One Class of Martin J. Gander Iterative Solvers for Helmholtz - - PowerPoint PPT Presentation

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 of Iterative Solvers for Helmholtz Problems: AILU Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer 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 of Iterative Solvers for Helmholtz Problems: AILU Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer 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 presence

• f rough media of geophysical interest. Moreover, its

performance 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 presence

• f rough media of geophysical interest. Moreover, its

performance is completely agnostic to the source.”

◮ “The resulting preconditioner has linear application

cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent

• f 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 presence

• f rough media of geophysical interest. Moreover, its

performance is completely agnostic to the source.”

◮ “The resulting preconditioner has linear application

cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent

• f the number of unknowns or the frequency.”

◮ “When combined with multifrontal methods, the solver

has nearlinear cost in examples, due to very small iteration numbers that are essentially independent of problem 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 presence

• f rough media of geophysical interest. Moreover, its

performance is completely agnostic to the source.”

◮ “The resulting preconditioner has linear application

cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent

• f the number of unknowns or the frequency.”

◮ “When combined with multifrontal methods, the solver

has nearlinear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains.”

◮ “The convergence of the method is proved for the case

• f constant wave number based on the analysis of the

fundamental 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 presence

• f rough media of geophysical interest. Moreover, its

performance is completely agnostic to the source.”

◮ “The resulting preconditioner has linear application

cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent

• f the number of unknowns or the frequency.”

◮ “When combined with multifrontal methods, the solver

has nearlinear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains.”

◮ “The convergence of the method is proved for the case

• f constant wave number based on the analysis of the

fundamental solution of the PML equation.”

◮ “Numerical results are presented to demonstrate the

efficiency as a preconditioner for solving the Helmholtz problems 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 x

u1,1 u1,J uJ,1 uJ,J

Au =        D1 U1 L1 D2 U2 ... ... ... LJ−2 DJ−1 UJ−1 LJ−1 DJ               u1 u2 . . . uJ−1 uJ        =        f1 f2 . . . fJ−1 fJ        = f where Dj = tridiag ( 1

h2 , − 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 −1

1 U1

I2 T −1

2 U2

... ... IJ−1 T −1

J−1UJ−1

IJ        where the Tj are the Schur complements that satisfy the recurrence relation T1 = D1, Tj = Dj − Lj−1T −1

j−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 Substitution

Using this factorization, we can solve by first solving by forward substitution the block lower triangular system        T1 L1 T2 ... ... LJ−2 TJ−1 LJ−1 TJ               v1 v2 . . . vJ−1 vJ        =        f1 f2 . . . fJ−1 fJ        and then solving by backward substitution the block upper triangular system        I1 T −1

1 U1

I2 T −1

2 U2

... ... IJ−1 T −1

J−1UJ−1

IJ               u1 u2 . . . uJ−1 uJ        =        v1 v2 . . . 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 a

sweeping 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 a

sweeping solve across the physical domain and back

◮ The forward substitution gives

v1 = T −1

1 f1,

v2 = T −1

2 (f2 − L1v1) = T −1 2 (f2 − L1T −1 1 f1) =: T −1 2 ˜

f2, v3 = T −1

3 (f3 − L2v2) = T −1 3 (f3 − L2T −1 2 ˜

f2) =: T −1

3 ˜

f3, . . . . . . . . . with the transferred source terms ˜ fj := fj − Lj−1T −1

j−1˜

fj−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 a

sweeping solve across the physical domain and back

◮ The forward substitution gives

v1 = T −1

1 f1,

v2 = T −1

2 (f2 − L1v1) = T −1 2 (f2 − L1T −1 1 f1) =: T −1 2 ˜

f2, v3 = T −1

3 (f3 − L2v2) = T −1 3 (f3 − L2T −1 2 ˜

f2) =: T −1

3 ˜

f3, . . . . . . . . . with the transferred source terms ˜ fj := fj − Lj−1T −1

j−1˜

fj−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 a b Γ Ω1 Ω2

New Schwarz algorithm uses different transmission conditions: (∆ + k2)un

1

= f in Ω1, ∂n1un

1 + DtN1(un 1)

= ∂n1un−1

2

+ DtN1(un−1

2

)

• n Γ,

(∆ + k2)un

2

= f in Ω2, ∂n2un

2 + DtN2(un 2)

= ∂n2un−1

1

+ DtN2(un−1

1

)

• n 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 a b Γ12 Γ21 Ω1 Ω2

New Schwarz algorithm uses different transmission conditions: (∆ + k2)un

1

= f in Ω1, ∂n1un

1 + DtN1(un 1)

= ∂n1un−1

2

+ DtN1(un−1

2

)

• n Γ12,

(∆ + k2)un

2

= f in Ω2, ∂n2un

2 + DtN2(un 2)

= ∂n2un−1

1

+ DtN2(un−1

1

)

• n Γ21.

This algorithm converges in two iterations, independently of the 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 a b Γ12 Γ23 Γ34 Γ45 Ω1 Ω2 Ω3 Ω4 Ω5

New Schwarz algorithm uses different transmission conditions: (∆ + k2)un

j

= f in Ωj, ∂nj un

j + DtNj(un j )

= ∂njun−1

j+1 + DtNj(un−1 j+1 )

• n Γj,j+1,

∂nj un

j + DtNj(un j )

= ∂njun−1

j−1 + DtNj(un−1 j−1 )

• n 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 a b Γ12 Γ23 Γ34 Γ45 Ω1 Ω2 Ω3 Ω4 Ω5

New Schwarz algorithm uses different transmission conditions: (∆ + k2)un

j

= f in Ωj, ∂nj un

j + DtNj(un j )

= ∂njun−1

j+1 + DtNj(un−1 j+1 )

• n Γj,j+1,

∂nj un

j + DtNj(un j )

= ∂njun−1

j−1 + DtNj(un−1 j−1 )

• n Γj,j−1,

With J subdomains, it converges in J iterations,

• r 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 a b Γ12 Γ23 Γ34 Γ45 Ω1 Ω2 Ω3 Ω4 Ω5

New Schwarz algorithm uses different transmission conditions: (∆ + k2)un

j

= f in Ωj, ∂nj un

j + DtNj(un j )

= ∂njun−1

j+1 + DtNj(un−1 j+1 )

• n Γj,j+1,

∂nj un

j + DtNj(un j )

= ∂njun−1

j−1 + DtNj(un−1 j−1 )

• n Γj,j−1,

With J subdomains, it converges in J iterations,

• r 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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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

1 1 0.05

y

0.5

x

0.5

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces x y 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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces x y 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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces x y 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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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-N´

u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces

• 10
• 5

10 -3 0.5 5 0.5

x y 1 1

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 Helmholtz

Helmholz with J subdomains, k constant per subdomain

J = 4 J = 8 α Iterative ρ GMRES Iterative ρ GMRES 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 4 0.005 3 3 7 0.13 4 3 5 6 5 10 0.40 7 5 7 0.01 4 4 8 0.25 5 4 5 14 6 24 0.68 9 6 8 0.05

• 1.52

7 5 8

• 11.48

15 11 16 0.1 11 10 26 0.69 8 6 10

• 2.74

17 13 18 1

• 3.86

20 14 20

• 188

39 32 45 k = [20 20 20 20] + α[0 20 10 − 10] 1 1 1 5.4e-15 1 1 1 1 1 1 5.3e-15 1 1 1 0.001 2 3 3 2.5e-2 4 3 3 2 4 4 1.1e-1 5 4 4 0.005 3 6 6 0.14 5 5 5

• 8
• 0.88

9 8 8 0.01 5 10 9 0.33 6 6 6

• 16
• 1.92

12 8 11 0.05

• 4.48

13 10 13

• 7.28

22 18 23 0.1

• 25
• 1.8

14 11 14

• 20.2

20 17 23 1

• 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 optimized Schwarz methods (DOSMs):

◮ Sweeping Preconditioner: DOSM with non-overlapping

subdomains with empty interior, PML or H-matrix transmission condition (TC) on the left and Dirichlet on the right

◮ Source transfer: DOSM with maximally overlapping

subdomains with PML TC in the forward sweep and source term set to zero in the overlap, and Dirichlet instead of PML on the right in the backward sweep.

◮ Single Layer Potential Method: DOSM with two

non-overlapping domain decompositions and PML TC

◮ Method of Polarized Traces: DOSM with

non-overlapping subdomains and PML TC Rigorous proofs in (G, Zhang 2017), available at www.unige.ch/∼gander