AILU: a preconditioner based on the analytic factorization of the elliptic operator. We investigate a new type of preconditioner for large systems of linear equations stemming from the discretization of elliptic symmetric partial differential equations. Instead of working at the matrix level, we construct an analytic factorization of the elliptic operator into two parabolic factors and we identify the two parabolic factors with the LU factors of an exact block LU decomposition at the matrix level. Since these factorizations are nonlocal, we introduce a second order local approximation of the parabolic factors. We analyze the approximate factorization at the continuous level and optimize its performance which leads to the new AILU (Analytic ILU) preconditioner with convergence rate effectiveness of the new approach.

References in zbMATH (referenced in 26 articles , 2 standard articles )

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  1. Bonev, Boris; Hesthaven, Jan S.: A hierarchical preconditioner for wave problems in quasilinear complexity (2022)
  2. Kim, Seungil; Zhang, Hui: Convergence analysis of the continuous and discrete non-overlapping double sweep domain decomposition method based on PMLs for the Helmholtz equation (2021)
  3. Appelö, Daniel; Garcia, Fortino; Runborg, Olof: WaveHoltz: iterative solution of the Helmholtz equation via the wave equation (2020)
  4. Lang, Chao; Gao, Rong; Qiu, Chujun: On block triangular preconditioned iteration methods for solving the Helmholtz equation (2020)
  5. Claeys, X.; Dolean, V.; Gander, M. J.: An introduction to multi-trace formulations and associated domain decomposition solvers (2019)
  6. Gander, Martin J.; Zhang, Hui: A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods (2019)
  7. Zepeda-Núñez, Leonardo; Demanet, Laurent: Nested domain decomposition with polarized traces for the 2D Helmholtz equation (2018)
  8. Chaouqui, Faycal; Gander, Martin J.; Santugini-Repiquet, Kévin: On nilpotent subdomain iterations (2017)
  9. Gander, Martin J.; Liu, Yongxiang: On the definition of Dirichlet and Neumann conditions for the biharmonic equation and its impact on associated Schwarz methods (2017)
  10. Gander, Martin J.; Solovyev, Sergey: A numerical study on the compressibility of subblocks of Schur complement matrices obtained from discretized Helmholtz equations (2017)
  11. Gander, Martin J.; Xu, Yingxiang: Optimized Schwarz methods with nonoverlapping circular domain decomposition (2017)
  12. Xi, Yuanzhe; Saad, Yousef: A rational function preconditioner for indefinite sparse linear systems (2017)
  13. Gander, Martin J.; Halpern, Laurence; Martin, Véronique: A new algorithm based on factorization for heterogeneous domain decomposition (2016)
  14. Liu, Fei; Ying, Lexing: Additive sweeping preconditioner for the Helmholtz equation (2016)
  15. Liu, Fei; Ying, Lexing: Recursive sweeping preconditioner for the three-dimensional Helmholtz equation (2016)
  16. Zepeda-Núñez, Leonardo; Demanet, Laurent: The method of polarized traces for the 2D Helmholtz equation (2016)
  17. Conen, Lea; Dolean, Victorita; Krause, Rolf; Nataf, Frédéric: A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator (2014)
  18. Du, Kui: A composite preconditioner for the electromagnetic scattering from a large cavity (2011)
  19. Niu, Qiang; Lu, Lin-Zhang: Fourier analysis of frequency filtering decomposition preconditioners (2010)
  20. Erlangga, Yogi A.: Advances in iterative methods and preconditioners for the Helmholtz equation (2008)

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