H.Tortel, A. Litman and I. Voznyuk Institut Fresnel, UMR-CNRS 7249, Marseille, France
Vocabulary related to electromagnetic fields Sources Free space
Vocabulary related to electromagnetic fields Incident )ield
Vocabulary related to electromagnetic fields Incident )ield
Vocabulary related to electromagnetic fields Total )ield Incident )ield
Vocabulary related to electromagnetic fields Total )ield – Incident )ield
Vocabulary related to electromagnetic fields Total )ield Scattered )ield – = = Incident )ield
Direct and Inverse problems. Definition Direct Know q Sources q Objects To find q Sca$ered field
Direct and Inverse problems. Definition Direct Know q Sources q Objects To find q Sca$ered field Inverse Know q Sources q Sca7ered field To find q Objects
2D & 3D Scattered field Helmholtz equations
2D & 3D Scattered field Helmholtz equations
Numerical techniques: DDM based on FEM Pros ü Well known ü Different media possible Anisotropic Inhomogeneous ü Arbitrary shaped objects
Numerical techniques: DDM based on FEM Cons Pros o Time ü Well known o Memory ü Different media possible o ParallelizaGon issues Anisotropic Inhomogeneous ü Arbitrary shaped objects
Numerical techniques: DDM based on FEM Cons Pros o Time ü Well known o Memory ü Different media possible o ParallelizaGon issues Anisotropic Inhomogeneous ü Arbitrary shaped objects Domain Decomposition technique Domain Decomposition Method [1] FETI method [2] References [1] Després 1991 FETI-DPEM2 [3] [2] Farhat et al 2001 [3] Lee and Jin 2007
FEM statement in 2D case
FEM statement in 2D case
Domain Decomposition
Domain Decomposition
Domain Decomposition
Notation of unknowns « c » - corner points in global numbering - interface points « r » - internal points
Notation of unknowns « c » - corner points in global numbering - interface points « r » - internal points
FETI main idea « c » - corner points in global numbering - interface points « r » - internal points
FETI main idea
FETI main idea
Connection between subdomains
Connection between subdomains
Interface problem of the proposed approach (a) (b) [2] [5] [3,4] “r” – Robin type “r” – Robin type “r” – Robin type “c” – Neumann type “c” – Neumann type “c” – Robin type References [2] Lee and Jin 2007 [3] Voznyuk et al 2013 [4] Voznyuk et al 2014 [5] Xue and Jin 2014
Resolution of the Interface problem
Resolution of the Interface problem
Last step: solution of independent problems
Testing methodology How to test the FETI-methods
Testing methodology How to test the FETI-methods q AnalyGcal funcGons
Testing methodology How to test the FETI-methods q AnalyGcal funcGons q Previously computed numerical soluGons
Testing methodology How to test the FETI-methods q AnalyGcal funcGons q Previously computed numerical soluGons q Comparison with FEM method
Testing methodology How to test the FETI-methods q AnalyGcal funcGons q Previously computed numerical soluGons q Comparison with FEM method q Measurements
Testing methodology How to test the FETI-methods q AnalyGcal funcGons q Previously computed numerical soluGons q Comparison with FEM method q Measurements Useful packages GMSH Mesh construcGon, fields and geometry representaGon METIS Domain DecomposiGon MUMPS FactorizaGon, ResoluGon of SLE MKL
3D Direct problem: physical statement 18 𝑑𝑛 1 2 , 8 𝑑 𝑛
3D Direct problem: Domain Decomposition
3D Direct problem: field distribution Without PML With PML The interface problem is solved with MUMPS based on LU-decomposition
FETI: implementation issues The Interface Problem (IP) Matrix BoIlenecks InverGng and storing matrices Storing the Interface Problem (IP) matrix CompuGng and storing LU of the IP matrix LU
Interface problem: iterative solution The Interface Problem (IP) Matrix GMRES iterative method
Interface problem: iterative solution The Interface Problem (IP) Matrix GMRES iterative method
Interface problem: iterative solution The Interface Problem (IP) Matrix The Reduced IP Matrix [1,2] GMRES References [1] Li and Jin 2007 iterative method [2] Xue and Jin 2012
Interface problem: iterative solution The Interface Problem (IP) Matrix The Reduced IP Matrix [1,2] GMRES References [1] Li and Jin 2007 iterative method [2] Xue and Jin 2012
Convergence: influence of PML Exact values of the Lagrange multipliers ( LU-decomposition ) Lagrange multipliers obtained after 10 iterations Presence of PML area
Convergence: influence of PML Exact values of the Lagrange multipliers Conclusion so far ( LU-decomposition ) q Bad influence of PML Lagrange multipliers obtained after 10 iterations q The error does not spread Presence of PML area
Acceleration of the convergence What we can play with q Robin-type boundary conditions Després approach [1] EMDA* approach [2] * Evanescent Modes Damping Algorithm References [1] Després 1991 [2] Boubendir et al 2000
Convergence: influence of the 𝛽 -parameter -parameter What we can play with q Change the iterative method? q Preconditioner ? q Geometrical/numerical issues? q Robin-type boundary conditions!? Després approach [1] EMDA* approach [2] * Evanescent Modes Damping Algorithm References [1] Després 1991 [2] Boubendir et al 2000
Convergence: influence of the 𝛽 -parameter -parameter What we can play with q Change the iterative method? q Preconditioner ? q Geometrical/numerical issues? q Robin-type boundary conditions!? Després approach [1] EMDA* approach [2] * Evanescent Modes Damping Algorithm References [1] Després 1991 [2] Boubendir et al 2000
Convergence: influence of the 𝛽 -parameter -parameter What we can play with q Change the iterative method? q Preconditioner ? q Geometrical/numerical issues? q Robin-type boundary conditions!? Després approach [1] EMDA* approach [2] MIXED approach [3] * Evanescent Modes Damping Algorithm References [1] Després 1991 [2] Boubendir et al 2000 [3] Voznyuk et al (submi7ed) 2014
3D Inverse problem: introduction Just an object ε r
3D Inversion: schematic configuration ?
3D Inversion: schematic configuration Source ? Receivers
3D Inversion: schematic configuration Source ? Receivers
3D Inversion: schematic configuration Source ? Receivers
3D Inversion: schematic configuration Source ? Receivers
3D Inversion: measurement setup E mes : N src × N rec Anechoic chamber ! 3D Fresnel database [1] q Set of homogeneous targets q 162 transmimng dipoles q 32 receiver posiGons q Distance: References [1] J.-M. Geffrin and P. Sabouroux 2009
3D Inversion: mathematical side E mes : N src × N rec ! Problem statement: Find such as the cost functional ‖ E mes − E calc ‖ J ( ε r , E ) = 2 is minimized W q W – covariance matrix, related to + noise q – scattered far-)ield, computed thanks to Near-to-Far-Field transformation
3D Inversion: mathematical side E mes : N src × N rec ! Problem statement: Find such as the cost functional ‖ E mes − E calc ‖ J ( ε r , E ) = 2 is minimized W q W – covariance matrix, related to + noise q – scattered far-)ield, computed thanks to Near-to-Far-Field transformation CharacterisOcs of problem q nonlinear q ill posed q underdetermined There is not a unique soluGon
3D Inversion: Lagrangian formalism E mes : N src × N rec ! Problem statement: Find such as the cost functional ‖ E mes − E calc ‖ J ( ε r , E ) = 2 is minimized W q An iterative quasi-Newton method based on L-BFGS [1] approach with Constraints constraints References [1] R. Byrd et al 1995
3D Inversion: Lagrangian formalism E mes : N src × N rec ! Problem statement: Find such as the cost functional ‖ E mes − E calc ‖ J ( ε r , E ) = 2 is minimized W q An iterative quasi-Newton method based on L-BFGS [1] approach with Constraints constraints Total Adjoint 8ield 8ield References [1] R. Byrd et al 1995
3D Inversion: Iterative FEM algorithm Initial guess FEM (Direct) Mis)it criterion Stop? End of process Max iteraGon FEM (Adjoint) No evoluGon Gradient evaluation Small misfit Small gradient Update direction New permittivity
3D Inversion: Iterative FETI algorithm Initial guess FETI Initialization Permanent FETI Initialization Initial guess non-Permanent FEM (Direct) FETI iterations (Direct) Mis)it criterion Mis)it criterion Stop? End of process Stop? Max iteraGon FEM (Adjoint) FETI iterations (Adjoint) No evoluGon Gradient evaluation Small misfit Gradient evaluation Small gradient Update direction Update direction New permittivity New permittivity
FETI optimization FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… N scr-1 N scr 1 s N scr Initial λ Initial λ Initial λ r r r One GMRES One GMRES One GMRES iteration iteration iteration η is η is η is reached? reached? reached?
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