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Uncertainty principles for far field patterns and applications to inverse source problems

Roland Griesmaier

roland.griesmaier@uni-wuerzburg.de

(joint work with J. Sylvester) Paris, September 2017

Uncertainty principles Roland Griesmaier

Outline

Source problems and far field patterns A regularized Picard criterion Uncertainty principles Corollaries of the uncertainty principles Numerical examples

Uncertainty principles Roland Griesmaier

Source problems and far field patterns

Uncertainty principles Roland Griesmaier

Far fields of compactly supported sources

−30 −20 −10 10 20 30 −30 −20 −10 10 20 30

k > 0 : wave number (= 2π/wave length) k2F : source term (∈ L2

0( R 2))

U : time-harmonic radiated wave Direct source problem: −∆U − k2U = k2F in R 2 and SRC Rescaling: Rewriting u(x) = U(kx) , f (x) = F(kx) we can w.l.o.g. set k = 1 (i.e., distances are measured in wavelengths)

Uncertainty principles Roland Griesmaier

Far fields of compactly supported sources

−30 −20 −10 10 20 30 −30 −20 −10 10 20 30

k = 1 : wave number (= 2π/wave length) f : source term (∈ L2

0( R 2))

u : time-harmonic radiated wave Direct source problem: −∆u − u = f in R 2 and SRC Far field expansion: u(x) = C eir √r α( x) + O(r −3/2) , r → ∞ , x = r x , where α(θ) =

• R 2 e−iθ·yf (y) dy =

f (θ) , θ ∈ S1

Uncertainty principles Roland Griesmaier

Facts about far fields

The far field radiated by a source f is its restricted Fourier transform: α = f |S1 Translations and Fourier transforms:

• f (· + c)(θ) = eic·θ

f (θ) , θ ∈ S1 , c ∈ R 2 , i.e., if f radiates α(θ), then f (· + c) radiates eic·θα(θ) Far field translation operator: Tc : L2(S1) → L2(S1) , (Tcα)(θ) := eic·θα(θ) Note that T ∗

c = T−c

Uncertainty principles Roland Griesmaier

A regularized Picard criterion

Uncertainty principles Roland Griesmaier

SVD of the restricted Fourier transform

Consider restriction of F to sources supported in BR(0): FBR(0) : L2(BR(0)) → L2(S1) , FBR (0)f := f

• S1

Singular value decomposition: (FBR(0)f )(θ) =

n

√ 2πsn(R)

• f (x), inJn(|x|)einϕx

sn(R)

• einθ

√ 2π

where s2

n(R) = BR(0) J2 n(x) dx

Asymptotically: limR→∞

s2

νR(R)

2R

=

• 1 − ν2

ν ≤ 1 ν ≥ 1 i.e., s2

n(R) ∼

• 2
• R2 − n2

n R n R

n 5 10 15 20 25 −10 10

R = 10

Uncertainty principles Roland Griesmaier

SVD of the restricted Fourier transform

Consider restriction of F to sources supported in BR(0): FBR(0) : L2(BR(0)) → L2(S1) , FBR (0)f := f

• S1

Singular value decomposition: (FBR(0)f )(θ) =

n

√ 2πsn(R)

• f (x), inJn(|x|)einϕx

sn(R)

• einθ

√ 2π

where s2

n(R) = BR(0) J2 n(x) dx

Asymptotically: limR→∞

s2

νR(R)

2R

=

• 1 − ν2

ν ≤ 1 ν ≥ 1 i.e., s2

n(R) ∼

• 2
• R2 − n2

n R n R

n 50 100 150 200 250 −100 100

R = 100

Uncertainty principles Roland Griesmaier

The Picard criterion

Fourier expansion of the far field: α(θ) =

n αn einθ √ 2π ,

θ ∈ S1 Radiated power of the far field: ∥α∥2

L2(S1) = n |αn|2

Picard criterion: α ∈ R(FBR(0)) ⇐ ⇒

1 2π

• n

|αn|2 s2

n (R) < ∞

Minimal power source: f ∗

α(x) = 1 √ 2π

• n

αn sn(R)2 inJn(|x|)einϕx ,

x ∈ BR(0) Input power required to radiate the far field: ∥f ∗

α∥2 L2(BR(0)) = 1 2π

• n

|αn|2 s2

n (R) Uncertainty principles Roland Griesmaier

A regularized Picard criterion

Picard criterion: α ∈ R(FBR(0)) ⇐ ⇒

1 2π

• n

|αn|2 s2

n (R) < ∞

Input power required to radiate the far field: ∥f ∗

α∥2 L2(BR (0))

Radiated power of the far field: ∥α∥2

L2(S1)

Regularizing assumptions: Not every source/farfield combination is equally relevant! physical sources have limited power P > 0 a receiver has a power threshold p > 0 Define N(R, P, p) := sup

2πs2

n (R)≥ p P

n The space of non-evanescent far fields is given by: VNE :=

• α ∈ L2(S1)
• α(θ) = N

n=−N αneinθ

For a wide range of p and P: N R

Uncertainty principles Roland Griesmaier

Questions

Uncertainty principles Roland Griesmaier

Far field splitting and data completion

−30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 −30 −20 −10 10 20 30

Far field splitting: Suppose γ = γ1 + · · · + γm , γj is radiated from Brj (cj ) i.e., γ = T ∗

c1α1 + · · · + T ∗ cmαm ,

αj is radiated from Brj (0) Can we stably recover the non-evanescent part of γ1, . . . , γm ? Data completion: Suppose we cannot measure γ on a subset Ω ⊂ S1, we measure

• γ = γ + β ,

β = −γ|Ω Can we stably recover the non-evanescent part of γ on Ω ?

Uncertainty principles Roland Griesmaier

Uncertainty principles

Uncertainty principles Roland Griesmaier

Far field translation

Translation of the far field: The far field translation operator Tc : L2(S1) → L2(S1) , (Tcα)(θ) := eic·θα(θ) acts on the Fourier coefficients {αn} of α as a convolution operator Tc : ℓ2 → ℓ2 , (Tc{αn})m =

n αm−n

inJn(|c|)einϕc We have estimates ∥Tc∥Lp,Lp = 1 and ∥Tc∥ℓ1,ℓ∞ ≤

1 |c|1/3

Uncertainty principles Roland Griesmaier

Uncertainty principles for far field translation

∥Tc∥Lp,Lp = 1 and ∥Tc∥ℓ1,ℓ∞ ≤

1 |c|1/3

Theorem: Let α, β ∈ L2(S1) and let c ∈ R 2. Then |⟨Tcα, β⟩| ≤

• ∥α∥ℓ0∥β∥ℓ0

|c|1/3 ∥α∥2∥β∥2 Proof: |⟨Tcα, β⟩| ≤ ∥Tcα∥ℓ∞∥β∥ℓ1 ≤ 1 |c|1/3 ∥α∥ℓ1∥β∥ℓ1 ≤ 1 |c|1/3

• ∥α∥ℓ0∥α∥2
• ∥β∥ℓ0∥β∥2
• Uncertainty principles

Roland Griesmaier

Uncertainty principles for far field translation

Assuming that the supports of the individual source components are well-separated, we can improve the first estimate: ∥Tc∥ℓ1[−N,N],ℓ∞[−M,M] ≤ 1 |c|1/2 if |c| > 2(M + N + 1) Theorem: Suppose that α ∈ ℓ2(−M, M), β ∈ ℓ2(−N, N) with M, N ≥ 1 and let c ∈ R 2 such that |c| > 2(M + N + 1) Then |⟨Tcα, β⟩| ≤

• (2N + 1)(2M + 1)

|c|1/2 ∥α∥2∥β∥2

Uncertainty principles Roland Griesmaier

Uncertainty principle for data completion

∥Tc∥Lp,Lp = 1 and ∥Tc∥ℓ1,ℓ∞ ≤

1 |c|1/3

Theorem: Let α, β ∈ L2(S1) and let c ∈ R 2. Then |⟨Tcα, β⟩| ≤

• ∥α∥ℓ0∥β∥L0

2π ∥α∥2∥β∥2 Proof: |⟨Tcα, β⟩| ≤ ∥Tcα∥L∞∥β∥L1 ≤ ∥α∥L∞∥β∥L1 ≤ 1 √ 2π ∥α∥ℓ1∥β∥L1 ≤ 1 √ 2π

• ∥α∥ℓ0∥α∥2
• ∥β∥L0∥β∥2
• Uncertainty principles

Roland Griesmaier

ℓ2 corollaries of the uncertainty principle

Uncertainty principles Roland Griesmaier

Stability of far field splitting by least squares

Theorem: Suppose that γ0, γ1 ∈ L2(S1), c1, c2 ∈ R 2 and N1, N2 ∈ N such that |c1 − c2| > 2(N1 + N2 + 1) and (2N1 + 1)(2N2 + 1) |c1 − c2| < 1 and let γ0 LS = T ∗

c1α0 1 + T ∗ c2α0 2 ,

α0

i ∈ ℓ2(−Ni, Ni)

γ1 LS = T ∗

c1α1 1 + T ∗ c2α1 2 ,

α1

i ∈ ℓ2(−Ni, Ni)

Then, for i = 1, 2 ∥α1

i − α0 i ∥2 2 ≤

• 1 − (2N1 + 1)(2N2 + 1)

|c1 − c2| −1 ∥γ1 − γ0∥2

2

Uncertainty principles Roland Griesmaier

Stability of data completion by least squares

Theorem: Suppose that γ0, γ1 ∈ L2(S1), c ∈ R 2, N ∈ N and Ω ⊂ S1 such that (2N + 1)|Ω| 2π < 1 and let γ0 LS = β0 + T ∗

c α0 ,

α0 ∈ ℓ2(−N, N) and β0 ∈ L2(Ω) γ1 LS = β1 + T ∗

c α1 ,

α1 ∈ ℓ2(−N, N) and β1 ∈ L2(Ω) Then ∥α1 − α0∥2

2 ≤

• 1 − (2N + 1)|Ω|

2π −1 ∥γ1 − γ0∥2

2

and ∥β1 − β0∥2

2 ≤

• 1 − (2N + 1)|Ω|

2π −1 ∥γ1 − γ0∥2

2

Uncertainty principles Roland Griesmaier

ℓ1 corollaries of the uncertainty principle

Uncertainty principles Roland Griesmaier

Stability of far field splitting by basis pursuit

Theorem: Suppose that γ0, α0

1, α0 2 ∈ L2(S1) and c1, c2 ∈ R 2 such that 4∥α0

i ∥ℓ0

|c1−c2|1/3 < 1

for i = 1, 2 and ∥γ0 − T ∗

c1α0 1 − T ∗ c2α0 2∥2 ≤ δ0

for some δ0 ≥ 0 If δ ≥ 0 and γ ∈ L2(S1) with δ ≥ δ0 + ∥γ − γ0∥2 and (α1, α2) = argmin ∥α1∥ℓ1 + ∥α2∥ℓ1 s.t. ∥γ − T ∗

c1α1 − T ∗ c2α2∥2 ≤ δ , α1, α2 ∈ L2(S1) ,

then, for i = 1, 2 ∥α0

i − αi∥2 2 ≤

• 1 −

4∥α0

i ∥ℓ0

|c1−c2|1/3

−1 4δ2

Uncertainty principles Roland Griesmaier

Stability of data completion by basis pursuit

Theorem: Suppose that γ0, α0 ∈ L2(S1), Ω ⊂ S1, β0 ∈ L2(Ω) and c ∈ R 2 such that

2 π ∥α0∥ℓ0|Ω| < 1

and ∥γ0 − T ∗

c α0 − β0∥2 ≤ δ0

for some δ0 ≥ 0 If δ ≥ 0 and γ ∈ L2(S1) with δ ≥ δ0 + ∥γ − γ0∥2 and α = argmin ∥α∥ℓ1 s.t. ∥γ − β − T ∗

c α∥2 ≤ δ , α ∈ L2(S1) , β ∈ L2(Ω)

then ∥α0 − α∥2

2 ≤

• 1 −

2∥α0∥ℓ0 |Ω| π

−1 4δ2 and ∥β0 − β∥2

2 ≤

• 1 −

2∥α0∥ℓ0 |Ω| π

−1 4δ2

Uncertainty principles Roland Griesmaier

Stability of data completion by basis pursuit

Corollary: Suppose that γ0, α0 ∈ L2(S1), Ω ⊂ S1, β0 ∈ L2(Ω) and c ∈ R 2 such that

4 √ 2π 1 τ 2 ∥α0∥ℓ0 < 1

and

4 √ 2π τ 2|Ω| < 1

for some τ > 0 and ∥γ0 − T ∗

c α0 − β0∥2 ≤ δ0

for some δ0 ≥ 0 If δ ≥ 0 and γ ∈ L2(S1) with δ ≥ δ0 + ∥γ − γ0∥2 and (α, β) = argmin 1

τ ∥α∥ℓ1 + τ∥β∥L1(S1) s.t. ∥γ − T ∗ c α − β∥2 ≤ δ , α, β ∈ L2(S1)

then ∥α0 − α∥2

2 ≤

• 1 −

4 √ 2π 1 τ 2 ∥α0∥ℓ0

−1 4δ2 and ∥β0 − β∥2

2 ≤

• 1 −

4 √ 2π τ 2|Ω|

−1 4δ2

Uncertainty principles Roland Griesmaier

Numerical examples

Uncertainty principles Roland Griesmaier

The setup

Geometry and a priori information −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 Exact farfield −6 −4 −2 2 4 6 π/2 π 3π/2 2π

Wavenumber: k = 1

Uncertainty principles Roland Griesmaier

Data completion using the ℓ2 approach

Observed far field data:

• γ = γ|S1\Ω,
• γ = β + 3

i=1 T ∗ ci αi ,

β = −γ|Ω Galerkin condition: ⟨β + T ∗

c1α1 + · · · + T ∗ c3α3, φ⟩ = ⟨

γ, φ⟩ for all φ ∈ L2(Ω) ⊕ T ∗

c1ℓ2(−N1, N1) ⊕ · · · ⊕ T ∗ c3ℓ2(−N3, N3)

Observed farfield −6 −4 −2 2 4 6 π/2 π 3π/2 2π Reconstructed missing data −6 −4 −2 2 4 6 π/2 π 3π/2 2π Absolute error −6 −4 −2 2 4 6 π/2 π 3π/2 2π

Condition number: 5.4 × 104

Uncertainty principles Roland Griesmaier

The setup

Geometry and a priori information −30 −20 −10 10 20 30 −30 −20 −10 10 20 30 Exact farfield −6 −4 −2 2 4 6 π/2 π 3π/2 2π

Wavenumber: k = 1

Uncertainty principles Roland Griesmaier

Data completion using the ℓ1 approach

Observed far field data:

• γ = γ|S1\Ω,
• γ = β + 3

i=1 T ∗ ci αi ,

β = −γ|Ω Tikhonov functional: Ψµ(α1, α2, α3) = ∥ γ − (I − PΩ)(

3

• i=1

T ∗

ci αi )∥2 ℓ2 + µ 3

• i=1

∥αi∥ℓ1 with [α1, α2, α3] ∈ ℓ2 × ℓ2 × ℓ2

Observed farfield −6 −4 −2 2 4 6 π/2 π 3π/2 2π Reconstructed missing data −6 −4 −2 2 4 6 π/2 π 3π/2 2π Absolute error −6 −4 −2 2 4 6 π/2 π 3π/2 2π

Apply (fast) iterated soft shrinkage ((F)ISTA) to minimize Ψµ; here µ = 10−3

Uncertainty principles Roland Griesmaier

Thanks for listening!

Uncertainty principles Roland Griesmaier