Non Uniform Discrete Fourier Transform for AS recall adaptive - PowerPoint PPT Presentation

NUDFT AFDTD Non Uniform Discrete Fourier Transform for AS recall adaptive acceleration of the NUDFT formulation Aitken-Schwarz DDM NUDFT for Aitken- Schwarz method A.Frullone, D.Tromeur-Dervout Numerical results Summary and CDCSP/ICJ UMR 5208 U.LYON 1-CNRS Future Work July, 5 2006 17th International Conference on Domain Decomposition Methods St.Wolfgang/Strobl - Austria

Introduction NUDFT AFDTD AS recall NUDFT formulation NUDFT for Aitken- Schwarz method Numerical results Summary and Future Work Aitken-Schwarz DDM for uniform grids 3D Poisson Pb 762Mdof/60s 5Mbit/s 1256 proc 3 cray T3E FFT of Schwarz DDM artificial interfaces ⇒ needs regular discretization of the interfaces Aitken acceleration of Fourier modes Barberou, Garbey, Hess, Resch, Rossi, Toivanen and Tromeur-Dervout, J. of Parallel and Distributed Computing, special issue on Grid computing, 63(5) :564-577, 2003 Aim : extension of this method to non uniform meshes

Introduction NUDFT AFDTD AS recall NUDFT formulation NUDFT for Aitken- Schwarz method Numerical results Summary and Future Work Aitken-Schwarz DDM for uniform grids 3D Poisson Pb 762Mdof/60s 5Mbit/s 1256 proc 3 cray T3E FFT of Schwarz DDM artificial interfaces ⇒ needs regular discretization of the interfaces Aitken acceleration of Fourier modes Barberou, Garbey, Hess, Resch, Rossi, Toivanen and Tromeur-Dervout, J. of Parallel and Distributed Computing, special issue on Grid computing, 63(5) :564-577, 2003 Aim : extension of this method to non uniform meshes

Outline NUDFT AFDTD AS recall Aitken-Schwarz recall NUDFT 1 formulation NUDFT for Aitken- New NUDFT formulation Schwarz 2 method Numerical results NUDFT for Aitken-Schwarz method 3 Summary and Future Work Numerical results 4 5 Summary and Future Work

Outline NUDFT AFDTD AS recall Aitken-Schwarz recall NUDFT 1 formulation NUDFT for Aitken- New NUDFT formulation Schwarz 2 method Numerical results NUDFT for Aitken-Schwarz method 3 Summary and Future Work Numerical results 4 5 Summary and Future Work

Acceleration of Schwarz Method for Elliptic Problems NUDFT AFDTD M.Garbey and D.Tromeur-Dervout : On some Aitken like acceleration of the Schwarz method , Int. J. for Numerical Methods in Fluids, 40(12) :1493-1513,2002 AS recall 1D additive Schwarz algorithm for linear differential operators : NUDFT formulation L [ u n + 1 ] = f in Ω 1 , u n + 1 1 | Γ 1 = u n 2 | Γ 1 , 1 NUDFT for L [ u n + 1 ] = f in Ω 2 , u n + 1 2 | Γ 2 = u n 1 | Γ 2 . Aitken- 2 Schwarz method the interface error operator T is linear , i.e Numerical u n + 1 1 | Γ 2 − U | Γ 2 = δ 1 ( u n 2 | Γ 1 − U | Γ 1 ) , results u n + 1 2 | Γ 1 − U | Γ 1 = δ 2 ( u n 1 | Γ 2 − U | Γ 2 ) . Summary and Future Work Consequently u 2 1 | Γ 2 − u 1 1 | Γ 2 = δ 1 ( u 1 2 | Γ 1 − u 0 2 | Γ 1 ) , u 2 2 | Γ 1 − u 1 2 | Γ 1 = δ 2 ( u 1 1 | Γ 2 − u 0 1 | Γ 2 ) , Computation of δ 1 / 2 : L [ v 1 / 2 ] = 0 in Ω 1 / 2 , v Γ 1 / 2 = 1 . thus δ 1 / 2 = v Γ 2 / 1 . iff δ 1 δ 2 � = 1 Aitken-Schwarz gives the solution with exactly 3 iterations and possibly 2 in the analytical case.

Acceleration of Schwarz Method for Elliptic Problems NUDFT AFDTD M.Garbey and D.Tromeur-Dervout : On some Aitken like acceleration of the Schwarz method , Int. J. for Numerical Methods in Fluids, 40(12) :1493-1513,2002 AS recall 1D additive Schwarz algorithm for linear differential operators : NUDFT formulation L [ u n + 1 ] = f in Ω 1 , u n + 1 1 | Γ 1 = u n 2 | Γ 1 , 1 NUDFT for L [ u n + 1 ] = f in Ω 2 , u n + 1 2 | Γ 2 = u n 1 | Γ 2 . Aitken- 2 Schwarz method the interface error operator T is linear , i.e Numerical u n + 1 1 | Γ 2 − U | Γ 2 = δ 1 ( u n 2 | Γ 1 − U | Γ 1 ) , results u n + 1 2 | Γ 1 − U | Γ 1 = δ 2 ( u n 1 | Γ 2 − U | Γ 2 ) . Summary and Future Work Consequently u 2 1 | Γ 2 − u 1 1 | Γ 2 = δ 1 ( u 1 2 | Γ 1 − u 0 2 | Γ 1 ) , u 2 2 | Γ 1 − u 1 2 | Γ 1 = δ 2 ( u 1 1 | Γ 2 − u 0 1 | Γ 2 ) , Computation of δ 1 / 2 : L [ v 1 / 2 ] = 0 in Ω 1 / 2 , v Γ 1 / 2 = 1 . thus δ 1 / 2 = v Γ 2 / 1 . iff δ 1 δ 2 � = 1 Aitken-Schwarz gives the solution with exactly 3 iterations and possibly 2 in the analytical case.

Acceleration of Schwarz Method for Elliptic Problems NUDFT AFDTD M.Garbey and D.Tromeur-Dervout : On some Aitken like acceleration of the Schwarz method , Int. J. for Numerical Methods in Fluids, 40(12) :1493-1513,2002 AS recall 1D additive Schwarz algorithm for linear differential operators : NUDFT formulation L [ u n + 1 ] = f in Ω 1 , u n + 1 1 | Γ 1 = u n 2 | Γ 1 , 1 NUDFT for L [ u n + 1 ] = f in Ω 2 , u n + 1 2 | Γ 2 = u n 1 | Γ 2 . Aitken- 2 Schwarz method the interface error operator T is linear , i.e Numerical u n + 1 1 | Γ 2 − U | Γ 2 = δ 1 ( u n 2 | Γ 1 − U | Γ 1 ) , results u n + 1 2 | Γ 1 − U | Γ 1 = δ 2 ( u n 1 | Γ 2 − U | Γ 2 ) . Summary and Future Work Consequently u 2 1 | Γ 2 − u 1 1 | Γ 2 = δ 1 ( u 1 2 | Γ 1 − u 0 2 | Γ 1 ) , u 2 2 | Γ 1 − u 1 2 | Γ 1 = δ 2 ( u 1 1 | Γ 2 − u 0 1 | Γ 2 ) , Computation of δ 1 / 2 : L [ v 1 / 2 ] = 0 in Ω 1 / 2 , v Γ 1 / 2 = 1 . thus δ 1 / 2 = v Γ 2 / 1 . iff δ 1 δ 2 � = 1 Aitken-Schwarz gives the solution with exactly 3 iterations and possibly 2 in the analytical case.

Acceleration of Schwarz Method for Elliptic Problems NUDFT AFDTD M.Garbey and D.Tromeur-Dervout : On some Aitken like acceleration of the Schwarz method , Int. J. for Numerical Methods in Fluids, 40(12) :1493-1513,2002 AS recall 1D additive Schwarz algorithm for linear differential operators : NUDFT formulation L [ u n + 1 ] = f in Ω 1 , u n + 1 1 | Γ 1 = u n 2 | Γ 1 , 1 NUDFT for L [ u n + 1 ] = f in Ω 2 , u n + 1 2 | Γ 2 = u n 1 | Γ 2 . Aitken- 2 Schwarz method the interface error operator T is linear , i.e Numerical u n + 1 1 | Γ 2 − U | Γ 2 = δ 1 ( u n 2 | Γ 1 − U | Γ 1 ) , results u n + 1 2 | Γ 1 − U | Γ 1 = δ 2 ( u n 1 | Γ 2 − U | Γ 2 ) . Summary and Future Work Consequently u 2 1 | Γ 2 − u 1 1 | Γ 2 = δ 1 ( u 1 2 | Γ 1 − u 0 2 | Γ 1 ) , u 2 2 | Γ 1 − u 1 2 | Γ 1 = δ 2 ( u 1 1 | Γ 2 − u 0 1 | Γ 2 ) , Computation of δ 1 / 2 : L [ v 1 / 2 ] = 0 in Ω 1 / 2 , v Γ 1 / 2 = 1 . thus δ 1 / 2 = v Γ 2 / 1 . iff δ 1 δ 2 � = 1 Aitken-Schwarz gives the solution with exactly 3 iterations and possibly 2 in the analytical case.

Acceleration of Schwarz Method for Elliptic Problems NUDFT AFDTD M.Garbey and D.Tromeur-Dervout : On some Aitken like acceleration of the Schwarz method , Int. J. for Numerical Methods in Fluids, 40(12) :1493-1513,2002 AS recall 1D additive Schwarz algorithm for linear differential operators : NUDFT formulation L [ u n + 1 ] = f in Ω 1 , u n + 1 1 | Γ 1 = u n 2 | Γ 1 , 1 NUDFT for L [ u n + 1 ] = f in Ω 2 , u n + 1 2 | Γ 2 = u n 1 | Γ 2 . Aitken- 2 Schwarz method the interface error operator T is linear , i.e Numerical u n + 1 1 | Γ 2 − U | Γ 2 = δ 1 ( u n 2 | Γ 1 − U | Γ 1 ) , results u n + 1 2 | Γ 1 − U | Γ 1 = δ 2 ( u n 1 | Γ 2 − U | Γ 2 ) . Summary and Future Work Consequently u 2 1 | Γ 2 − u 1 1 | Γ 2 = δ 1 ( u 1 2 | Γ 1 − u 0 2 | Γ 1 ) , u 2 2 | Γ 1 − u 1 2 | Γ 1 = δ 2 ( u 1 1 | Γ 2 − u 0 1 | Γ 2 ) , Computation of δ 1 / 2 : L [ v 1 / 2 ] = 0 in Ω 1 / 2 , v Γ 1 / 2 = 1 . thus δ 1 / 2 = v Γ 2 / 1 . iff δ 1 δ 2 � = 1 Aitken-Schwarz gives the solution with exactly 3 iterations and possibly 2 in the analytical case.

The algorithm in 2D or 3D writes : step1 : reconstruct P from datas given by two Schwarz NUDFT iterates AFDTD step2 : apply one additive Schwarz iterate to the Poisson AS recall problem with block solver of choice i.e multigrids, FFT etc... NUDFT formulation step3 : NUDFT for Aitken- compute the Fourier expansion ˆ u n Schwarz j | Γ i , n = 0 , 1 of the method traces on the artificial interface Γ i , i = 1 .. nd for the Numerical initial boundary condition u 0 | Γ i and the Schwarz iterate results Summary and solution u 1 | Γ i . Future Work apply generalized Aitken acceleration based on u ∞ = ( Id − P ) − 1 (ˆ u 1 − P ˆ u 0 ) ˆ u ∞ in order to get ˆ | Γ i . recompose the trace u ∞ | Γ i in physical space. step4 : compute in parallel the solution in each subdomains Ω j , with new inner BCs and blocksolver of choice.

The algorithm in 2D or 3D writes : step1 : reconstruct P from datas given by two Schwarz NUDFT iterates AFDTD step2 : apply one additive Schwarz iterate to the Poisson AS recall problem with block solver of choice i.e multigrids, FFT etc... NUDFT formulation step3 : NUDFT for Aitken- compute the Fourier expansion ˆ u n Schwarz j | Γ i , n = 0 , 1 of the method traces on the artificial interface Γ i , i = 1 .. nd for the Numerical initial boundary condition u 0 | Γ i and the Schwarz iterate results Summary and solution u 1 | Γ i . Future Work apply generalized Aitken acceleration based on u ∞ = ( Id − P ) − 1 (ˆ u 1 − P ˆ u 0 ) ˆ u ∞ in order to get ˆ | Γ i . recompose the trace u ∞ | Γ i in physical space. step4 : compute in parallel the solution in each subdomains Ω j , with new inner BCs and blocksolver of choice.