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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works Increasing stability in the inverse source problem with attenuation and many frequencies Shuai Lu (Fudan University) Joint work

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Increasing stability in the inverse source problem with attenuation and many frequencies

Shuai Lu (Fudan University)

Joint work with Gang Bao, Jin Cheng, Victor Isakov and William Rundell IAS Workshop on Inverse Problems, Imaging and Partial Differential Equations May 20–24, 2019 – The Hong Kong University of Science and Technology

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Outline

1

Introduction Related work Non-radiating source Helmholtz equation Main result

2

Increasing stability with attenuation Analytic functions Exact observability bounds Proof of the main theorem

3

Numerical algorithm for source identification Regularization method Numerical examples

4

Ongoing works

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Outline

1

Introduction Related work Non-radiating source Helmholtz equation Main result

2

Increasing stability with attenuation Analytic functions Exact observability bounds Proof of the main theorem

3

Numerical algorithm for source identification Regularization method Numerical examples

4

Ongoing works

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Related work

Multifrequency (highfrequency) regimes Inverse medium problems Chen & Rokhlin (97); Chen (97a, 97b); Bao, Chen &

Ma(00); Bao & Liu (03), Bao & Li (07); Bao & Triki (10); Nagayasu, Uhlmann & Wang (13); Bao & Triki (19)...

Inverse obstacle/interface problems Coifman, Goldberg, Hrycak & Rokhlin

(99); Bao, Hou & Li (07); Bao & Lin (10, 11); Sini & Th` anh (12); Borges & Greengard (15); Rondi & Sini (15)...

Inverse source problems Bao, Lin & Triki (10, 11); Bao, Lu, Rundell & Xu;

Cheng, Isakov & Lu (16); Li & Yuan (17); & Isakov & Lu (18)...

Bao, Gang; Li, Peijun; Lin, Junshan; Triki, Faouzi Inverse scattering problems with multi-frequencies. Inverse Problems 31 (2015), no. 9, 093001, 21 pp.

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Related work

Multifrequency (highfrequency) regimes Continuation problems Subbarayappa & Isakov (07, 10); Isakov & Kindermann

(11); Isakov (14)...

Schr¨

  • dinger problems Isakov (11); Isakov, Nagayasu, Uhlmann & Wang (14);

Isakov & Wang (14); Isakov, Lai & Wang (16); Isakov, Lu & Xu (19)...

Multifrequency electric impedance tomography Ammari & Triki (17);

Cheng, Choulli & Lu (19)...

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Types of measurements

Frequency sweep:

Inverse medium problems; inverse obstacle/interface problems; inverse source problems; multifrequency electric impedance tomography

Dirichlet-to-Neumann map at a fixed high wavenumber

Inverse medium problems; Schr¨

  • dinger problems

Cauchy data at a fixed high wavenumber

Continuation problems

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Types of measurements

Frequency sweep:

Inverse medium problems; inverse obstacle/interface problems; inverse source problems; multifrequency electric impedance tomography

Dirichlet-to-Neumann map at a fixed high wavenumber

Inverse medium problems; Schr¨

  • dinger problems

Cauchy data at a fixed high wavenumber

Continuation problems

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Forward source problems

Helmholtz equation with inner source terms Radiated field u(x;k) satisfies Helmholtz equation and Sommerfeld radiation condition ∆u(x;k)+k2u(x;k) = −f(x), x ∈ Rd lim

rx→∞r

d−1 2

x

∂u(x,k) ∂rx −iku(x,k)

  • = 0

where k 0 is the wavenumber. We assume f(x) ∈ L2(Rd) and compactly supported in a bounded open domain Ω. Outgoing radiated field u(x;k) is generated by the source function f(x) and measured at ∂Ω.

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Forward map

∂Ω

Definition: Forward operator Lk If we assume f †(x) ∈ L2(Ω) satisfying suppf † ⊂⊂ Ω , (Lkf †)(x) = u(x;k)|∂Ω = g(x;k) =

  • Ω f †(y)Φ(x,y;k)dy,

x ∈ ∂Ω where Φ(x,y;k) is the fundamental solution. Lk : L2(Ω) → H

1 2 (∂Ω).

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Non-radiating source

Non-radiating source at a wavenumber k The outgoing radiated field u(x;k) ∈ H1(Ω) corresponding to the non-zero source f(x) ∈ L2(Ω) satisfies u(x;k)|∂Ω = 0. We call f(x) the non-radiating source for the wavenumber k. For instance ∅ =

  • f(x) ∈ L2(Ω) | f(x) = −∆w(x)−k2w(x), w(x) ∈ C∞

0 (Ω)

  • .

Bleistein and Cohen [J. Math. Phy. 1977]; Kim and Wolf [Optics Comm. 1986]; Marengo and Ziolkowski [PRL 1999].

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Spherical Fourier transform

Spherical Fourier basis

Let Ω = Bρ in R3 with ρ > 0. Choose ψnmℓ(x) = (Nnℓ)−1/2 jn(ζnℓrx)Ym

n (ˆ

x) with ζnℓ = znl

ρ , ℓ = 1,2,..., znℓ is the ℓ-th positive zero of the spherical

Bessel function jn(r), Nnℓ = ρ3

2

d

drjn(ζnℓρ)

2, rx = |x| and ˆ x = x

|x|.

Spherical Fourier transform

For all f ∈ L2(Ω), anmℓ =

  • Ω f(y)ψnmℓ(y)dy.

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Spherical Fourier transform

Notice the fact that f(x) =

∞

∑

n=0 ∞

∑

ℓ=1 n

∑

m=−n

anmℓψnmℓ(x), Φ(x,y;k) = eik|x−y| 4π|x−y| = ik

∞

∑

n=0 n

∑

m=−n

h(1)

n (krx)Ym n (ˆ

x)jn(kry)Ym

n (ˆ

y). by choosing k = ζnℓ, we obtain (Lkψn′m′ℓ′)(x) =

∞

∑

n=0 n

∑

m=−n

ikh(1)

n (krx)Ym n (ˆ

x)

  • Ω ψn′m′ℓ′(y)jn(kry)Ym

n (ˆ

y)dy

  • =

∞

∑

n=0 n

∑

m=−n

ikh(1)

n (krx)Ym n (ˆ

x)

  • NS

n′ℓ′

1

2 δn=n′,m=m′δk=ζn′ℓ′

= ikh(1)

n′ (krx)Ym′ n′ (ˆ

x)

  • NS

n′ℓ′

1

2 δk=ζn′ℓ′,

x ∈ ∂Ω.

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Singular value decomposition (SVD)

SVD of the forward operator Lk Assume Ω = Bρ in R3, the singular value system of the forward

  • perator Lk : L2(Ω) → H

1 2 (∂Ω) is {σnmℓ,ψnmℓ,φnmℓ} when k = ζnℓ.

Here ψnmℓ(x) = (Nnℓ)−1/2 jn(ζnℓrx)Ym

n (ˆ

x) with x ∈ Ω. Furthermore, for all ℓ = 1,2,···, φnmℓ(x) = Ym

n (ˆ

x) with x = ρˆ x, and for all m = 0,±1,··· ,±n, n = 0,1,···, the singular values σnmℓ are σnmℓ(k) =

  • ikh(1)

n (kρ)

  • (Nnℓ)

1 2 δk=ζnℓ 0. 5 10 15 20

σnml

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Inverse source problems

Ideal inverse source problems Identify the source function f(x) from the measured data (Dirichlet, Neumann, Cauchy) on the boundary ∂Ω for wavenumbers k ∈ (0,+∞). Real situation Identify the source function f(x) from the measured data (Dirichlet, Neumann, Cauchy) on the boundary ∂Ω for wavenumbers k ∈ (0,K), K >> 0.

Hoenders and Ferwerda [PRL 2001]; Devaney et. al [SIAP 2007]; Bao, Lin and Triki [JDE 2010].

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Inverse source problems

Ideal inverse source problems Identify the source function f(x) from the measured data (Dirichlet, Neumann, Cauchy) on the boundary ∂Ω for wavenumbers k ∈ (0,+∞). Real situation Identify the source function f(x) from the measured data (Dirichlet, Neumann, Cauchy) on the boundary ∂Ω for wavenumbers k ∈ (0,K), K >> 0.

Hoenders and Ferwerda [PRL 2001]; Devaney et. al [SIAP 2007]; Bao, Lin and Triki [JDE 2010].

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Helmholtz equation

Helmholtz equation with attenuation Radiated field u(x;k) satisfies Helmholtz equation (∆+k2 +ikb)u = −f1 −bf0 +ikf0 inR3, 0 < k < K, with the exponential decay at infinity: |u(x)|+|∇u(x)| ≤ C(u)e−δ(u)|x| and some δ(u) > 0. We assume f0, f1 ∈ H3(R3) and compactly supported in a bounded

  • pen domain Ω.

For special case of b = 0, we refer to

[J. Cheng, V. Isakov and S. Lu, 16]

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Main theorem

There exists a generic constant C only depending on the domain Ω such that f02

(1)(Ω)+f12 (0)(Ω) ≤ CeCb2

  • ε2 +

M2

3

1+K

4 3 |E| 1 2

  • for all u ∈ H2(Ω) solving the above Helmholtz equation, with 1 < K.

Here ε2 =

K

  • ω2u(,ω)2

(0)(∂Ω)+∇u(,ω)2 (0)(∂Ω)

  • dω,

E = −lnε and M3 = max

  • f0(3)(Ω)+f1(3)(Ω),1
  • where ·(ℓ)(Ω) is

the standard Sobolev norm in Hℓ(Ω).

[V. Isakov and S. Lu, 18]

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Outline

1

Introduction Related work Non-radiating source Helmholtz equation Main result

2

Increasing stability with attenuation Analytic functions Exact observability bounds Proof of the main theorem

3

Numerical algorithm for source identification Regularization method Numerical examples

4

Ongoing works

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Solution of the Helmholtz equation u∗(x,κ) := u(x,k), κ := k

  • 1+ b

k i, then the Helmholtz equation becomes ∆u∗ +κ2u∗ = −f1 −bf0 +ikf0 with a solution u∗(x,κ) = 1 4π

  • Ω(f1 +bf0 −ikf0)(y)eiκ|x−y|

|x−y| dy which exponentially decays for large |x| because the imaginary part Im(κ) > 0.

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Frequency domain = ⇒ Time domain

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Wave equation

Wave equation The hyperbolic initial value problem ∂ 2

t U −∆U +b∂tU = 0 on Ω×(0,+∞),

U(x,0) = f0, ∂tU(x,0) = f1 on Ω. Fourier transform u(x,k) = 1 √ 2π

+∞

−∞ U(x,t)eiktdt.

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Why time domain?

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Exact observability bound

Exact observability bound Let the observation time 4(D+1) < T < 5(D+1). Then there exists a generic constant C depending on the domain Ω such that

f02

(1)(Ω)+f12 (0)(Ω) ≤ CeCb2

∂tU2

(0)(∂Ω×(0,T))+∇U2 (0)(∂Ω×(0,T))

  • for all U ∈ H2(Ω×(0,+∞)) solving the hyperbolic initial problem.

Proof based on the Carleman estimate. Klibanov & Malinsky 1991; Tataru 1995.

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The main mathematical problem!

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Equivalency of the boundary data ∂tU2

(0)(∂Ω×(0,T)) = 2

∞

0 ω2

  • ∂Ω u(·,ω)u(·,−ω)dΓdω;

∇U2

(0)(∂Ω×(0,T)) = 2

∞

  • ∂Ω ∇u(·,ω)·∇u(·,−ω)dΓdω.

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Analytic functions

Measurement ε2 = I0(K)+I1(K). Analytic functions I0(k) := 2

k

0 ω2

  • ∂Ω u(·,ω)u(·,−ω)dΓdω,

I1(k) := 2

k

  • ∂Ω ∇u(·,ω)·∇u(·,−ω)dΓdω,

which are analytic functions with respect to the wave number k ∈ C\[−bi,0].

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Harmonic measure

Boundness of both values Let suppf0,suppf1 ⊂ Ω and f1 ∈ H1(Ω), f0 ∈ H1(Ω). S is the sector

  • |argk| < π

4

  • r {|k2| < k1} with k = k1 +k2i. Then, in S, following

bounds holds true

|I0(k)| ≤ 24π|∂Ω|D 1 3|k|3f12

(0)(Ω)+

1 3|k|3b2 + 1 5|k|5

  • f02

(0)(Ω)

  • eD(4k1+b),

|I1(k)| ≤ 24π|∂Ω|D

  • |k|f12

(1)(Ω)+

  • |k|b2 + 1

3|k|3

  • f02

(1)(Ω)

  • eD(4k1+b),

where |∂Ω| is the area of ∂Ω and D = sup|x−y| over x,y ∈ Ω.

K Known data [0, K] Unknown data [K, ∞)

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Harmonic measure

  • I0(k)e−(4D+1)k
  • ≤ 24π|∂Ω|DeDb
  • 2

√ 2 3 k3

1f12 (0)(Ω)+

  • 2

√ 2 3 k2

1b2 + 4

√ 2 5 k5

1

  • f02

(0)(Ω)

  • e−k1

≤ Cb2eDbM2

0,

with M0 = max

  • f0(0)(Ω)+f1(0)(Ω),1
  • . Noticing that
  • I0(k)e−(4D+1)k
  • ≤ ε2 on [0,K],

we conclude that

  • I0(k)e−(4D+1)k
  • ≤ Cb2eDbε2µ(k)M2

0,

when K < k < +∞

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Harmonic measure

Lower bound of the Harmonic measure Let S be the sector {k : − π

4 < argk < π 4 } and µ(k) be the harmonic

measure of the interval [0,K] in S\[0,K]. If 0 < k < 2

1 4 K, µ(k) > 1

2.

If 2

1 4 K < k, µ(k) > 1

π

k

K

4 −1 − 1

2 .

K Known data [0, K] Unknown data [K, ∞)

[J. Cheng, V. Isakov and S. Lu, 16]

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Auxiliary equalities

Auxiliary equalities

∞

−∞ ω2u(·,ω)2 (0)(∂Ω)dω = I0(k)+

  • k<|ω| ω2u(·,ω)2

(0)(∂Ω)dω,

∞

−∞ ∇u(·,ω)2 (0)(∂Ω)dω = I1(k)+

  • k<|ω| ∇u(·,ω)2

(0)(∂Ω)dω.

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Auxiliary lemma

Boundness of the high frequency modes Let u be a solution to the Helmholtz equation with f1 ∈ H3(Ω) and f0 ∈ H3(Ω), suppf0,suppf1 ⊂ Ω. Then

  • k<|ω| ω2u(·,ω)2

(0)(∂Ω)dω +

  • k<|ω| ∇u(·,ω)2

(0)(∂Ω)

≤ Cb5k−2 f02

(3)(Ω)+f12 (3)(Ω)

  • .

Proof based on the decay property of the Cauchy problem for a semi-linear wave equation. Kawashima et. al 1995.

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Increasing stability

Increasing stability theorem

There exists a generic constant C only depending on the domain Ω such that f02

(1)(Ω)+f12 (0)(Ω) ≤ CeCb2

  • ε2 +

M2

3

1+K

4 3 |E| 1 2

  • for all u ∈ H2(Ω) solving the above Helmholtz equation, with 1 < K. Here

ε2 =

K

  • ω2u(,ω)2

(0)(∂Ω)+∇u(,ω)2 (0)(∂Ω)

  • dω,

E = −lnε and M3 = max

  • f0(3)(Ω)+f1(3)(Ω),1
  • where ·(ℓ)(Ω) is the

standard Sobolev norm in Hℓ(Ω).

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Outline

1

Introduction Related work Non-radiating source Helmholtz equation Main result

2

Increasing stability with attenuation Analytic functions Exact observability bounds Proof of the main theorem

3

Numerical algorithm for source identification Regularization method Numerical examples

4

Ongoing works

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Regularization method

Radiated field u(x;k) satisfies Helmholtz equation and Sommerfeld radiation condition (∆+k2 +ikb)u = −f inR3, with f ∈ L2(Ω) compactly supported in Ω. Inverse source problems Identify the source f from the near field data u = u0, ∂νu = u1 on ∂Ω, with 1 < k < K.

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Regularization method

Algorithm 1: Recursive Kaczmarz-Landweber Iteration Input: Wave number set K with J discrete wave numbers kj ∈ [1,K], j = {1,2,...,J}; measurement data

  • uκj,

∂uκj ∂ν

  • |Γ for all κj =
  • k2

j +ikjb with kj ∈ K .

Output: Reconstructed source function f J

2N with J & 2N denoting the Kaczmarz & Landweber

iteration number. 1: Set the initial guess f 0

2N = 0, j,n = 0.

2: For j = 1,2,...,J (Kaczmarz iteration) 3: k = kj; f j

0 = f j−1 2N ;

4: For n=1,2,. . . N (Landweber iteration for Dirichlet & Neumann measurement) 5: f j

n = f j n−1 + µ(L1 κ)∗(uκ −L1 κf j n−1),

f j

n = f j n−1 + µ(L2 κ)∗ ∂uκ ∂ν −L2 κf j n−1

  • ;

6: n = n+1; 7: End 8: End. [G. Bao, S. Lu, W. Rundell and B. Xu, 15]

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Convergence

Convergence theorem Let the exact source function f † ∈ Hτ

0(Ω) with τ > d

  • 2. Then for a

sufficiently large wavenumber interval, given any ε > 0, there exist finite iterations such that the final iterate in the recursive algorithm approximates the exact solution f i

j −f † ≤ ε.

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Error estimate

Error estimate theorem Let the exact source function f † ∈ Hτ

0(Ω) with τ > d 2 and δ be the

noise level at each wavenumber. The stopping criteria for the recursive algorithm is well-defined as the smallest integer jDP with Lkf i,δ

jDP −gδ k ≤ rδ, r > 1. Then following error estimate holds true

f i,δ

jDP −f † ≤ Cδ

τ τ+1 +ε

for near field datum.

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Exact Solution

f(x,y,z) =1.1e−200((y−0.12)2+(x−0.01)2+(z−0.00)2) +0.5e−180((y−0.01)2+(x+0.01)2+(z−0.00)2) +2e−180((y−0.21)2+(x−0.11)2+(z−0.00)2) +0.8e−180((y+0.05)2+(x+0.05)2+(z−0.00)2) +0.6e−160((y−0.12)2+(x−0.04)2+(z−0.12)2) +0.7e−170((y−0.02)2+(x−0.06)2+(z−0.07)2) +0.8e−160((y+0.12)2+(x+0.04)2+(z+0.12)2) +0.7e−190((y+0.02)2+(x+0.06)2+(z+0.07)2) +1e−180((y−0.06)2+(x−0.08)2+(z−0.15)2) +1.1e−190((y−0.09)2+(x−0.1)2+(z−0.03)2) +0.6e−200((y+0.06)2+(x+0.08)2+(z+0.15)2) +1.3e−210((y+0.09)2+(x+0.1)2+(z+0.03)2) +1.2e−220((y−0.13)2+(x+0.02)2+(z+0.20)2) +0.5e−230((y+0.02)2+(x−0.08)2+(z−0.2)2) +0.7e−150((y−0.21)2+(x−0.04)2+(z−0.03)2) +0.9e−200((y−0.05)2+(x+0.06)2+(z+0.09)2) +0.9e−250((y+0.02)2+(x+0.16)2+(z+0.14)2) +1.5e−240((y−0.01)2+(x−0.11)2+(z−0.14)2).

Shuai Lu (Fudan University) Increasing stability in the inverse source problem 37 / 45

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Approximate Solution: K = 200 and exact measurement

−0.2 0.2 −0.2 0.2 0.5 1 1.5 2 Exact source: slice at z=0 (b=1) −0.2 0.2 −0.2 0.2 0.5 1 1.5 2 Approximate source: slice at z=0 (b=1) 50 100 150 200 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 Error slope for exact data (b=1) Logscale of relative error Frequency

Figure: (b = 1) Left: exact source: slide of z = 0. Middle: approximate source: slide of z = 0 for exact data. Right: error slope for relative error.

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Approximate Solution: K = 200 and noisy measurement

−0.2 0.2 −0.2 0.2 0.5 1 1.5 2 Approximate source for 0.01 noise level (b=1) −0.2 0.2 −0.2 0.2 0.5 1 1.5 2 Approximate source for 0.10 noise level (b=1) 50 100 150 200 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 Error slope for 0.01, 0.10 noise level (b=1) Logscale of relative error Frequency 0.01 noise level 0.10 noise level

Figure: (b = 1) Left: approximate source: slide of z = 0 for 0.01 noise level. Middle: approximate source: slide of z = 0 for 0.1 noise level. Right: error slope for relative error for both noises.

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Different choices of b b value Relative error for exact data b = 0.1 0.0307 b = 1 0.0373 b = 5 0.0974 b = 10 0.2451 b = 15 0.5866 b = 20 0.9118 b = 30 0.9492

5 10 15 20 25 30 2 4 6 8 10 12 14 Fitting of absolute error with discrete b b Absolute error Absolute error with discrete b The fitting line of e−1.0049+0.2711b −0.0052 b

2

Table: Left: Choices of b and it relative error for exact data. Right: Fitting function ec1+c2b+c3b2 with c1 = −1.0049, c2 = 0.2711 and c3 = −0.0052 for the absolute error.

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Outline

1

Introduction Related work Non-radiating source Helmholtz equation Main result

2

Increasing stability with attenuation Analytic functions Exact observability bounds Proof of the main theorem

3

Numerical algorithm for source identification Regularization method Numerical examples

4

Ongoing works

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Ongoing work

Ongoing work: Linearized inverse Schr¨

  • dinger potential problem

Recover the potential function c(x) from Dirichlet-to-Neumann map

  • −∆u−(k2 −c)u = 0

in Ω ⊂ Rn, u = g0

  • n ∂Ω,

Bounded exponential functions; Infinite measurement at a fixed (large) wavenumber (DtN map); Numerical reconstruction algorithm based on the linearization.

Isakov (11); Isakov, Nagayasu, Uhlmann & Wang (14); Isakov & Wang (14); Isakov, Lai & Wang (16); Isakov, Lu & Xu (19, submitted)

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Ongoing work

Ongoing work: Linearized inverse Schr¨

  • dinger potential problem

Linearized inverse Schr¨

  • dinger potential problem:

(I0)

  • −∆u0 −k2u0 = 0

in Ω, u0 = g0

  • n ∂Ω,

(1a) (I1)

  • −∆u1 −k2u1 = −cu0

in Ω, u1 = 0

  • n ∂Ω.

(1b) H¨

  • lder type stability!!

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Ongoing work

Ongoing: stochastic source Parameter identification of stochastic differential equations ∆u+k2u = f(x)+g(x) ˙ Wx; Uncertainty quantification & Data assimilation       

  • u

u′ ′ =

  • 1

−k2

  • u

u′

  • +
  • f(x)
  • +
  • g(x) ˙

Wx

  • v = [1

0]

  • u

u′

  • .

Fluctuation dissipation theorem (linear response)

Li (11); Bao, Chow, Li and Zhou (14); Iglesias, Law and Stuart (13); Branicki and Majda (14); Iglesias, Lin, Lu and Stuart (17); Ding, Lu and Cheng (18); Niu, Lu and Cheng (19).

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Introduction Increasing stability with attenuation Numerical algorithm for source identification Ongoing works

Thank you for your attention.

Shuai Lu School of Mathematical Sciences, Fudan University Email: slu@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/shuailu/

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