iterativesolvers oneclassof martinj.gander ... · has nearlinear cost in examples, due to very...
TRANSCRIPT
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
One Class ofIterative Solvers for Helmholtz Problems:
AILU Factorizations, SweepingPreconditioners, Source Transfer, SingleLayer Potentials, Polarized Traces, and
Optimal and Optimized Schwarz Methods
Martin J. Gander
University of Geneva
Paris, September 2017
in collaboration with Hui Zhang
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
One Class ofIterative Solvers for Helmholtz Problems:
AILU Factorizations, SweepingPreconditioners, Source Transfer, SingleLayer Potentials, Polarized Traces, and
Optimal and Optimized Schwarz Methods
Martin J. Gander
University of Geneva
Paris, September 2017
in collaboration with Hui Zhang
Why it is difficult to solve the Helmholtz equation with
classical iterative methods (Ernst and G 2012)
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Quotes from Key References (2013-2017)
“The method has sublinear runtime even in the presenceof rough media of geophysical interest. Moreover, itsperformance is completely agnostic to the source.”
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Quotes from Key References (2013-2017)
“The method has sublinear runtime even in the presenceof rough media of geophysical interest. Moreover, itsperformance is completely agnostic to the source.”
“The resulting preconditioner has linear applicationcost, and the preconditioned iterative solver convergesin a number of iterations that is essentially independentof the number of unknowns or the frequency.”
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Quotes from Key References (2013-2017)
“The method has sublinear runtime even in the presenceof rough media of geophysical interest. Moreover, itsperformance is completely agnostic to the source.”
“The resulting preconditioner has linear applicationcost, and the preconditioned iterative solver convergesin a number of iterations that is essentially independentof the number of unknowns or the frequency.”
“When combined with multifrontal methods, the solverhas nearlinear cost in examples, due to very smalliteration numbers that are essentially independent ofproblem size and number of subdomains.”
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Quotes from Key References (2013-2017)
“The method has sublinear runtime even in the presenceof rough media of geophysical interest. Moreover, itsperformance is completely agnostic to the source.”
“The resulting preconditioner has linear applicationcost, and the preconditioned iterative solver convergesin a number of iterations that is essentially independentof the number of unknowns or the frequency.”
“When combined with multifrontal methods, the solverhas nearlinear cost in examples, due to very smalliteration numbers that are essentially independent ofproblem size and number of subdomains.”
“The convergence of the method is proved for the caseof constant wave number based on the analysis of thefundamental solution of the PML equation.”
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Quotes from Key References (2013-2017)
“The method has sublinear runtime even in the presenceof rough media of geophysical interest. Moreover, itsperformance is completely agnostic to the source.”
“The resulting preconditioner has linear applicationcost, and the preconditioned iterative solver convergesin a number of iterations that is essentially independentof the number of unknowns or the frequency.”
“When combined with multifrontal methods, the solverhas nearlinear cost in examples, due to very smalliteration numbers that are essentially independent ofproblem size and number of subdomains.”
“The convergence of the method is proved for the caseof constant wave number based on the analysis of thefundamental solution of the PML equation.”
“Numerical results are presented to demonstrate theefficiency as a preconditioner for solving the Helmholtzproblems considered in the paper.”
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Helmholtz Model Problem and Discretization
(∆ + k2)u = f in Ω := (0, a)× (0, b)
y
xu1,1
u1,J
uJ,1
uJ,J
Au =
D1 U1
L1 D2 U2
. . .. . .
. . .
LJ−2 DJ−1 UJ−1
LJ−1 DJ
u1u2...
uJ−1
uJ
=
f1f2...
fJ−1
fJ
= f
where Dj = tridiag ( 1h2,− 4
h2+ k2, 1
h2), Lj = Uj = diag ( 1
h2).
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Block LU Factorization
The block LU factorization
A = LU
leads to the two factors
T1
L1 T2
. . .. . .
LJ−2 TJ−1
LJ−1 TJ
I1 T−11 U1
I2 T−12 U2
. . .. . .
IJ−1 T−1J−1UJ−1
IJ
where the Tj are the Schur complements that satisfy therecurrence relation
T1 = D1, Tj = Dj − Lj−1T−1j−1Uj−1 for j ≥ 2
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Forward and Backward SubstitutionUsing this factorization, we can solve by first solving byforward substitution the block lower triangular system
T1
L1 T2
. . .. . .
LJ−2 TJ−1
LJ−1 TJ
v1v2...
vJ−1
vJ
=
f1f2...
fJ−1
fJ
and then solving by backward substitution the block uppertriangular system
I1 T−11 U1
I2 T−12 U2
. . .. . .
IJ−1 T−1J−1UJ−1
IJ
u1u2...
uJ−1
uJ
=
v1v2...
vJ−1
vJ
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Observations
The forward and backward substitution represent asweeping solve across the physical domain and back
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Observations
The forward and backward substitution represent asweeping solve across the physical domain and back
The forward substitution gives
v1 =T−11 f1,
v2 =T−12 (f2 − L1v1) =T−1
2 (f2 − L1T−11 f1) =:T−1
2 f2,
v3 =T−13 (f3 − L2v2) =T−1
3 (f3 − L2T−12 f2) =:T−1
3 f3,...
......
with the transferred source terms
fj := fj − Lj−1T−1j−1fj−1.
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Observations
The forward and backward substitution represent asweeping solve across the physical domain and back
The forward substitution gives
v1 =T−11 f1,
v2 =T−12 (f2 − L1v1) =T−1
2 (f2 − L1T−11 f1) =:T−1
2 f2,
v3 =T−13 (f3 − L2v2) =T−1
3 (f3 − L2T−12 f2) =:T−1
3 f3,...
......
with the transferred source terms
fj := fj − Lj−1T−1j−1fj−1.
Note that vJ = uJ , so after the forward substitution,the last set of unknowns is already the exact solution.
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
A New Schwarz Method (Nataf, Rogier 1994)
x
y
0 a
b
ΓΩ1 Ω2
New Schwarz algorithm uses different transmission conditions:
(∆ + k2)un1 = f in Ω1,
∂n1un1 +DtN1(u
n1 ) = ∂n1u
n−12 +DtN1(u
n−12 ) on Γ,
(∆ + k2)un2 = f in Ω2,
∂n2un2 +DtN2(u
n2 ) = ∂n2u
n−11 +DtN2(u
n−11 ) on Γ,
This algorithm converges in two iterations,
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
A New Schwarz Method (Nataf, Rogier 1994)
x
y
0 a
b
Γ12Γ21Ω1 Ω2
New Schwarz algorithm uses different transmission conditions:
(∆ + k2)un1 = f in Ω1,
∂n1un1 +DtN1(u
n1 ) = ∂n1u
n−12 +DtN1(u
n−12 ) on Γ12,
(∆ + k2)un2 = f in Ω2,
∂n2un2 +DtN2(u
n2 ) = ∂n2u
n−11 +DtN2(u
n−11 ) on Γ21.
This algorithm converges in two iterations, independently ofthe overlap (G, Halpern, Nataf 1999)!
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
A New Schwarz Method (Nataf, Rogier 1994)
x
y
0 a
b
Γ12 Γ23 Γ34 Γ45
Ω1 Ω2 Ω3 Ω4 Ω5
New Schwarz algorithm uses different transmission conditions:
(∆ + k2)unj = f in Ωj ,
∂njunj +DtNj(u
nj ) = ∂nju
n−1j+1 +DtNj(u
n−1j+1 ) on Γj ,j+1,
∂njunj +DtNj(u
nj ) = ∂nju
n−1j−1 +DtNj(u
n−1j−1 ) on Γj ,j−1,
With J subdomains, it converges in J iterations,
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
A New Schwarz Method (Nataf, Rogier 1994)
x
y
0 a
b
Γ12 Γ23 Γ34 Γ45
Ω1 Ω2 Ω3 Ω4 Ω5
New Schwarz algorithm uses different transmission conditions:
(∆ + k2)unj = f in Ωj ,
∂njunj +DtNj(u
nj ) = ∂nju
n−1j+1 +DtNj(u
n−1j+1 ) on Γj ,j+1,
∂njunj +DtNj(u
nj ) = ∂nju
n−1j−1 +DtNj(u
n−1j−1 ) on Γj ,j−1,
With J subdomains, it converges in J iterations,or in one forward and backward sweep.
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
A New Schwarz Method (Nataf, Rogier 1994)
x
y
0 a
b
Γ12 Γ23 Γ34 Γ45
Ω1 Ω2 Ω3 Ω4 Ω5
New Schwarz algorithm uses different transmission conditions:
(∆ + k2)unj = f in Ωj ,
∂njunj +DtNj(u
nj ) = ∂nju
n−1j+1 +DtNj(u
n−1j+1 ) on Γj ,j+1,
∂njunj +DtNj(u
nj ) = ∂nju
n−1j−1 +DtNj(u
n−1j−1 ) on Γj ,j−1,
With J subdomains, it converges in J iterations,or in one forward and backward sweep.
Continuous analog of the block LU decomposition !
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz G, Halpern, Nataf 1999: Optimal Convergence for
Overlapping and Non-Overlapping Schwarz Waveform
Relaxation G, Nataf 2000: AILU: a preconditioner based on the
analytic factorization of the elliptic operator G, Magoules, Nataf 2002: Optimized Schwarz
methods without overlap for the Helmholtz equation G 2006: Optimized Schwarz methods
01
1
0.05
y
0.5
x
0.50 0
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.)
Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
x
y
0 1
Ω1 Ω2 Ω3 Ω4 Ω5
PML PML
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.)
Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
x
y
0 1
Ω1 Ω2 Ω3 Ω4 Ω5
∂x+DtNleft ∂x+DtNright
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.)
Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
x
y
0 1
Ω1 Ω2 Ω3 Ω4 Ω5
∂x+DtNleft ∂x+DtNright
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Methods Based on Optimal Schwarz (cont.) Enquist, Ying 2010: Sweeping Preconditioner for the
Helmholtz Equation
Chen, Xiang 2012: A Source Transfer DD Method for
Helmholtz Equations in Unbounded Domain
Stolk 2013: A rapidly converging domain
decomposition method for the Helmholtz equation
Zepeda-Nunez, Hewett, Demanet 2014:Preconditioning the 2D Helmholtz equation with
polarized traces
-10
-5
10-3
0.5
5
0.5
xy 00
0
11
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Limitations of this Approach for HelmholtzHelmholz with J subdomains, k constant per subdomain
J = 4 J = 8α Iterative ρ GMRES Iterative ρ GMRES0 1 1 1 4.4e-15 1 1 1 1 1 1 8.3e-15 1 1 1
0.001 2 3 3 2.5e-2 3 3 3 3 3 4 7.4e-2 4 3 40.005 3 3 7 0.13 4 3 5 6 5 10 0.40 7 5 70.01 4 4 8 0.25 5 4 5 14 6 24 0.68 9 6 80.05 - - - 1.52 7 5 8 - - - 11.48 15 11 160.1 11 10 26 0.69 8 6 10 - - - 2.74 17 13 181 - - - 3.86 20 14 20 - - - 188 39 32 45
k = [20 20 20 20] + α[0 20 10 − 10]
0 1 1 1 5.4e-15 1 1 1 1 1 1 5.3e-15 1 1 10.001 2 3 3 2.5e-2 4 3 3 2 4 4 1.1e-1 5 4 40.005 3 6 6 0.14 5 5 5 - 8 - 0.88 9 8 80.01 5 10 9 0.33 6 6 6 - 16 - 1.92 12 8 110.05 - - - 4.48 13 10 13 - - - 7.28 22 18 230.1 - 25 - 1.8 14 11 14 - - - 20.2 20 17 231 - - - 9.62 31 24 36 - - - 8.93 66 55 70
k = [40 40 40 40] + α[0 40 20 − 20]
Iterative Solvers
for Helmholtz
Martin J. Gander
Quotes
Basic Algorithms
Model Problem
Block LU
New Schwarz
Optimized Schwarz
Helmholtz
OSM Based
Limitations
Conclusion
Conclusions
All these recent preconditioners are variants of optimizedSchwarz methods (DOSMs):
Sweeping Preconditioner: DOSM with non-overlappingsubdomains with empty interior, PML or H-matrixtransmission condition (TC) on the left and Dirichlet onthe right
Source transfer: DOSM with maximally overlappingsubdomains with PML TC in the forward sweep andsource term set to zero in the overlap, and Dirichletinstead of PML on the right in the backward sweep.
Single Layer Potential Method: DOSM with twonon-overlapping domain decompositions and PML TC
Method of Polarized Traces: DOSM withnon-overlapping subdomains and PML TC
Rigorous proofs in (G, Zhang 2017), available atwww.unige.ch/∼gander