H.Tortel, A. Litman and I. Voznyuk Institut Fresnel, UMR-CNRS 7249, - - PowerPoint PPT Presentation

H.Tortel, A. Litman and I. Voznyuk Institut Fresnel, UMR-CNRS 7249, Marseille, France

Sources

Vocabulary related to electromagnetic fields

Free space

Incident )ield

Vocabulary related to electromagnetic fields

Incident )ield

Vocabulary related to electromagnetic fields

Incident )ield Total )ield

Vocabulary related to electromagnetic fields

–

Total )ield Incident )ield

Vocabulary related to electromagnetic fields

= Total )ield Incident )ield

– =

Scattered )ield

Vocabulary related to electromagnetic fields

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

Direct and Inverse problems. Definition

2D & 3D Scattered field Helmholtz equations

2D & 3D Scattered field Helmholtz equations

Pros ü Well known ü Different media possible Anisotropic Inhomogeneous ü Arbitrary shaped objects

Numerical techniques: DDM based on FEM

Pros ü Well known ü Different media possible Anisotropic Inhomogeneous ü Arbitrary shaped objects Cons

• Time
• Memory
• ParallelizaGon issues

Numerical techniques: DDM based on FEM

Domain Decomposition technique

Domain Decomposition Method [1] FETI method [2]

FETI-DPEM2 [3]

References [1] Després 1991 [2] Farhat et al 2001 [3] Lee and Jin 2007

Pros ü Well known ü Different media possible Anisotropic Inhomogeneous ü Arbitrary shaped objects Cons

• Time
• Memory
• ParallelizaGon issues

Numerical techniques: DDM based on FEM

FEM statement in 2D case

FEM statement in 2D case

Domain Decomposition

Domain Decomposition

Domain Decomposition

« c » « r »

• corner points in

global numbering

• interface points
• internal points

Notation of unknowns

Notation of unknowns

« c » « r »

• corner points in

global numbering

• interface points
• internal points

« c » « r »

• corner points in

global numbering

• interface points
• internal points

FETI main idea

FETI main idea

FETI main idea

Connection between subdomains

Connection between subdomains

Interface problem of the proposed approach

“r” – Robin type “c” – Robin type

[3,4]

(a)

“r” – Robin type “c” – Neumann type

(b)

“r” – Robin type “c” – Neumann type

[2] [5] References [2] Lee and Jin 2007 [3] [4] Voznyuk et al Voznyuk et al 2013 2014 [5] Xue and Jin 2014

Resolution of the Interface problem

Resolution of the Interface problem

Last step: solution of independent problems

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 MKL FactorizaGon, ResoluGon of SLE

Testing methodology

18 𝑑𝑛 1 2 , 8 𝑑 𝑛

3D Direct problem: physical statement

3D Direct problem: Domain Decomposition

The interface problem is solved with MUMPS based on LU-decomposition Without PML With PML 3D Direct problem: field distribution

LU

BoIlenecks InverGng and storing matrices Storing the Interface Problem (IP) matrix CompuGng and storing LU of the IP matrix

The Interface Problem (IP) Matrix

FETI: implementation issues

GMRES

iterative method The Interface Problem (IP) Matrix

Interface problem: iterative solution

GMRES

iterative method The Interface Problem (IP) Matrix

Interface problem: iterative solution

GMRES

iterative method The Reduced IP Matrix [1,2] The Interface Problem (IP) Matrix

References [1] Li and Jin 2007 [2] Xue and Jin 2012

Interface problem: iterative solution

GMRES

iterative method The Reduced IP Matrix [1,2] The Interface Problem (IP) Matrix

References [1] Li and Jin 2007 [2] Xue and Jin 2012

Interface problem: iterative solution

Exact values of the Lagrange multipliers (LU-decomposition) Lagrange multipliers obtained after 10 iterations Presence of PML area

Convergence: influence of PML

Conclusion so far q Bad influence of PML q The error does not spread Exact values of the Lagrange multipliers (LU-decomposition) Lagrange multipliers obtained after 10 iterations Presence of PML area

Convergence: influence of PML

References [1] Després 1991 [2] Boubendir et al 2000 * Evanescent Modes Damping Algorithm

What we can play with

q Robin-type boundary conditions Després approach [1] EMDA* approach [2]

Acceleration of the convergence

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]

Convergence: influence of the 𝛽-parameter

• parameter

References [1] Després 1991 [2] Boubendir et al 2000 * Evanescent Modes Damping Algorithm

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]

Convergence: influence of the 𝛽-parameter

• parameter

References [1] Després 1991 [2] Boubendir et al 2000 * Evanescent Modes Damping Algorithm

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]

References [1] Després 1991 [2] Boubendir et al 2000 [3] Voznyuk et al (submi7ed) 2014 * Evanescent Modes Damping Algorithm

Convergence: influence of the 𝛽-parameter

• parameter

Just an object

3D Inverse problem: introduction

εr

?

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: schematic configuration

Anechoic chamber

References [1] J.-M. Geffrin and P. Sabouroux 2009

3D Fresnel database [1] q Set of homogeneous targets q 162 transmimng dipoles q 32 receiver posiGons q Distance:

3D Inversion: measurement setup

! E mes :N src × N rec

3D Inversion: mathematical side

Problem statement:

Find such as the cost functional is minimized q W – covariance matrix, related to + noise q – scattered far-)ield, computed thanks to Near-to-Far-Field transformation

! E mes :N src × N rec

J (εr ,E)= ‖E mes −E calc‖

W 2

3D Inversion: mathematical side

Problem statement:

Find such as the cost functional is minimized 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

! E mes :N src × N rec

J (εr ,E)= ‖E mes −E calc‖

W 2

3D Inversion: Lagrangian formalism

Constraints Problem statement:

Find such as the cost functional is minimized q An iterative quasi-Newton method based on L-BFGS [1] approach with constraints

References [1]

• R. Byrd et al

1995

! E mes :N src × N rec

J (εr ,E)= ‖E mes −E calc‖

W 2

3D Inversion: Lagrangian formalism

Total 8ield Adjoint 8ield Problem statement:

Find such as the cost functional is minimized q An iterative quasi-Newton method based on L-BFGS [1] approach with constraints

Constraints

References [1]

• R. Byrd et al

1995

! E mes :N src × N rec

J (εr ,E)= ‖E mes −E calc‖

W 2

Initial guess FEM (Direct) Mis)it criterion Stop? Gradient evaluation Update direction New permittivity FEM (Adjoint) End of process Max iteraGon No evoluGon Small misfit Small gradient

3D Inversion: Iterative FEM algorithm

Initial guess FEM (Direct) Mis)it criterion Stop? Gradient evaluation Update direction New permittivity FEM (Adjoint) End of process Max iteraGon No evoluGon Small misfit Small gradient Initial guess FETI Initialization Permanent FETI iterations (Direct) FETI Initialization non-Permanent Mis)it criterion Stop? Gradient evaluation Update direction New permittivity FETI iterations (Adjoint)

3D Inversion: Iterative FETI algorithm

FETI optimization

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached?

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached? Cost function Inversion iterations

FETI optimization: GMRES stopping criterion

Cost function GMRES stopping criterion 𝜃 Cost function Inversion iterations

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached?

FETI optimization: GMRES stopping criterion

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached? Without initialization

FETI optimization: starting point

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached? Without initialization lambdaR initialization

FETI optimization: starting point

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached? Without initialization lambdaR initialization Sources initialization

FETI optimization: starting point

Initial λ One GMRES iteration One GMRES iteration One GMRES iteration

η is reached? η is reached? FETI (GMRES) iteraOons (Direct or Adjoint) 1 s = 2… Nscr-1 Nscr

1 r

Initial λ

s r

Initial λ

Nscr r

η is reached? Direct problem Adjoint problem Without initialization lambdaR initialization Sources initialization

FETI optimization: starting point

Cost FuncOon convergence FETI iteraOons Profile cut comparison

3D Inversion from measurements

3D Inversion: Conclusion

Conclusion q Successful combinaGon of the Inversion algorithm and FETI method q Specific implementaGon of FETI method q Tests on experimental fields

Conclusions q Development of the 3D FETI method q ImplementaGon to the Large-Scale v Direct problems v Inverse quanGtaGve problems PerspecOve q Play with the transmission condiGons q Introduce a-priori informaGon in inversion

General conclusions and perspecGves

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