdoktorn)






























You are here.

What is applied mathematics?

”Applied maths is about using mathematics to solvereal world problems neither seeking nor avoiding

mathematical difficulties.“

–Lord Rayleigh


What is applied mathematics?

”Applied maths is about using mathematics to solvereal world problems neither seeking nor avoiding

mathematical difficulties.“

–Lord Rayleigh


Seismic Full-Waveform Inversion

sender receivers Interplay of...

measurements,

ωk

Solve the linear systems of equations,

(K + iωkC − ω2kM)xk = b,

efficiently (= fast and at low memory) for multiple frequencies.




measurements,

ωk







measurements,

seismology,

ωk







measurements,

seismology,

ωk







measurements,

seismology,

computer simulations→ matrix computations

ωk







measurements,

seismology,


ωk

”xk = oil = $“

”Solve the linear systems of equations,


efficiently (= fast and at low memory) for multiple frequencies.“




measurements,

seismology,


ωk

”xk = oil = $“

Density distribution Simulations ω1···ωN




measurements,

seismology,


ωk

”xk = oil = $“

Density distribution Simulations ω1···ωN


Numerical Linear Algebra

A very classical linear algebra problem,

® + o = 1500

® + KK = 7.5

oo−KKK = 160

A more formal way of writing this,1 1 01 0 20 2 -3

︸︷︷︸

123

®oK

=

15007.5160




® + o = 1500

® + KK = 7.5

oo−KKK = 160


︸︷︷︸

123

®oK

=

15007.5160




® + o = 1500

® + KK = 7.5

oo−KKK = 160


︸︷︷︸

123

®oK

=

15007.5160




® + o = 1500

® + KK = 7.5

oo−KKK = 160


︸︷︷︸

123

®oK

=

15007.5160




® + o = 1500

® + KK = 7.5

oo−KKK = 160


︸︷︷︸

=:A

®oK

︸︷︷︸

=:x

=

15007.5160

︸︷︷︸

=:b




® + o = 1500

® + KK = 7.5

oo−KKK = 160


︸︷︷︸

=:A

®oK

︸︷︷︸

=:x

=

15007.5160

︸︷︷︸

=:b

The matrix A is symmetric




® + o = 1500

® + KK = 7.5

oo−KKK = 160

A more formal way of writing this,∗ ∗∗ ∗∗ ∗

︸︷︷︸

=:A

®oK

︸︷︷︸

=:x

=

15007.5160

︸︷︷︸

=:b

The matrix A is symmetric and sparse.




® + o = 1500

® + KK = 7.5

oo−KKK = 160

A more formal way of writing this,∗ ∗ ∗∗

︸︷︷︸

=:A

®oK

︸︷︷︸

=:x

=

15007.5160

︸︷︷︸

=:b

The matrix A is symmetric and sparse.



The matrix A can be...

sym

met

ric

Hermitian

skew

-sym

met

ric

positive (semi-)definite

indefinite

square

rect

angu

lar

invertible

diagonalizable

SPDsequentially semi-seperable

block tri-diagonal

upper Hessenberg

spar

se

dense

Port-Hamiltonian

nilpotent

ill-conditionedlow-rank




sym

met

ric

Hermitian

skew

-sym

met

ric


indefinite

square

rect

angu

lar

invertible

diagonalizable


block tri-diagonal

upper Hessenberg

spar

se

dense

Port-Hamiltonian

nilpotent





sym

met

ric

Hermitian

skew

-sym

met

ric


indefinite

square

rect

angu

lar

invertible

diagonalizable


block tri-diagonal

upper Hessenberg

spar

se

dense

Port-Hamiltonian

nilpotent



Shifted systems vs. matrix equation

Two main approaches for solving,

(K + iωkC − ω2kM)xk = b, k > 1.

Shifted systems([iC KI 0

]- ωk

[M 00 I

])[ωkxk

xk

]=

[b0

]Matrix equation

KX + iCXΩ−MXΩ2 = B




(K + iωkC − ω2kM)xk = b, k > 1.


]- ωk

[M 00 I

])[ωkxk

xk

]=

[b0

]

Most work for x0 (at ω = 0)

Requires preconditioning

Matrix equation





(K + iωkC − ω2kM)xk = b, k > 1.


]- ωk

[M 00 I

])[ωkxk

xk

]=

[b0

]



Matrix equation


Solve for X = [x1, ..., xN ]all-at-once





(K + iωkC − ω2kM)xk = b, k > 1.


]- ωk

[M 00 I

])[ωkxk

xk

]=

[b0

]



Matrix equation


Solve for X = [x1, ..., xN ]all-at-once



Preconditioning Let A := K + iωC − ω2M

Solve large-scale linear system,

Ax = b, with A ∈ CN×N ,N 1 (∗)

with an iterative method, i.e. compute xi with xi → x asi →∞.

Instead of (∗), solve the system

P−1Ax = P−1b,

where P is a preconditioner.

Number of Iterations

0 20 40 60 80 100

Re

sid

ua

l n

orm

10-6

10-4

10-2

100

102

No Preconditioner

Good Preconditioner

Better Preconditioner


Preconditioning

However, it’s often not that simple!([iC KI 0

]- ωk

[M 00 I

])[ωkxk

xk

]=

[b0

]

Main challenges:

multiple linear systems

single preconditioner

wide frequency range

preserve structure

τ∗ = ?


Preconditioning

However, it’s often not that simple!([iC KI 0

]- ωk

[M 00 I

])[ωkxk

xk

]=

[b0

]

Main challenges:

multiple linear systems

single preconditioner

wide frequency range

preserve structure

τ∗ = ?


Spectral analysis

real part-1 -0.5 0 0.5 1

ima

g p

art

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

”ε > 0“


Spectral analysis

real part-1.5 -1 -0.5 0 0.5 1 1.5

ima

g p

art

-0.5

0

0.5

1

1.5

2

”ε > 0“


Spectral analysis

real part-1.5 -1 -0.5 0 0.5 1 1.5

ima

g p

art

-0.5

0

0.5

1

1.5

2

”ε > 0“


Spectral analysis

real part-1.5 -1 -0.5 0 0.5 1 1.5

ima

g p

art

-0.5

0

0.5

1

1.5

2

”ε > 0“


Spectral analysis

real part-1.5 -1 -0.5 0 0.5 1 1.5

ima

g p

art

-0.5

0

0.5

1

1.5

2

”ε > 0“

c

R

ck−1

Rk

ck

ϕk−1

c1


Spectral analysis

Thm.: Optimal seed shift for multi-shift GMRES [B/vG, 2016]

(i) For λ ∈ Λ[AB−1] it holds =(λ) ≥ 0.

(ii) The preconditioned spectra are enclosed by circles of radiiRk and center points ck .

(iii) The points ckNk=1 ⊂ C described in statement (ii) lie ona circle with center c and radius R.

(iv) Consider the preconditioner P(τ∗) = A− τ∗B. An optimalseed frequency τ∗ for preconditioned multi-shift GMRES isgiven by,

τ∗(ε, ω1, ωN) = minτ∈C

maxk=1,..,N

(Rk(τ)

|ck |

)= ... =

=2ω1ωN

ω1 + ωN− i

√[ε2(ω1 + ωN)2 + (ωN − ω1)2]ω1ωN

ω1 + ωN


Spectral analysis

Thm.: Optimal seed shift for multi-shift GMRES [B/vG, 2016]

(i) For λ ∈ Λ[AB−1] it holds =(λ) ≥ 0.

(ii) The preconditioned spectra are enclosed by circles of radiiRk and center points ck .

(iii) The points ckNk=1 ⊂ C described in statement (ii) lie ona circle with center c and radius R.

(iv) Consider the preconditioner P(τ∗) = A− τ∗B. An optimalseed frequency τ∗ for preconditioned multi-shift GMRES isgiven by,

τ∗(ε, ω1, ωN) = minτ∈C

maxk=1,..,N

(Rk(τ)

|ck |

)= ... =

=2ω1ωN

ω1 + ωN− i

√[ε2(ω1 + ωN)2 + (ωN − ω1)2]ω1ωN

ω1 + ωN


Spectral analysis

Proof:


Spectral analysis

Proof: Not now.


Spectral analysis

Proof: There is an App for that.


https://manuelmbaumann.github.io/optimal-tau/

Convergence behavior and spectral bounds

For any τ ...

0 10 20 30 40 50 60 70Number of matrix-vector multiplications

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Relative residu

al norm

f = 1.0 Hzf = 3.0 Hzf = 5.0 Hzf = 7.0 Hzf = 9.0 Hz

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5real part

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

imag

par

t



Convergence behavior and spectral bounds

For the optimal τ∗...

0 5 10 15 20 25 30 35 40 45Number of matrix-vector multiplications

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Relative residu

al norm


−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5real part

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

imag

par

t



Lot’s of details...


What happens today?

15:00 – 16:00 Formal PhD defense

16:15 – 17:30 Reception (in this building)

21:00 – ?? More reception (borrel) at Prinsenkwartier

?−→


Thank you all for coming!

Documents

Fast Iterative Solution of the Time-Harmonic Elastic Wave ...manuelbaumann.de/projects/mmb_defense.pdf · Fast Iterative Solution of the Time-Harmonic Elastic Wave Equation at Multiple