Convergence of Truncated T-Matrix Approximation M. Ganesh 1 , S.C. - PowerPoint PPT Presentation

Convergence of Truncated T-Matrix Approximation M. Ganesh 1 , S.C. Hawkins 2 , Ralf Hiptmair 3 1 Department of Mathematics, Colorado School of Mines 2 Department of Mathematics, Macquarie University, Syndney 3 Seminar for Applied Mathematics, ETH Zürich Workshop on Wave Propagation and Scattering, Inverse Problems and Applications November 21-25, 2011, RICAM, Linz Current research R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 1 / 20

Acoustic Scattering Sound-soft acoustic scattering (at a particle D ): u inc ∆ u s + k 2 u s = 0 in R d \ D , R u s = − u inc on ∂ D , ρ D + radiation conditions at ∞ . 0 D 2 diam ( D ) ≤ ρ D < R . 1 R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 2 / 20

Acoustic Scattering Sound-soft acoustic scattering (at a particle D ): u inc ∆ u s + k 2 u s = 0 in R d \ D , R u s = − u inc on ∂ D , ρ D + radiation conditions at ∞ . 0 D 2 diam ( D ) ≤ ρ D < R . 1 ρ D ˆ = size of particle R ˆ = “separation distance” R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 2 / 20

Acoustic Scattering Sound-soft acoustic scattering (at a particle D ): u inc ∆ u s + k 2 u s = 0 in R d \ D , R u s = − u inc on ∂ D , ρ D + radiation conditions at ∞ . 0 D 2 diam ( D ) ≤ ρ D < R . 1 ρ D ˆ = size of particle R ˆ = “separation distance” | x |→∞ | x | ( d − 1 ) / 2 e − ik | x | u s ( x ) , u ∞ ( � x ) = x = x / | x | � Far field (pattern): lim R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 2 / 20

Acoustic Scattering Sound-soft acoustic scattering (at a particle D ): u inc ∆ u s + k 2 u s = 0 in R d \ D , R u s = − u inc on ∂ D , ρ D + radiation conditions at ∞ . 0 D 2 diam ( D ) ≤ ρ D < R . 1 ρ D ˆ = size of particle R ˆ = “separation distance” | x |→∞ | x | ( d − 1 ) / 2 e − ik | x | u s ( x ) , u ∞ ( � x ) = x = x / | x | � Far field (pattern): lim S ∞ : { u inc } �→ { u ∞ } Scattering map: R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 2 / 20

What Next ? Acoustic Scattering 1 T-Matrix 2 Convergence Theory 3 Numerical Experiments 4 R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 3 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ basis fields for u inc (incoming waves) � | ℓ | ( kR ) J | ℓ | ( k | x | ) Y ℓ ( � x ) , H ( 1 ) d = 2 , E ℓ ( x ) = � ℓ ∈ I d . h ( 1 ) | ℓ | ( kR ) j | ℓ | ( k | x | ) Y ℓ ( � x ) , d = 3 , R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ basis fields for u inc (incoming waves) � | ℓ | ( kR ) J | ℓ | ( k | x | ) Y ℓ ( � x ) , H ( 1 ) d = 2 , E ℓ ( x ) = � ℓ ∈ I d . h ( 1 ) | ℓ | ( kR ) j | ℓ | ( k | x | ) Y ℓ ( � x ) , d = 3 , H ( 1 ) ( h ( 1 ) ℓ ), J ℓ ( j ℓ ) ˆ = (spherical) Hankel-/Bessel-functions ℓ R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ basis fields for u inc (incoming waves) � | ℓ | ( kR ) J | ℓ | ( k | x | ) Y ℓ ( � x ) , H ( 1 ) d = 2 , E ℓ ( x ) = � ℓ ∈ I d . h ( 1 ) | ℓ | ( kR ) j | ℓ | ( k | x | ) Y ℓ ( � x ) , d = 3 , H ( 1 ) ( h ( 1 ) ℓ ), J ℓ ( j ℓ ) ˆ = (spherical) Hankel-/Bessel-functions ℓ � { ℓ = ℓ : ℓ ∈ Z } , for d = 2 , I d := { ℓ = ( ℓ, j ) : ℓ ∈ N 0 , | j | ≤ ℓ } , for d = 3 . R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ basis fields for u inc (incoming waves) � | ℓ | ( kR ) J | ℓ | ( k | x | ) Y ℓ ( � x ) , H ( 1 ) d = 2 , E ℓ ( x ) = � ℓ ∈ I d . h ( 1 ) | ℓ | ( kR ) j | ℓ | ( k | x | ) Y ℓ ( � x ) , d = 3 , H ( 1 ) ( h ( 1 ) ℓ ), J ℓ ( j ℓ ) ˆ = (spherical) Hankel-/Bessel-functions ℓ � { ℓ = ℓ : ℓ ∈ Z } , for d = 2 , I d := { ℓ = ( ℓ, j ) : ℓ ∈ N 0 , | j | ≤ ℓ } , for d = 3 . circular ( d = 2) Y ℓ ( � x ) ˆ = L 2 ( S d − 1 ) -orthogonal harmonics spherical ( d = 3) R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ basis fields for u inc (incoming waves) � | ℓ | ( kR ) J | ℓ | ( k | x | ) Y ℓ ( � x ) , H ( 1 ) d = 2 , E ℓ ( x ) = � ℓ ∈ I d . h ( 1 ) | ℓ | ( kR ) j | ℓ | ( k | x | ) Y ℓ ( � x ) , d = 3 , H ( 1 ) ( h ( 1 ) ℓ ), J ℓ ( j ℓ ) ˆ = (spherical) Hankel-/Bessel-functions ℓ � { ℓ = ℓ : ℓ ∈ Z } , for d = 2 , I d := { ℓ = ( ℓ, j ) : ℓ ∈ N 0 , | j | ≤ ℓ } , for d = 3 . circular ( d = 2) Y ℓ ( � x ) ˆ = L 2 ( S d − 1 ) -orthogonal harmonics spherical ( d = 3) R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Incoming) ☛ ✟ expansion of u inc / u s into cylindrical/radial waves Idea: ✡ ✠ basis fields for u inc (incoming waves) � | ℓ | ( kR ) J | ℓ | ( k | x | ) Y ℓ ( � x ) , H ( 1 ) d = 2 , E ℓ ( x ) = � ℓ ∈ I d . h ( 1 ) | ℓ | ( kR ) j | ℓ | ( k | x | ) Y ℓ ( � x ) , d = 3 , H ( 1 ) ( h ( 1 ) ℓ ), J ℓ ( j ℓ ) ˆ = (spherical) Hankel-/Bessel-functions ℓ � { ℓ = ℓ : ℓ ∈ Z } , for d = 2 , I d := { ℓ = ( ℓ, j ) : ℓ ∈ N 0 , | j | ≤ ℓ } , for d = 3 . circular ( d = 2) Y ℓ ( � x ) ˆ = L 2 ( S d − 1 ) -orthogonal harmonics spherical ( d = 3) E ℓ = entire Helmholtz solutions ∀ ℓ � R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 4 / 20

Basis Fields (Outgoing) basis fields for scattered waves  √ π k ( − i ) | ℓ | ( 1 − i ) H ( 1 ) | ℓ | ( k | x | ) Y ℓ ( � x ) , for d = 2 ,   E ℓ ( x ) = ℓ ∈ I d .  k  ( − i ) | ℓ | + 1 h ( 1 ) | ℓ | ( k | x | ) Y ℓ ( � x ) , for d = 3 , R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 5 / 20

Basis Fields (Outgoing) basis fields for scattered waves  √ π k ( − i ) | ℓ | ( 1 − i ) H ( 1 ) | ℓ | ( k | x | ) Y ℓ ( � x ) , for d = 2 ,   E ℓ ( x ) = ℓ ∈ I d .  k  ( − i ) | ℓ | + 1 h ( 1 ) | ℓ | ( k | x | ) Y ℓ ( � x ) , for d = 3 , E ℓ = radiating Helmholtz solutions on R d \ { 0 } R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 5 / 20

Basis Fields (Outgoing) basis fields for scattered waves  √ π k ( − i ) | ℓ | ( 1 − i ) H ( 1 ) | ℓ | ( k | x | ) Y ℓ ( � x ) , for d = 2 ,   E ℓ ( x ) = ℓ ∈ I d .  k  ( − i ) | ℓ | + 1 h ( 1 ) | ℓ | ( k | x | ) Y ℓ ( � x ) , for d = 3 , E ℓ = radiating Helmholtz solutions on R d \ { 0 } E ∞ = Y ℓ Far fields: ℓ ( ➣ purpose of scaling factors) R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 5 / 20

(Infinite) T-Matrix: Definition Expansions: � � u inc = p ℓ � E ℓ u ∞ = a ℓ ′ E ∞ ← → ℓ ′ . ℓ ′ ∈ I d ℓ ∈ I d R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 6 / 20

(Infinite) T-Matrix: Definition Expansions: � � u inc = p ℓ � E ℓ u ∞ = a ℓ ′ E ∞ ← → ℓ ′ . ℓ ′ ∈ I d ℓ ∈ I d R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 6 / 20

(Infinite) T-Matrix: Definition Expansions: � � u inc = p ℓ � E ℓ u ∞ = a ℓ ′ E ∞ ← → ℓ ′ . ℓ ′ ∈ I d ℓ ∈ I d � a ℓ ′ = S d − 1 u ∞ ( � x ) Y ℓ ′ ( � x ) d � x = R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 6 / 20

(Infinite) T-Matrix: Definition Expansions: � � u inc = p ℓ � E ℓ u ∞ = a ℓ ′ E ∞ ← → ℓ ′ . ℓ ′ ∈ I d ℓ ∈ I d � � S d − 1 S ∞ ( u inc )( � a ℓ ′ = S d − 1 u ∞ ( � x ) Y ℓ ′ ( � x ) d � x = x ) Y ℓ ′ ( � x ) d � x R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 6 / 20

(Infinite) T-Matrix: Definition Expansions: � � u inc = p ℓ � E ℓ u ∞ = a ℓ ′ E ∞ ← → ℓ ′ . ℓ ′ ∈ I d ℓ ∈ I d � � S d − 1 S ∞ ( u inc )( � a ℓ ′ = S d − 1 u ∞ ( � x ) Y ℓ ′ ( � x ) d � x = x ) Y ℓ ′ ( � x ) d � x � � p ℓ S d − 1 [ S ∞ ( � E ℓ )]( � x ) Y ℓ ′ ( � x ) d � x . = ℓ ∈ I d R. Hiptmair (SAM, ETHZ) T -Matrix Approximation RICAM Workshop, Nov 2011 6 / 20