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Iterative Solvers for Helmholtz One Class of Martin J. Gander Iterative Solvers for Helmholtz Problems: Quotes AILU Factorizations, Sweeping Basic Algorithms Model Problem Block LU Preconditioners, Source Transfer, Single New Schwarz

  1. Iterative Solvers A New Schwarz Method (Nataf, Rogier 1994) for Helmholtz Martin J. Gander y b Quotes Basic Algorithms Model Problem Ω 1 Γ Ω 2 Block LU New Schwarz Optimized Schwarz x Helmholtz 0 a OSM Based Limitations New Schwarz algorithm uses different transmission conditions: Conclusion (∆ + k 2 ) u n = f in Ω 1 , 1 ∂ n 1 u n 1 + DtN 1 ( u n ∂ n 1 u n − 1 + DtN 1 ( u n − 1 1 ) = ) on Γ , 2 2 (∆ + k 2 ) u n = in Ω 2 , f 2 ∂ n 2 u n − 1 + DtN 2 ( u n − 1 ∂ n 2 u n 2 + DtN 2 ( u n 2 ) = ) on Γ , 1 1 This algorithm converges in two iterations,

  2. Iterative Solvers A New Schwarz Method (Nataf, Rogier 1994) for Helmholtz y Martin J. Gander b Quotes Basic Algorithms Model Problem Ω 1 Γ 21 Γ 12 Ω 2 Block LU New Schwarz Optimized Schwarz x Helmholtz 0 a OSM Based Limitations New Schwarz algorithm uses different transmission conditions: Conclusion (∆ + k 2 ) u n = f in Ω 1 , 1 ∂ n 1 u n − 1 + DtN 1 ( u n − 1 ∂ n 1 u n 1 + DtN 1 ( u n 1 ) = ) on Γ 12 , 2 2 (∆ + k 2 ) u n = f in Ω 2 , 2 ∂ n 2 u n − 1 + DtN 2 ( u n − 1 ∂ n 2 u n 2 + DtN 2 ( u n 2 ) = ) on Γ 21 . 1 1 This algorithm converges in two iterations, independently of the overlap ( G, Halpern, Nataf 1999 )!

  3. Iterative Solvers A New Schwarz Method (Nataf, Rogier 1994) for Helmholtz Martin J. Gander y b Quotes Basic Algorithms Γ 12 Γ 23 Γ 34 Γ 45 Model Problem Block LU Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 New Schwarz Optimized Schwarz x Helmholtz 0 a OSM Based Limitations New Schwarz algorithm uses different transmission conditions: Conclusion (∆ + k 2 ) u n = in Ω j , f j ∂ n j u n − 1 j +1 + DtN j ( u n − 1 ∂ n j u n j + DtN j ( u n j ) = j +1 ) on Γ j , j +1 , ∂ n j u n j + DtN j ( u n ∂ n j u n − 1 j − 1 + DtN j ( u n − 1 j ) = j − 1 ) on Γ j , j − 1 , With J subdomains, it converges in J iterations,

  4. Iterative Solvers A New Schwarz Method (Nataf, Rogier 1994) for Helmholtz Martin J. Gander y b Quotes Basic Algorithms Γ 12 Γ 23 Γ 34 Γ 45 Model Problem Block LU Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 New Schwarz Optimized Schwarz x Helmholtz 0 a OSM Based Limitations New Schwarz algorithm uses different transmission conditions: Conclusion (∆ + k 2 ) u n = in Ω j , f j ∂ n j u n − 1 j +1 + DtN j ( u n − 1 ∂ n j u n j + DtN j ( u n j ) = j +1 ) on Γ j , j +1 , ∂ n j u n j + DtN j ( u n ∂ n j u n − 1 j − 1 + DtN j ( u n − 1 j ) = j − 1 ) on Γ j , j − 1 , With J subdomains, it converges in J iterations, or in one forward and backward sweep.

  5. Iterative Solvers A New Schwarz Method (Nataf, Rogier 1994) for Helmholtz Martin J. Gander y b Quotes Basic Algorithms Γ 12 Γ 23 Γ 34 Γ 45 Model Problem Block LU Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 New Schwarz Optimized Schwarz x Helmholtz 0 a OSM Based Limitations New Schwarz algorithm uses different transmission conditions: Conclusion (∆ + k 2 ) u n = in Ω j , f j ∂ n j u n − 1 j +1 + DtN j ( u n − 1 ∂ n j u n j + DtN j ( u n j ) = j +1 ) on Γ j , j +1 , ∂ n j u n j + DtN j ( u n ∂ n j u n − 1 j − 1 + DtN j ( u n − 1 j ) = j − 1 ) on Γ j , j − 1 , With J subdomains, it converges in J iterations, or in one forward and backward sweep. Continuous analog of the block LU decomposition !

  6. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  7. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  8. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  9. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  10. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  11. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  12. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  13. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  14. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  15. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  16. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  17. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  18. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  19. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  20. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  21. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  22. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  23. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  24. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  25. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  26. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  27. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  28. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  29. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  30. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  31. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  32. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  33. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  34. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  35. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  36. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  37. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  38. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  39. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  40. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  41. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  42. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  43. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  44. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  45. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  46. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  47. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  48. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  49. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  50. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  51. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  52. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  53. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  54. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  55. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  56. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  57. Iterative Solvers Methods Based on Optimal Schwarz for Helmholtz Martin J. Gander ◮ G, Halpern, Nataf 1999: Optimal Convergence for Overlapping and Non-Overlapping Schwarz Waveform Quotes Relaxation Basic Algorithms Model Problem ◮ G, Nataf 2000: AILU: a preconditioner based on the Block LU New Schwarz analytic factorization of the elliptic operator Optimized Schwarz ◮ G, Magoules, Nataf 2002: Optimized Schwarz Helmholtz OSM Based methods without overlap for the Helmholtz equation Limitations Conclusion ◮ G 2006: Optimized Schwarz methods 0.05 0 1 1 0.5 0.5 0 0 y x

  58. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Quotes Helmholtz Equation Basic Algorithms Model Problem ◮ Chen, Xiang 2012: A Source Transfer DD Method for Block LU New Schwarz Helmholtz Equations in Unbounded Domain Optimized Schwarz Helmholtz ◮ Stolk 2013: A rapidly converging domain OSM Based decomposition method for the Helmholtz equation Limitations Conclusion ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces y PML PML x 1 0 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5

  59. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Quotes Helmholtz Equation Basic Algorithms Model Problem ◮ Chen, Xiang 2012: A Source Transfer DD Method for Block LU New Schwarz Helmholtz Equations in Unbounded Domain Optimized Schwarz Helmholtz ◮ Stolk 2013: A rapidly converging domain OSM Based decomposition method for the Helmholtz equation Limitations Conclusion ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces y ∂ x + DtN right ∂ x + DtN left x 1 0 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5

  60. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Quotes Helmholtz Equation Basic Algorithms Model Problem ◮ Chen, Xiang 2012: A Source Transfer DD Method for Block LU New Schwarz Helmholtz Equations in Unbounded Domain Optimized Schwarz Helmholtz ◮ Stolk 2013: A rapidly converging domain OSM Based decomposition method for the Helmholtz equation Limitations Conclusion ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Preconditioning the 2D Helmholtz equation with polarized traces y ∂ x + DtN right ∂ x + DtN left x 1 0 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5

  61. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  62. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  63. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  64. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  65. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  66. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  67. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  68. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  69. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  70. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  71. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  72. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  73. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  74. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  75. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  76. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  77. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  78. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  79. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  80. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  81. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  82. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  83. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  84. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  85. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  86. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

  87. Iterative Solvers Methods Based on Optimal Schwarz (cont.) for Helmholtz Martin J. Gander ◮ Enquist, Ying 2010: Sweeping Preconditioner for the Helmholtz Equation Quotes ◮ Chen, Xiang 2012: A Source Transfer DD Method for Basic Algorithms Model Problem Helmholtz Equations in Unbounded Domain Block LU New Schwarz ◮ Stolk 2013: A rapidly converging domain Optimized Schwarz Helmholtz decomposition method for the Helmholtz equation OSM Based Limitations ◮ Zepeda-N´ u˜ nez, Hewett, Demanet 2014: Conclusion Preconditioning the 2D Helmholtz equation with polarized traces 10 -3 5 0 -5 -10 1 1 0.5 0.5 0 0 y x

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