Fast Iterative Solution of the Time-Harmonic Elastic Wave...
Transcript of Fast Iterative Solution of the Time-Harmonic Elastic Wave...
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)
Manuel Baumann PhD Defense Talk 2 / 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)
Manuel Baumann PhD Defense Talk 2 / 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)
Manuel Baumann PhD Defense Talk 2 / 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)
Manuel Baumann PhD Defense Talk 2 / 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)
Manuel Baumann PhD Defense Talk 2 / 13
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
Manuel Baumann PhD Defense Talk 2 / 13
What is applied mathematics?
”Applied maths is about using mathematics to solvereal world problems neither seeking nor avoiding
mathematical difficulties.“
–Lord Rayleigh
Manuel Baumann PhD Defense Talk 3 / 13
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.
Manuel Baumann PhD Defense Talk 3 / 13
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.
Manuel Baumann PhD Defense Talk 3 / 13
Seismic Full-Waveform Inversion
sender receivers Interplay of...
measurements,
seismology,
ωk
Solve the linear systems of equations,
(K + iωkC − ω2kM)xk = b,
efficiently (= fast and at low memory) for multiple frequencies.
Manuel Baumann PhD Defense Talk 3 / 13
Seismic Full-Waveform Inversion
sender receivers Interplay of...
measurements,
seismology,
ωk
Solve the linear systems of equations,
(K + iωkC − ω2kM)xk = b,
efficiently (= fast and at low memory) for multiple frequencies.
Manuel Baumann PhD Defense Talk 3 / 13
Seismic Full-Waveform Inversion
sender receivers Interplay of...
measurements,
seismology,
computer simulations→ matrix computations
ωk
Solve the linear systems of equations,
(K + iωkC − ω2kM)xk = b,
efficiently (= fast and at low memory) for multiple frequencies.
Manuel Baumann PhD Defense Talk 3 / 13
Seismic Full-Waveform Inversion
sender receivers Interplay of...
measurements,
seismology,
computer simulations→ matrix computations
ωk
”xk = oil = $“
”Solve the linear systems of equations,
(K + iωkC − ω2kM)xk = b,
efficiently (= fast and at low memory) for multiple frequencies.“
Manuel Baumann PhD Defense Talk 3 / 13
Seismic Full-Waveform Inversion
sender receivers Interplay of...
measurements,
seismology,
computer simulations→ matrix computations
ωk
”xk = oil = $“
Density distribution Simulations ω1···ωN
Manuel Baumann PhD Defense Talk 3 / 13
Seismic Full-Waveform Inversion
sender receivers Interplay of...
measurements,
seismology,
computer simulations→ matrix computations
ωk
”xk = oil = $“
Density distribution Simulations ω1···ωN
Manuel Baumann PhD Defense Talk 3 / 13
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
Manuel Baumann PhD Defense Talk 4 / 13
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
Manuel Baumann PhD Defense Talk 4 / 13
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
Manuel Baumann PhD Defense Talk 4 / 13
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
Manuel Baumann PhD Defense Talk 4 / 13
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
︸ ︷︷ ︸
=:A
®oK
︸ ︷︷ ︸
=:x
=
15007.5160
︸ ︷︷ ︸
=:b
Manuel Baumann PhD Defense Talk 4 / 13
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
︸ ︷︷ ︸
=:A
®oK
︸ ︷︷ ︸
=:x
=
15007.5160
︸ ︷︷ ︸
=:b
The matrix A is symmetric
Manuel Baumann PhD Defense Talk 3 / 13
Numerical Linear Algebra
A very classical linear algebra problem,
® + 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.
Manuel Baumann PhD Defense Talk 3 / 13
Numerical Linear Algebra
A very classical linear algebra problem,
® + 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.
Manuel Baumann PhD Defense Talk 3 / 13
Numerical Linear Algebra
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
Manuel Baumann PhD Defense Talk 4 / 13
Numerical Linear Algebra
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
Manuel Baumann PhD Defense Talk 4 / 13
Numerical Linear Algebra
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
Manuel Baumann PhD Defense Talk 4 / 13
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
Manuel Baumann PhD Defense Talk 5 / 13
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
]
Most work for x0 (at ω = 0)
Requires preconditioning
Matrix equation
KX + iCXΩ−MXΩ2 = B
Manuel Baumann PhD Defense Talk 5 / 13
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
]
Most work for x0 (at ω = 0)
Requires preconditioning
Matrix equation
KX + iCXΩ−MXΩ2 = B
Solve for X = [x1, ..., xN ]all-at-once
Requires preconditioning
Manuel Baumann PhD Defense Talk 5 / 13
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
]
Most work for x0 (at ω = 0)
Requires preconditioning
Matrix equation
KX + iCXΩ−MXΩ2 = B
Solve for X = [x1, ..., xN ]all-at-once
Requires preconditioning
Manuel Baumann PhD Defense Talk 5 / 13
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
Manuel Baumann PhD Defense Talk 6 / 13
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∗ = ?
Manuel Baumann PhD Defense Talk 7 / 13
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∗ = ?
Manuel Baumann PhD Defense Talk 7 / 13
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“
Manuel Baumann PhD Defense Talk 8 / 13
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“
Manuel Baumann PhD Defense Talk 8 / 13
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“
Manuel Baumann PhD Defense Talk 8 / 13
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“
Manuel Baumann PhD Defense Talk 8 / 13
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
Manuel Baumann PhD Defense Talk 8 / 13
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
Manuel Baumann PhD Defense Talk 9 / 13
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
Manuel Baumann PhD Defense Talk 9 / 13
Spectral analysis
Proof:
Manuel Baumann PhD Defense Talk 10 / 13
Spectral analysis
Proof: Not now.
Manuel Baumann PhD Defense Talk 10 / 13
Spectral analysis
Proof: There is an App for that.
Manuel Baumann PhD Defense Talk 10 / 13
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
f = 1.0 Hzf = 3.0 Hzf = 5.0 Hzf = 7.0 Hzf = 9.0 Hz
Manuel Baumann PhD Defense Talk 11 / 13
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
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
f = 1.0 Hzf = 3.0 Hzf = 5.0 Hzf = 7.0 Hzf = 9.0 Hz
Manuel Baumann PhD Defense Talk 11 / 13
Lot’s of details...
Manuel Baumann PhD Defense Talk 12 / 13
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
?−→
Manuel Baumann PhD Defense Talk 13 / 13
Thank you all for coming!