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Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with : Fernando Reitich, University of Minnesota

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Page 1: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Fatih EcevitBogazici University, Istanbul

Convergent Scattering Algorithms

Joint work with: Fernando Reitich, University of Minnesota

Page 2: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Integral Equation Formulations

Radiation Condition:

Single layer potential:

current

Single layer density:

Page 3: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Integral Equation Formulations: AnsatzSingle layer density:

Page 4: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Integral Equation Formulations: AnsatzSingle layer density:

Page 5: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Integral Equation Formulations: AnsatzSingle layer density:

Page 6: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

: open subset of: open conic subset of i.e.

: Hoermander Class of order and

(multi-indices), s.t. compact,

A little bit of microlocal analysis:Hoermander Classes

invariant under diffeomorphisms in the x variable

generalizes to the case where is a smooth manifold

Page 7: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

: open subset of: open conic subset of i.e.

where as

A little bit of microlocal analysis:Asymptotic Expansions

Asymptotic Expansion of :

for

andwhere

Page 8: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

: compact, smooth, strictly convex

compact

Asymptotic Expansions of

Theorem (R.Melrose & M.Taylor - ‘85) :

i.e.

On the illuminated region

i.e.

On the shadow region

decays rapidly in the sense of Schwarz as

Page 9: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

compact

i.e.

i.e.

On a vicinity of the shadow boundary

Positive on the illuminated regionNegative on the shadow regionVanishes precisely to first order at the shadow boundary{

Page 10: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Theorem (Domínguez, Graham, Smyshlyaev ‘07): … derivative estimates

… arclength parametrization

… shadow boundaries

… resembles the behavior of

Page 11: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Several Numerical Algorithms

Domínguez, Graham, Smyshlyaev … 2007 …

Bruno, Geuzaine, Monro, Reitich … 2004 …

Bruno, Geuzaine (3D) ……………. 2007 …

Huybrechs, Vandewalle …….…… 2007 …

Domínguez, E., Graham, ………… 2007 …

Page 12: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Multiple Scattering Configurations

Page 13: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Multiple Scattering Configurations

Page 14: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Multiple Scattering FormulationIntegral Equation of the 2nd Kind:

Disjoint Scatterers:Component form:

Multiply with theinverse of thediagonal operator

Invert the diagonal:

Page 15: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Multiple Scattering FormulationIntegral Equation of the 2nd Kind:

Disjoint Scatterers:Component form:

Invert the diagonal:

Page 16: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Multiple Scattering FormulationIntegral Equation of the 2nd Kind:

Disjoint Scatterers:… Operator equation of the 2nd kind

… Neumann series

twice the normal derivative (evaluated on )of the field scattered from

is the superposition over all infinite pathsof the solutions of the integral equations

Page 17: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Multiple Scattering FormulationIntegral Equation of the 2nd Kind:

Disjoint Scatterers:Reduction to the Interaction of Two-substructures:

Page 18: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Generalized Phase Extraction: (for a collection of convex obstacles)

… given by GO

Multiple Scattering FormulationIntegral Equation of the 2nd Kind:

Disjoint Scatterers:Reduction to the Interaction of Two-substructures:

Page 19: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Visibility:

No-occlusion:

Page 20: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Broken rays: well-defined, existence, uniqueness

Page 21: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Broken rays: well-defined, existence, uniqueness

Page 22: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Broken rays: illuminated regions (IL)shadow regions (SR)shadow boundaries (SB)

Page 23: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

… convexwave fronts

Page 24: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Asymptotic Expansions of Scattered Fields

Theorem (E., Reitich 2009):

Page 25: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota
Page 26: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Theorem (E., Reitich 2009):planewave incidence…ansatz………………….

On the illuminated region:

… Hoermander classes

……… asymptotic expansions

Page 27: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Theorem (E., Reitich 2009):planewave incidence…ansatz………………….

Over the entire boundary:

……………………. Hoermander classes

… asymptotic expansions

Page 28: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Theorem (E., Reitich 2009):planewave incidence…ansatz………………….

Over the entire boundary:

……………………. Hoermander classes

… asymptotic expansions

On the illuminated region:

… Hoermander classes

……… asymptotic expansions

Extends in the same way to 3D

Page 29: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Theorem (E., Reitich 2009): … derivative estimates

… arclength parametrization

… shadow boundaries

… resembles the behavior of

…extension of single scattering results in DGS (2006) to multiple scattering

Page 30: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

GeneralizedGeometrical OpticsApproximations

Page 31: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota
Page 32: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota
Page 33: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Asymptotic Expansions in 2DTheorem: (E., Reitich) For any , the iterated density satisfies

on any compact subset of the m-th illuminated region as Here, isdefined over the entire boundary by

withand

and defined recursively as

where

and

Page 34: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Acoustic Asymptotic Expansions in 3DTheorem: (Anand, Boubendir, E., Reitich)density satisfies

on any compact subset of the m-th illuminated region as Here, isdefined over the entire boundary by

withand

and defined recursively as

where

and

For any , the iterated

Page 35: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Acoustic Asymptotic Expansions in 3DTheorem: (Anand, Boubendir, E., Reitich)

where

and

Here, defining

we have set

and

Finally

Page 36: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Electromagnetic Asymptotic Expansions in 3D

Radiation Condition: Silver-Muller radiation condition

in

Perfect Conductor: on

onthe scattered electromagnetic field can be recovered through theclassical Stratton-Chu formulae.

Page 37: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Electromagnetic Asymptotic Expansions in 3DTheorem: (E., Hackbusch)

on any compact subset of the m-th illuminated region asHere, is defined over the entire boundary by

with

For any , the iterated surface current satisfies

and

Page 38: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Rate of Convergence on Periodic Orbits Periodic Phase on:

Periodic Phase Minimizer:

with Rate of Convergence:

Solutions of explicitquadratic equations

curvatures

principalcurvatures matrix

rotation

3D:

2D:

Page 39: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Rate of Convergence on Periodic Orbits In Summary:

depend only on the geometry and the direction of incidence.The constants involved in the order terms, and

Numerically for a fixed periodic orbit:

Displayed in Numerical Examples:

Page 40: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

2 Periodic Example:

PlanewaveIllumination

Numerical Examples in 2D

Page 41: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

2 Periodic Example:

PlanewaveIllumination

Numerical Examples in 2D

Page 42: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

2 Periodic Example:

PlanewaveIllumination

Numerical Examples in 2D

Page 43: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

2 Periodic Example:

Point SourceIllumination

Numerical Examples in 2D

Page 44: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

3 Periodic Example:

PlanewaveIllumination

Numerical Examples in 2D

Page 45: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

3 Periodic Example:

Point SourceIllumination

Numerical Examples in 2D

Page 46: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

2 Periodic Example:

0.07240.07400.07850.0718

Iteration 1 Iteration 2 Iteration 3

Iteration 10

Numerical Examples in 3D

Page 47: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Numerical Examples in 3D

Page 48: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

Numerical Examples in 3D

Page 49: Fatih Ecevit Bogazici University, Istanbul · Fatih Ecevit Bogazici University, Istanbul Convergent Scattering Algorithms Joint work with: Fernando Reitich, University of Minnesota

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