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