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

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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

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SLIDE 1

Convergence of Truncated T-Matrix Approximation

  • M. Ganesh1, S.C. Hawkins2, Ralf Hiptmair3

1Department of Mathematics, Colorado School of Mines 2Department of Mathematics, Macquarie University, Syndney 3Seminar 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

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SLIDE 2

Acoustic Scattering

D R ρD uinc Sound-soft acoustic scattering (at a particle D): ∆us + k2us = 0 in Rd \ D , us = −uinc on ∂D , + radiation conditions at ∞ .

1 2 diam(D) ≤ ρD < R .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 2 / 20

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SLIDE 3

Acoustic Scattering

D R ρD uinc Sound-soft acoustic scattering (at a particle D): ∆us + k2us = 0 in Rd \ D , us = −uinc on ∂D , + radiation conditions at ∞ .

1 2 diam(D) ≤ ρD < R .

ρD ˆ = size of particle R ˆ = “separation distance”

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 2 / 20

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SLIDE 4

Acoustic Scattering

D R ρD uinc Sound-soft acoustic scattering (at a particle D): ∆us + k2us = 0 in Rd \ D , us = −uinc on ∂D , + radiation conditions at ∞ .

1 2 diam(D) ≤ ρD < R .

ρD ˆ = size of particle R ˆ = “separation distance” Far field (pattern): u∞( x) = lim

|x|→∞ |x|(d−1)/2e−ik|x|us(x),

  • x = x/|x|
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 2 / 20

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SLIDE 5

Acoustic Scattering

D R ρD uinc Sound-soft acoustic scattering (at a particle D): ∆us + k2us = 0 in Rd \ D , us = −uinc on ∂D , + radiation conditions at ∞ .

1 2 diam(D) ≤ ρD < R .

ρD ˆ = size of particle R ˆ = “separation distance” Far field (pattern): u∞( x) = lim

|x|→∞ |x|(d−1)/2e−ik|x|us(x),

  • x = x/|x|

Scattering map: S∞ : {uinc} → {u∞}

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 2 / 20

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SLIDE 6

What Next ?

1

Acoustic Scattering

2

T-Matrix

3

Convergence Theory

4

Numerical Experiments

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 3 / 20

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SLIDE 7

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 8

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves basis fields for uinc (incoming waves)

  • Eℓ(x) =
  • H(1)

|ℓ| (kR) J|ℓ|(k|x|)Yℓ(

x), d = 2 , h(1)

|ℓ| (kR) j|ℓ|(k|x|)Yℓ(

x), d = 3 , ℓ ∈ Id .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 9

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves basis fields for uinc (incoming waves)

  • Eℓ(x) =
  • H(1)

|ℓ| (kR) J|ℓ|(k|x|)Yℓ(

x), d = 2 , h(1)

|ℓ| (kR) j|ℓ|(k|x|)Yℓ(

x), d = 3 , ℓ ∈ Id . H(1)

ℓ

(h(1)

ℓ ), Jℓ (jℓ) ˆ

= (spherical) Hankel-/Bessel-functions

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 10

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves basis fields for uinc (incoming waves)

  • Eℓ(x) =
  • H(1)

|ℓ| (kR) J|ℓ|(k|x|)Yℓ(

x), d = 2 , h(1)

|ℓ| (kR) j|ℓ|(k|x|)Yℓ(

x), d = 3 , ℓ ∈ Id . H(1)

ℓ

(h(1)

ℓ ), Jℓ (jℓ) ˆ

= (spherical) Hankel-/Bessel-functions Id := {ℓ = ℓ : ℓ ∈ Z}, for d = 2 , {ℓ = (ℓ, j) : ℓ ∈ N0, |j| ≤ ℓ}, for d = 3.

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 11

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves basis fields for uinc (incoming waves)

  • Eℓ(x) =
  • H(1)

|ℓ| (kR) J|ℓ|(k|x|)Yℓ(

x), d = 2 , h(1)

|ℓ| (kR) j|ℓ|(k|x|)Yℓ(

x), d = 3 , ℓ ∈ Id . H(1)

ℓ

(h(1)

ℓ ), Jℓ (jℓ) ˆ

= (spherical) Hankel-/Bessel-functions Id := {ℓ = ℓ : ℓ ∈ Z}, for d = 2 , {ℓ = (ℓ, j) : ℓ ∈ N0, |j| ≤ ℓ}, for d = 3. Yℓ( x) ˆ =L2(Sd−1)-orthogonal circular (d = 2) spherical (d = 3) harmonics

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 12

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves basis fields for uinc (incoming waves)

  • Eℓ(x) =
  • H(1)

|ℓ| (kR) J|ℓ|(k|x|)Yℓ(

x), d = 2 , h(1)

|ℓ| (kR) j|ℓ|(k|x|)Yℓ(

x), d = 3 , ℓ ∈ Id . H(1)

ℓ

(h(1)

ℓ ), Jℓ (jℓ) ˆ

= (spherical) Hankel-/Bessel-functions Id := {ℓ = ℓ : ℓ ∈ Z}, for d = 2 , {ℓ = (ℓ, j) : ℓ ∈ N0, |j| ≤ ℓ}, for d = 3. Yℓ( x) ˆ =L2(Sd−1)-orthogonal circular (d = 2) spherical (d = 3) harmonics

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 13

Basis Fields (Incoming)

☛ ✡ ✟ ✠ Idea: expansion of uinc/us into cylindrical/radial waves basis fields for uinc (incoming waves)

  • Eℓ(x) =
  • H(1)

|ℓ| (kR) J|ℓ|(k|x|)Yℓ(

x), d = 2 , h(1)

|ℓ| (kR) j|ℓ|(k|x|)Yℓ(

x), d = 3 , ℓ ∈ Id . H(1)

ℓ

(h(1)

ℓ ), Jℓ (jℓ) ˆ

= (spherical) Hankel-/Bessel-functions Id := {ℓ = ℓ : ℓ ∈ Z}, for d = 2 , {ℓ = (ℓ, j) : ℓ ∈ N0, |j| ≤ ℓ}, for d = 3. Yℓ( x) ˆ =L2(Sd−1)-orthogonal circular (d = 2) spherical (d = 3) harmonics

  • Eℓ = entire Helmholtz solutions ∀ℓ
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 4 / 20

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SLIDE 14

Basis Fields (Outgoing)

basis fields for scattered waves Eℓ(x) =     

√ πk (−i)|ℓ|(1−i) H(1) |ℓ| (k|x|)Yℓ(

x) , for d = 2 ,

k (−i)|ℓ|+1 h(1) |ℓ| (k|x|)Yℓ(

x) , for d = 3 , ℓ ∈ Id .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 5 / 20

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SLIDE 15

Basis Fields (Outgoing)

basis fields for scattered waves Eℓ(x) =     

√ πk (−i)|ℓ|(1−i) H(1) |ℓ| (k|x|)Yℓ(

x) , for d = 2 ,

k (−i)|ℓ|+1 h(1) |ℓ| (k|x|)Yℓ(

x) , for d = 3 , ℓ ∈ Id . Eℓ = radiating Helmholtz solutions

  • n Rd \ {0}
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 5 / 20

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SLIDE 16

Basis Fields (Outgoing)

basis fields for scattered waves Eℓ(x) =     

√ πk (−i)|ℓ|(1−i) H(1) |ℓ| (k|x|)Yℓ(

x) , for d = 2 ,

k (−i)|ℓ|+1 h(1) |ℓ| (k|x|)Yℓ(

x) , for d = 3 , ℓ ∈ Id . Eℓ = radiating Helmholtz solutions

  • n Rd \ {0}

Far fields: E∞

ℓ

= Yℓ (➣ purpose of scaling factors)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 5 / 20

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SLIDE 17

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 18

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 19

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

aℓ′ =

  • Sd−1 u∞(

x) Y ℓ′( x) d x =

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 20

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

aℓ′ =

  • Sd−1 u∞(

x) Y ℓ′( x) d x =

  • Sd−1 S∞(uinc)(

x) Y ℓ′( x) d x

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 21

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

aℓ′ =

  • Sd−1 u∞(

x) Y ℓ′( x) d x =

  • Sd−1 S∞(uinc)(

x) Y ℓ′( x) d x =

  • ℓ∈Id

pℓ

  • Sd−1[S∞(

Eℓ)]( x) Y ℓ′( x) d x .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 22

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

aℓ′ =

  • Sd−1 u∞(

x) Y ℓ′( x) d x =

  • Sd−1 S∞(uinc)(

x) Y ℓ′( x) d x =

  • ℓ∈Id

pℓ

  • Sd−1[S∞(

Eℓ)]( x) Y ℓ′( x) d x .

  • a = T

p with (T)ℓ′,ℓ =

  • Sd−1[S∞(

Eℓ)]( x) Y ℓ′( x) d x, ℓ′, ℓ ∈ Id .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 23

(Infinite) T-Matrix: Definition

Expansions: uinc =

  • ℓ∈Id

pℓ Eℓ ← → u∞ =

  • ℓ′∈Id

aℓ′E∞

ℓ′ .

aℓ′ =

  • Sd−1 u∞(

x) Y ℓ′( x) d x =

  • Sd−1 S∞(uinc)(

x) Y ℓ′( x) d x =

  • ℓ∈Id

pℓ

  • Sd−1[S∞(

Eℓ)]( x) Y ℓ′( x) d x .

  • a =

T-Matrix

T p with (T)ℓ′,ℓ =

  • Sd−1[S∞(

Eℓ)]( x) Y ℓ′( x) d x, ℓ′, ℓ ∈ Id .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 6 / 20

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SLIDE 24

Truncated T-Matrix

Discretization of scattering map S∞ by truncating T-matrix

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 7 / 20

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SLIDE 25

Truncated T-Matrix

Discretization of scattering map S∞ by truncating T-matrix u∞

(N′,N) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

tℓ′,ℓpℓ   E∞

ℓ′ =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T)ℓ′,ℓpℓ   Yℓ′ ,

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 7 / 20

slide-26
SLIDE 26

Truncated T-Matrix

Discretization of scattering map S∞ by truncating T-matrix u∞

(N′,N) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

tℓ′,ℓpℓ   E∞

ℓ′ =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T)ℓ′,ℓpℓ   Yℓ′ , with resolution levels given by Id,N :=

  • {ℓ = ℓ : −N ≤ |ℓ| ≤ N},

for d = 2 , {ℓ = (ℓ, j) : 0 ≤ ℓ ≤ N, |j| ≤ ℓ}, for d = 3 .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 7 / 20

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SLIDE 27

Truncated T-Matrix

Discretization of scattering map S∞ by truncating T-matrix u∞

(N′,N) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

tℓ′,ℓpℓ   E∞

ℓ′ =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T)ℓ′,ℓpℓ   Yℓ′ , with resolution levels given by Id,N :=

  • {ℓ = ℓ : −N ≤ |ℓ| ≤ N},

for d = 2 , {ℓ = (ℓ, j) : 0 ≤ ℓ ≤ N, |j| ≤ ℓ}, for d = 3 . Idea: ➊ offline stage: precompute entries (T)ℓ′,ℓ, ℓ′ ∈ Id,N′, ℓ ∈ Id,N (approximately)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 7 / 20

slide-28
SLIDE 28

Truncated T-Matrix

Discretization of scattering map S∞ by truncating T-matrix u∞

(N′,N) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

tℓ′,ℓpℓ   E∞

ℓ′ =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T)ℓ′,ℓpℓ   Yℓ′ , with resolution levels given by Id,N :=

  • {ℓ = ℓ : −N ≤ |ℓ| ≤ N},

for d = 2 , {ℓ = (ℓ, j) : 0 ≤ ℓ ≤ N, |j| ≤ ℓ}, for d = 3 . Idea: ➊ offline stage: precompute entries (T)ℓ′,ℓ, ℓ′ ∈ Id,N′, ℓ ∈ Id,N (approximately) ➣ use any forward field solver

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 7 / 20

slide-29
SLIDE 29

Truncated T-Matrix

Discretization of scattering map S∞ by truncating T-matrix u∞

(N′,N) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

tℓ′,ℓpℓ   E∞

ℓ′ =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T)ℓ′,ℓpℓ   Yℓ′ , with resolution levels given by Id,N :=

  • {ℓ = ℓ : −N ≤ |ℓ| ≤ N},

for d = 2 , {ℓ = (ℓ, j) : 0 ≤ ℓ ≤ N, |j| ≤ ℓ}, for d = 3 . Idea: ➊ offline stage: precompute entries (T)ℓ′,ℓ, ℓ′ ∈ Id,N′, ℓ ∈ Id,N (approximately) ➣ use any forward field solver ➋ online stage: fast computation of approximate far field u∞

(N′,N)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 7 / 20

slide-30
SLIDE 30

T-Matrix: Applications

Scattering from particles in random orientation

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 8 / 20

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SLIDE 31

T-Matrix: Applications

Scattering from particles in random orientation Scattering by ensembles of particles

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 8 / 20

slide-32
SLIDE 32

T-Matrix: Applications

Scattering from particles in random orientation Scattering by ensembles of particles

  • A. DOICU, T. WRIEDT, AND Y. EREMIN, Light Scattering by Systems of

Particles, vol. 124 of Springer Series in Optical Sciences, Springer, Heidelberg, 2006.

  • M. J. GROTE AND C. KIRSCH, Dirichlet-to-Neumann boundary conditions

for multiple scattering problems, J. Comput. Phys., 201 (2004),

  • pp. 630–650.
  • P. A. MARTIN, Multiple scattering, vol. 107 of Encyclopedia of

Mathematics and its Applications, Cambridge University Press, Cambridge, 2006.

  • M. I. MISHCHENKO, L. D. TRAVIS, AND D. W. MACKOWSKI, T-matrix

computations of light scattering by nonspherical particles: A review, Journal of Quantitative Spectroscopy and Radiative Transfer, 55 (1996),

  • pp. 535 – 575.
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 8 / 20

slide-33
SLIDE 33

What Next ?

1

Acoustic Scattering

2

T-Matrix

3

Convergence Theory

4

Numerical Experiments

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 9 / 20

slide-34
SLIDE 34

Truncation Error Estimate

Smoothness space for incident fields: for weight sequence (wℓ)ℓ, wℓ ≥ w > 0, Xw =   ψ =

  • ℓ∈Id

qℓ Eℓ : ψ2

Xw :=

  • ℓ∈Id

wℓ|qℓ|2 < ∞    .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 10 / 20

slide-35
SLIDE 35

Truncation Error Estimate

Smoothness space for incident fields: for weight sequence (wℓ)ℓ, wℓ ≥ w > 0, Xw =   ψ =

  • ℓ∈Id

qℓ Eℓ : ψ2

Xw :=

  • ℓ∈Id

wℓ|qℓ|2 < ∞    . ✬ ✫ ✩ ✪

  • Lemma. If uinc ∈ Xw, then
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤

  • uinc

2

Xw

  

  • ℓ∈Id

1 wℓ

  • (Id − PN′)S∞

Eℓ

  • 2

L2(Sd−1) +

  • ℓ∈Id\Id,N

1 wℓ

  • S∞

Eℓ

  • 2

L2(Sd−1)

  

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 10 / 20

slide-36
SLIDE 36

Truncation Error Estimate

Smoothness space for incident fields: for weight sequence (wℓ)ℓ, wℓ ≥ w > 0, Xw =   ψ =

  • ℓ∈Id

qℓ Eℓ : ψ2

Xw :=

  • ℓ∈Id

wℓ|qℓ|2 < ∞    . ✬ ✫ ✩ ✪

  • Lemma. If uinc ∈ Xw, then
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤

  • uinc

2

Xw

  

  • ℓ∈Id

1 wℓ

  • (Id − PN′)S∞

Eℓ

  • 2

L2(Sd−1) +

  • ℓ∈Id\Id,N

1 wℓ

  • S∞

Eℓ

  • 2

L2(Sd−1)

  

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 10 / 20

slide-37
SLIDE 37

Truncation Error Estimate

Smoothness space for incident fields: for weight sequence (wℓ)ℓ, wℓ ≥ w > 0, Xw =   ψ =

  • ℓ∈Id

qℓ Eℓ : ψ2

Xw :=

  • ℓ∈Id

wℓ|qℓ|2 < ∞    . ✬ ✫ ✩ ✪

  • Lemma. If uinc ∈ Xw, then
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤

  • uinc

2

Xw

  

  • ℓ∈Id

1 wℓ

  • (Id −

Orthogonal projection PN′)S∞ Eℓ

  • 2

L2(Sd−1) +

  • ℓ∈Id\Id,N

1 wℓ

  • S∞

Eℓ

  • 2

L2(Sd−1)

  

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 10 / 20

slide-38
SLIDE 38

Truncation Error Estimate

Smoothness space for incident fields: for weight sequence (wℓ)ℓ, wℓ ≥ w > 0, Xw =   ψ =

  • ℓ∈Id

qℓ Eℓ : ψ2

Xw :=

  • ℓ∈Id

wℓ|qℓ|2 < ∞    . ✬ ✫ ✩ ✪

  • Lemma. If uinc ∈ Xw, then
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤

  • uinc

2

Xw

  

  • ℓ∈Id

1 wℓ

  • (Id −

Orthogonal projection PN′)S∞ Eℓ

  • 2

L2(Sd−1) +

  • ℓ∈Id\Id,N

1 wℓ

  • S∞

Eℓ

  • 2

L2(Sd−1)

  

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 10 / 20

slide-39
SLIDE 39

Estimates: Error Terms

★ ✧ ✥ ✦

  • Lemma. For all ℓ ∈ Id, and assuming the threshold condition N′ > Rk/2,
  • (Id − PN′)S∞

Eℓ

  • 2

L2(Sd−1) =

  • ℓ′∈Id\Id,N′

|tℓ′,ℓ|2 ≤ C Rke 2N′ 2N′ ρD R 2|ℓ| |ℓ|d/2−1 .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 11 / 20

slide-40
SLIDE 40

Estimates: Error Terms

★ ✧ ✥ ✦

  • Lemma. For all ℓ ∈ Id, and assuming the threshold condition N′ > Rk/2,
  • (Id − PN′)S∞

Eℓ

  • 2

L2(Sd−1) =

  • ℓ′∈Id\Id,N′

|tℓ′,ℓ|2 ≤ C Rke 2N′ 2N′ ρD R 2|ℓ| |ℓ|d/2−1 . ✛ ✚ ✘ ✙

  • Lemma. For all ℓ ∈ Id
  • S∞

Eℓ

  • 2

L2(Sd−1) ≤ C

ρD R 2|ℓ| |ℓ|d/2−1 . Proof uses

  • Eℓ↾∂D
  • H

1 2 (∂D) ≤ C

ρD R |ℓ| |ℓ|d/4−1/2.

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 11 / 20

slide-41
SLIDE 41

Main Theorem

✬ ✫ ✩ ✪

  • Theorem. For uinc ∈ Xw, assuming the threshold condition N′ > Rk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤ C

  • uinc

2

Xw

   ρKe 2N′ 2N′

ℓ∈Id

|ℓ|d/2−1 wℓ ρD R 2|ℓ| +

  • ℓ∈Id\Id,N

|ℓ|d/2−1 wℓ ρD R 2|ℓ|    .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 12 / 20

slide-42
SLIDE 42

Main Theorem

✬ ✫ ✩ ✪

  • Theorem. For uinc ∈ Xw, assuming the threshold condition N′ > Rk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤ C

  • uinc

2

Xw

   ρKe 2N′ 2N′

ℓ∈Id

|ℓ|d/2−1 wℓ ρD R 2|ℓ| +

  • ℓ∈Id\Id,N

|ℓ|d/2−1 wℓ ρD R 2|ℓ|    . weights wℓ make the difference !

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 12 / 20

slide-43
SLIDE 43

Main Theorem

✬ ✫ ✩ ✪

  • Theorem. For uinc ∈ Xw, assuming the threshold condition N′ > Rk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤ C

  • uinc

2

Xw

   ρKe 2N′ 2N′

ℓ∈Id

|ℓ|d/2−1 wℓ ρD R 2|ℓ| +

  • ℓ∈Id\Id,N

|ℓ|d/2−1 wℓ ρD R 2|ℓ|    . weights wℓ make the difference !

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 12 / 20

slide-44
SLIDE 44

Main Theorem

✬ ✫ ✩ ✪

  • Theorem. For uinc ∈ Xw, assuming the threshold condition N′ > Rk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤ C

  • uinc

2

Xw

   ρKe 2N′ 2N′

ℓ∈Id

|ℓ|d/2−1 wℓ ρD R 2|ℓ| +

  • ℓ∈Id\Id,N

|ℓ|d/2−1 wℓ ρD R 2|ℓ|    . weights wℓ make the difference ! (wℓ fast increasing ➣ fast convergence)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 12 / 20

slide-45
SLIDE 45

Main Theorem

✬ ✫ ✩ ✪

  • Theorem. For uinc ∈ Xw, assuming the threshold condition N′ > Rk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1)

≤ C

  • uinc

2

Xw

   ρKe 2N′ 2N′

ℓ∈Id

|ℓ|d/2−1 wℓ ρD R 2|ℓ| +

  • ℓ∈Id\Id,N

|ℓ|d/2−1 wℓ ρD R 2|ℓ|    . weights wℓ make the difference ! (wℓ fast increasing ➣ fast convergence)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 12 / 20

slide-46
SLIDE 46

Plane Wave Incidence

Expansion uinc =

ℓ∈Id pℓ

Eℓ for plane wave: uinc(x) = exp(ikx · d)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 13 / 20

slide-47
SLIDE 47

Plane Wave Incidence

Expansion uinc =

ℓ∈Id pℓ

Eℓ for plane wave: uinc(x) = exp(ikx · d) ⇒ pℓ =          √ 2π i|ℓ| H(1)

ℓ (kR)

Yℓ( d), for d = 2, 4π i|ℓ| h(1)

ℓ (kR)

Yℓ( d), for d = 3 .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 13 / 20

slide-48
SLIDE 48

Plane Wave Incidence

Expansion uinc =

ℓ∈Id pℓ

Eℓ for plane wave: uinc(x) = exp(ikx · d) ⇒ pℓ =          √ 2π i|ℓ| H(1)

ℓ (kR)

Yℓ( d), for d = 2, 4π i|ℓ| h(1)

ℓ (kR)

Yℓ( d), for d = 3 . Weights from Hankel function asymptotic estimates wℓ = |ℓ|−d 2|ℓ| Rke 2|ℓ| κ = Rk ✄

1 2 3 4 5 6 7 8 9 10 10

−10

10

−5

10 10

5

10

10

10

15

10

20

|l| wl

Weights: incident plane wave in 2D κ = 1.00 κ = 2.00 κ = 4.00 κ = 8.00 κ = 16.00

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 13 / 20

slide-49
SLIDE 49

Plane Wave Incidence

Expansion uinc =

ℓ∈Id pℓ

Eℓ for plane wave: uinc(x) = exp(ikx · d) ⇒ pℓ =          √ 2π i|ℓ| H(1)

ℓ (kR)

Yℓ( d), for d = 2, 4π i|ℓ| h(1)

ℓ (kR)

Yℓ( d), for d = 3 . ✬ ✫ ✩ ✪ Theorem. uinc(x) = exp(ikx · d), d ∈ Sd−1 for d = 2, 3 threshold conditions N′ > Rk/2 and N > Rk/2 + 3d

4

  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1) ≤ C

  • eRkκe

2 3d/2−1 κe 2N′ 2N′ + N

3d/2−1 κe

2N 2N , κ := Rk.

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 13 / 20

slide-50
SLIDE 50

Plane Wave Incidence

Expansion uinc =

ℓ∈Id pℓ

Eℓ for plane wave: uinc(x) = exp(ikx · d) ⇒ pℓ =          √ 2π i|ℓ| H(1)

ℓ (kR)

Yℓ( d), for d = 2, 4π i|ℓ| h(1)

ℓ (kR)

Yℓ( d), for d = 3 . ✬ ✫ ✩ ✪ Theorem. uinc(x) = exp(ikx · d), d ∈ Sd−1 for d = 2, 3 threshold conditions N′ > Rk/2 and N > Rk/2 + 3d

4

  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1) ≤ C

  • eRkκe

2 3d/2−1 κe 2N′ 2N′ + N

3d/2−1 κe

2N 2N , κ := Rk. super-exponential convergence

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 13 / 20

slide-51
SLIDE 51

Point Source Excitation

Expansion for point source in x0, |x0| > R, uinc(x) =        H(1)

0 (k|x − x0|),

d = 2, exp(ik|x − x0|) k|x − x0| , d = 3.

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 14 / 20

slide-52
SLIDE 52

Point Source Excitation

Expansion for point source in x0, |x0| > R, uinc(x) =        H(1)

0 (k|x − x0|),

d = 2, exp(ik|x − x0|) k|x − x0| , d = 3. ⇒ pℓ =                √ 2π H(1)

|ℓ| (k|x0|)

H(1)

|ℓ| (kR)

Yℓ( x0) , 4πik h(1)

|ℓ| (k|x0|)

h(1)

|ℓ| (kR)

Yℓ( x 0) .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 14 / 20

slide-53
SLIDE 53

Point Source Excitation

Expansion for point source in x0, |x0| > R, uinc(x) =        H(1)

0 (k|x − x0|),

d = 2, exp(ik|x − x0|) k|x − x0| , d = 3. ⇒ pℓ =                √ 2π H(1)

|ℓ| (k|x0|)

H(1)

|ℓ| (kR)

Yℓ( x0) , 4πik h(1)

|ℓ| (k|x0|)

h(1)

|ℓ| (kR)

Yℓ( x 0) . (“slowly increasing”) weights: wℓ := 1 |ℓ|d |x0| R 2|ℓ|

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 14 / 20

slide-54
SLIDE 54

Point Source Excitation

Expansion for point source in x0, |x0| > R, uinc(x) =        H(1)

0 (k|x − x0|),

d = 2, exp(ik|x − x0|) k|x − x0| , d = 3. ⇒ pℓ =                √ 2π H(1)

|ℓ| (k|x0|)

H(1)

|ℓ| (kR)

Yℓ( x0) , 4πik h(1)

|ℓ| (k|x0|)

h(1)

|ℓ| (kR)

Yℓ( x 0) . (“slowly increasing”) weights: wℓ := 1 |ℓ|d |x0| R 2|ℓ| ★ ✧ ✥ ✦

  • Theorem. Assuming threshold condition N′ > ρDk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1) ≤

C

  • 1 − (ρD/|x0|)2q+1

ρDke 2N′ 2N′ + Nq ρD |x0| 2N .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 14 / 20

slide-55
SLIDE 55

Point Source Excitation

Expansion for point source in x0, |x0| > R, uinc(x) =        H(1)

0 (k|x − x0|),

d = 2, exp(ik|x − x0|) k|x − x0| , d = 3. ⇒ pℓ =                √ 2π H(1)

|ℓ| (k|x0|)

H(1)

|ℓ| (kR)

Yℓ( x0) , 4πik h(1)

|ℓ| (k|x0|)

h(1)

|ℓ| (kR)

Yℓ( x 0) . (“slowly increasing”) weights: wℓ := 1 |ℓ|d |x0| R 2|ℓ| ★ ✧ ✥ ✦

  • Theorem. Assuming threshold condition N′ > ρDk/2,
  • u∞ − u∞

(N′,N)

  • 2

L2(Sd−1) ≤

C

  • 1 − (ρD/|x0|)2q+1

exponential convergence ρDke 2N′ 2N′ + Nq ρD |x0| 2N .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 14 / 20

slide-56
SLIDE 56

Approximate T-Matrix

Discretization error ➣ approximate far field

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-57
SLIDE 57

Approximate T-Matrix

Discretization error ➣ approximate far field

  • S∞

h

Eℓ − S∞ Eℓ

  • L2(Sd−1) ≤ ǫ(ℓ) ≤ ǫ ,

ℓ ∈ Id,N .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-58
SLIDE 58

Approximate T-Matrix

Discretization error ➣ approximate far field

  • S∞

h

Eℓ − S∞ Eℓ

  • L2(Sd−1) ≤ ǫ(ℓ) ≤ ǫ ,

ℓ ∈ Id,N . u∞

(N′,N,h) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T h)ℓ′,ℓ pℓ   Yℓ′ with (T h)ℓ′,ℓ := S∞

h

Eℓ, Yℓ′ .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-59
SLIDE 59

Approximate T-Matrix

Discretization error ➣ approximate far field

  • S∞

h

Eℓ − S∞ Eℓ

  • L2(Sd−1) ≤ ǫ(ℓ) ≤ ǫ ,

ℓ ∈ Id,N . u∞

(N′,N,h) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T h)ℓ′,ℓ pℓ   Yℓ′ with (T h)ℓ′,ℓ := S∞

h

Eℓ, Yℓ′ . ➥ additive contribution to truncation error ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-60
SLIDE 60

Approximate T-Matrix

Discretization error ➣ approximate far field

  • S∞

h

Eℓ − S∞ Eℓ

  • L2(Sd−1) ≤ ǫ(ℓ) ≤ ǫ ,

ℓ ∈ Id,N . u∞

(N′,N,h) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T h)ℓ′,ℓ pℓ   Yℓ′ with (T h)ℓ′,ℓ := S∞

h

Eℓ, Yℓ′ . ➥ additive contribution to truncation error ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-61
SLIDE 61

Approximate T-Matrix

Discretization error ➣ approximate far field

  • S∞

h

Eℓ − S∞ Eℓ

  • L2(Sd−1) ≤ ǫ(ℓ) ≤ ǫ ,

ℓ ∈ Id,N . u∞

(N′,N,h) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T h)ℓ′,ℓ pℓ   Yℓ′ with (T h)ℓ′,ℓ := S∞

h

Eℓ, Yℓ′ . ➥ additive contribution to truncation error ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .

Plane wave incidence ∆ ≤ C

  • · · · + ǫ2

Point source excitation ∆ ≤ C (1 − (R/|x0|)2)2d−1

  • · · · + ǫ2
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-62
SLIDE 62

Approximate T-Matrix

Discretization error ➣ approximate far field

  • S∞

h

Eℓ − S∞ Eℓ

  • L2(Sd−1) ≤ ǫ(ℓ) ≤ ǫ ,

ℓ ∈ Id,N . u∞

(N′,N,h) =

  • ℓ′∈Id,N′

 

ℓ∈Id,N

(T h)ℓ′,ℓ pℓ   Yℓ′ with (T h)ℓ′,ℓ := S∞

h

Eℓ, Yℓ′ . ➥ additive contribution to truncation error ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .

Plane wave incidence ∆ ≤ C

  • · · · + ǫ2

Point source excitation ∆ ≤ C (1 − (R/|x0|)2)2d−1

  • · · · + ǫ2
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 15 / 20

slide-63
SLIDE 63

Optimization of Offline Stage

Recall: impact of discretization error in offline stage ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 16 / 20

slide-64
SLIDE 64

Optimization of Offline Stage

Recall: impact of discretization error in offline stage ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .

Constraint: Wtot ˆ = total resources available for offline stage

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 16 / 20

slide-65
SLIDE 65

Optimization of Offline Stage

Recall: impact of discretization error in offline stage ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .

Constraint: Wtot ˆ = total resources available for offline stage Assumption: algebraically convergent method for S∞ Wℓ ∼ cǫ(ℓ)−1/α , α > 0, c > 0, ℓ ∈ Id,N , (Wℓ ˆ = cost of approximating S∞ Eℓ with error ǫ(ℓ))

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 16 / 20

slide-66
SLIDE 66

Optimization of Offline Stage

Recall: impact of discretization error in offline stage ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .

Constraint: Wtot ˆ = total resources available for offline stage Assumption: algebraically convergent method for S∞ Wℓ ∼ cǫ(ℓ)−1/α , α > 0, c > 0, ℓ ∈ Id,N , (Wℓ ˆ = cost of approximating S∞ Eℓ with error ǫ(ℓ)) Optimization:

  • ℓ∈Id,N

W −2α

ℓ

wℓ → min :

  • ℓ∈Id,N

Wℓ = Wtot .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 16 / 20

slide-67
SLIDE 67

Optimization of Offline Stage

Recall: impact of discretization error in offline stage ∆ :=

  • u∞ − u∞

(N′,N,h)

  • 2

L2(Sd−1) ≤ C

  • uinc

2

Xw ·

  • · · · +
  • ℓ∈Id,N

[ǫ(ℓ)]2 wℓ

  • .

Constraint: Wtot ˆ = total resources available for offline stage Assumption: algebraically convergent method for S∞ Wℓ ∼ cǫ(ℓ)−1/α , α > 0, c > 0, ℓ ∈ Id,N , (Wℓ ˆ = cost of approximating S∞ Eℓ with error ǫ(ℓ)) Optimization:

  • ℓ∈Id,N

W −2α

ℓ

wℓ → min :

  • ℓ∈Id,N

Wℓ = Wtot . Wℓ = Wtot · (wℓ)1/(2α+1)/

  • ℓ∈Id,N

(wℓ)1/(2α+1) .

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 16 / 20

slide-68
SLIDE 68

What Next ?

1

Acoustic Scattering

2

T-Matrix

3

Convergence Theory

4

Numerical Experiments

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 17 / 20

slide-69
SLIDE 69

Numerical Experiments: Setting

2D Cassini oval 3D erythrocyte shape Highly accurate solution of forward problem: 2D: method of fundamental solutions (A. Barnett and T. Betcke, JCP 2008) 3D: spectral BEM (M. Ganesh & I. Graham, JCP 2004)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 18 / 20

slide-70
SLIDE 70

Empiric Truncation Error

2D Cassini oval

5 6 7 8 9 10 11 12 13 14 15 10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 truncation parameter N far field error eN plane wave point source plane wave bound point source bound

3D erythrocyte shape

◮ k = 10, ρD = 1 2, R = 0.55, |x0| = 3 ◮ “Exact” far field from highly accurate numerical solution ◮ Curves for theoretical bounds with constant C fitted to data

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 19 / 20

slide-71
SLIDE 71

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20

slide-72
SLIDE 72

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • M. GANESH, S. HAWKINS, AND R. HIPTMAIR, Convergence analysis with

parameter estimates for a reduced basis acoustic scattering t-matrix method, Report 2011-04, SAM, ETH Zürich, Zürich, Switzerland, 2011. http://www.sam.math.ethz.ch/reports/2011/04, To appear in IMA J.

  • Numer. Anal.
  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20

slide-73
SLIDE 73

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • M. GANESH, S. HAWKINS, AND R. HIPTMAIR, Convergence analysis with

parameter estimates for a reduced basis acoustic scattering t-matrix method, Report 2011-04, SAM, ETH Zürich, Zürich, Switzerland, 2011. http://www.sam.math.ethz.ch/reports/2011/04, To appear in IMA J.

  • Numer. Anal.

(Open) issues: A priori error analysis T-matrix based multi-particle scattering

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20

slide-74
SLIDE 74

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • M. GANESH, S. HAWKINS, AND R. HIPTMAIR, Convergence analysis with

parameter estimates for a reduced basis acoustic scattering t-matrix method, Report 2011-04, SAM, ETH Zürich, Zürich, Switzerland, 2011. http://www.sam.math.ethz.ch/reports/2011/04, To appear in IMA J.

  • Numer. Anal.

(Open) issues: A priori error analysis T-matrix based multi-particle scattering (adaptive) control of (accuracy of) forward solvers in offline stage

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20

slide-75
SLIDE 75

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • M. GANESH, S. HAWKINS, AND R. HIPTMAIR, Convergence analysis with

parameter estimates for a reduced basis acoustic scattering t-matrix method, Report 2011-04, SAM, ETH Zürich, Zürich, Switzerland, 2011. http://www.sam.math.ethz.ch/reports/2011/04, To appear in IMA J.

  • Numer. Anal.

(Open) issues: A priori error analysis T-matrix based multi-particle scattering (adaptive) control of (accuracy of) forward solvers in offline stage Compression of T-matrix: optimal (non-rectangular) truncation ?

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20

slide-76
SLIDE 76

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • M. GANESH, S. HAWKINS, AND R. HIPTMAIR, Convergence analysis with

parameter estimates for a reduced basis acoustic scattering t-matrix method, Report 2011-04, SAM, ETH Zürich, Zürich, Switzerland, 2011. http://www.sam.math.ethz.ch/reports/2011/04, To appear in IMA J.

  • Numer. Anal.

(Open) issues: A priori error analysis T-matrix based multi-particle scattering (adaptive) control of (accuracy of) forward solvers in offline stage Compression of T-matrix: optimal (non-rectangular) truncation ? explicit k-dependence in estimates (for convex D)

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20

slide-77
SLIDE 77

Conclusion

Exponential (in truncation parameter N) convergence of T-matrix method

  • M. GANESH, S. HAWKINS, AND R. HIPTMAIR, Convergence analysis with

parameter estimates for a reduced basis acoustic scattering t-matrix method, Report 2011-04, SAM, ETH Zürich, Zürich, Switzerland, 2011. http://www.sam.math.ethz.ch/reports/2011/04, To appear in IMA J.

  • Numer. Anal.

(Open) issues: A priori error analysis T-matrix based multi-particle scattering (adaptive) control of (accuracy of) forward solvers in offline stage Compression of T-matrix: optimal (non-rectangular) truncation ? explicit k-dependence in estimates (for convex D)

Thank you !

  • R. Hiptmair (SAM, ETHZ)

T

  • Matrix Approximation

RICAM Workshop, Nov 2011 20 / 20