Inverse Source Problems for Wave Propagation

Peijun Li

Department of Mathematics

Purdue University

Joint Work with

G. Bao, C. Chen, G. Yuan, Y. Zhao

Outline

Motivation and model problems

Inverse random source

Increasing stability

Ongoing and future work

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Source scattering problems

Source scattering problems are concerned with the relationship betweenradiating sources and wave fields.

Direct problem: To determine the wave field from the given sourceand the differential equation governing the wave motion.

Inverse problem: To determine the radiating source which producesthe measured wave field.

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Wave equations for source scattering

The Helmholtz equation - acoustic wave

∆u + κ2u = f in Rd .

The Navier equation - elastic wave

µ∆u + (λ+ µ)∇∇ · u + ω2u = f in Rd .

The Maxwell equations - electromagnetic wave

∇× E − iωµH = 0, ∇×H + iωεE = J in R3.

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Problem geometry

ΓR

source

Ω

BR

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Applications

Antenna synthesis

Tomography (PAT)

Medical imaging (MEG, EEG, ENG)

Fluorescence microscopy

Neuroscience (brain imaging)

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Related work

Deterministic problems

Devaney and Sherman (’82), Marengo and Devaney (’99), Ammari,Bao, and Fleming (’02), Fokas, Kurylev, and Marinakis (’04), Hauer,Kuhn, Potthast (’05), Albanese and Monk (’06), Devaney, Marengo,and Li (’07), Eller and Valdivia (’09), Bao, Lin, and Triki (’10), Badiaand Nara (’11), Tittelfitz (’15), Bao, Lu, Rundell, and Xu (’15),Zhang and Guo (’15), Bao, L., Lin, and Triki (’15), Cheng, Isakov,and Lu (’16)

Stochastic problems

Devaney (’79), L. (’11), Bao and Xu (’13), Bao, Chow, L., and Zhou(’14), Bao, Chen, and L. (’16)

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Inverse random source problem

Homogeneous media

Inhomogeneous media

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Why stochastic modeling?

Uncertainties are widely introduced to the mathematical models for threemajor reasons:

Randomness directly appears in the studied systems

Incomplete knowledge of the systems may be modeled by uncertainties

Stochastic techniques are introduced to couple the interferencebetween different scales

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Available framework

Homogenization theory: J. Keller, J. Lions, G. Papanicolaou

Bayesian statistics: D. Donoho, A. Stuart, E. Somersalo

Wiener chaos expansion: R. Cameron, W. Martin, G. Karniadakis

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Model problem

The Helmholtz equation

∆u(x , κ) + κ2u(x , κ) = f (x), x ∈ R2.

The source functionf (x) = g(x) + σ(x)Wx .

The Sommerfeld radiation condition

limr→∞

r1/2(∂ru − iκu) = 0, r = |x |.

The direct problem: Given g and σ, to determine the random wave field u.

The inverse problem: To recover g and σ2 from u|ΓRat κj , j = 1, . . . ,m.

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White noise

Wx is the 2-parameter Brownian motion on (R2, B(R2), µ).

White noise

Wx :=∂2Wx

∂x1∂x2.

Stochastic integral ∫R2

φ(x)dWx =

∫R2

Wx∂2φ(x)

∂x1∂x2dx .

Proposition

E

∫R2

φ(x)dWx = 0, E∣∣∣∫

R2

φ(x)dWx

∣∣∣2 =

∫R2

|φ(x)|2dx .

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Deterministic direct problem

Consider the deterministic scattering problem∆u + κ2u = g , x ∈ R2,

∂ru − iκu = o(r−1/2), r →∞.

Given g ∈ L2(Ω), it has a unique solution

u(x) =

∫ΩG (x , y)g(y)dy ,

where the Green function

G (x , y) = − i

4H

(1)0 (κ|x − y |).

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Stochastic direct problem

Consider the stochastic scattering problem∆u + κ2u = g + σWx , x ∈ R2,

∂ru − iκu = o(r−1/2), r →∞.

Theorem (Bao-Chen-L)

There exists a unique continuous stochastic process (mild solution) u,which satisfies

u(x) =

∫ΩG (x , y)g(y)dy +

∫ΩG (x , y)σ(y)dWy .

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Function regularity

g ∈ L2(Ω).

σ is chosen such that the stochastic integral∫ΩG (x , y , κ)σ(y)dWy

satisfies

E∣∣∫

ΩG (x , y , κ)σ(y)dWy

∣∣2 =

∫Ω|G (x , y , κ)|2σ2(y)dy <∞.

Hence σ ∈ Lp(Ω), p > 2 and σ ∈ C 0,η(Ω), 0 < η < 1.

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Constructive proof

Step 1: continuous modification of the Gaussian random field

v(x) =

∫ΩG (x , y)σ(y)dWy .

Step 2: construct an approximation sequence

W nx =

n∑j=1

|Kj |−12 ξjχKj

(x), ξj = |Kj |−12

∫Kj

dWx , ξj ∼ N (0, 1).

Step 3: consider an approximated solution

un(x) =

∫ΩG (x , y)g(y)dy +

∫ΩG (x , y)σ(y)dW n

x .

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Stochastic inverse problem

Recall the mild solution

u(x , κj) =

∫ΩG (x , y , κj)g(y)dy +

∫ΩG (x , y , κj)σ(y)dWy .

Taking the expectation yields

Eu(x , κj) =

∫ΩG (x , y , κj)g(y)dy .

Consider real and imaginary parts

ReG (x , y , κj) =1

4Y0(κj |x − y |), ImG (x , y , κj) = −1

4J0(κj |x − y |).

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Reconstruction of the mean

Real-valued Fredholm integral equations

EReu(x , κj) =1

4

∫ΩY0(κj |x − y |)g(y)dy ,

EImu(x , κj) = −1

4

∫ΩJ0(κj |x − y |)g(y)dy .

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Reconstruction of the variance

Taking the variance of the mild solution yields

VReu(x , κj) =1

16

∫ΩY 2

0 (κj |x − y |)σ2(y)dy ,

VImu(x , κj) =1

16

∫ΩJ2

0 (κj |x − y |)σ2(y)dy .

VReu(x , κj)−VImu(x , κj) =1

16

∫Ω

(Y 2

0 (κj |x − y |)− J20 (κj |x − y |)

)σ2(y)dy

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Numerical method - Kaczmarz algorithm

Consider the linear system of equations

Ajq = pj , j = 1, . . . ,m.

A1q = p1

A2q = p2A3q = p3

q0

q1q2

q3

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Regularized Kaczmarz algorithm

Let q0 = 0, do k = 0, 1, . . .q0 = qk ,

qj = qj−1 + ATj (µI + AjA

Tj )−1(pj − Ajqj−1), j = 1, . . . ,m,

qk+1 = qm,

where µ > 0 is a regularization parameter.

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Direct solver

Consider the approximated scattering problem∆u + κ2u = g + σW n

x , x ∈ R2,

∂ru − iκu = o(r−1/2), r →∞,

where

W nx =

n∑j=1

|Kj |−12 ξjχKj

(x).

Direct solver: FEM with PML, Monte Carlo.

Parameters: κj = (j − 0.5)π, j = 1, . . . , 5, µ = 1.0× 10−7, outloop k = 5.

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Numerical result

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Inhomogeneous media: direct problem

Consider the stochastic scattering problem∆u + κ2(1 + q)u = g + σWx , x ∈ R2,

∂ru − iκu = o(r−1/2), r →∞.

Theorem (Bao-Chen-L)

The stochastic scattering problem admits a unique continuous stochasticprocess u, which satisfies

u(x) = −κ2

∫ΩG (x , y)q(y)u(y)dy

+

∫ΩG (x , y)g(y)dy +

∫ΩG (x , y)σ(y)dWy .

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Inhomogeneous media: inverse problem

Consider the inhomogeneous stochastic Helmholtz equation

∆u(x , κ) + κ2(1 + q(x))u(x , κ) = g(x) + σ(x)Wx in BR .

Let v be the eigenfunction for the following problem:∆v(x , κ) + κ2(1 + q(x))v(x , κ) = 0 inBR ,

v(x , κ) = 0 on ΓR .

We have from the integation by parts that

−∫

ΓR

∂νv(x , κ)u(x , κ)dγ =

∫BR

g(x)v(x , κ)dx +

∫BR

σ(x)v(x , κ)dWx .

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Numerical result - homogeneous medium

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Numerical result - inhomogeneous medium

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Increasing stability

Continuous frequency data

Discrete frequency data

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Problem geometry

ΓR

source

Ω

BR

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Model problem

The Helmholtz equation

∆u(x , κ) + κ2u(x , κ) = f (x), x ∈ R2.

The Sommerfeld radiation condition

limr→∞

r1/2(∂ru − iκu) = 0, r = |x |.

Given f ∈ L2(Ω), it has a unique solution:

u(x , κ) =

∫ΩG (x , y ;κ)f (y)dy ,

where

G (x , y ;κ) = − i

4H

(1)0 (κ|x − y |).

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Transparent boundary condition

Given function u on ΓR , it has the Fourier series expansion:

u(R, θ) =∑n∈Z

un(R)e inθ, un(R) =1

2π

∫ 2π

0u(R, θ)e−inθdθ.

Introduce the DtN operator T : H12 (ΓR)→ H−

12 (ΓR)

(Tu)(R, θ) = κ∑n∈Z

H(1)′n (κR)

H(1)n (κR)

un(R)e inθ.

Transparent boundary condition:

∂νu = Tu on ΓR .

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Problem formulation - continuous frequency data

Consider reduced problem∆u + κ2u = f in BR ,

∂νu = Tu on ΓR .

Define boundary data

‖u(·, κ)‖2ΓR

=

∫ΓR

(|Tu(x , κ)|2 + κ2|u(x , κ)|2

)dγ(x).

ISP 1. Let f be a complex function with a compact support Ω ⊂ BR . TheISP is to determine f from the data u(x , κ), x ∈ ΓR , κ ∈ (0,K ), whereK > 1 is a positive constant.

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Main result - ISP 1

Define

FM = f ∈ Hm(Ω) : ‖f ‖Hm(BR) ≤ M, suppf = Ω ⊂ BR,

where m > 2 is an integer and M > 1 is a constant.

Theorem (L-Yuan)

Let f ∈ FM and u be the solution of the scattering problem correspondingto f . Then

‖f ‖2L2(Ω) . ε2 +

M2(K

23 | ln ε|

14

(6m−15)3

)2m−5,

where

ε =

(∫ K

0κ‖u(·, κ)‖2

ΓRdκ

) 12

.

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Sketch of proof

Step 1: energy estimate

‖f ‖2L2(Ω) .

∫ ∞0

κ‖u(·, κ)‖2ΓRdκ.

Step 2: low frequency estimate

I1(s) =

∫ s

0κ3

∫ΓR

∣∣∣∣∫ΩH

(1)0 (κ|x − y |)f (y)dy

∣∣∣∣2 dγ(x)dκ,

I2(s) =

∫ s

0κ

∫ΓR

∣∣∣∣∫Ω∂νxH

(1)0 (κ|x − y |)f (y)dy

∣∣∣∣2 dγ(x)dκ.

Step 3: high frequency tail estimate∫ +∞

sκ‖u(·, κ)‖2

ΓRdκ . s−(2m−5)‖f ‖2

Hm(Ω).

Step 4: link between low and high frequencies.

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Problem formulation - discrete frequency data

Consider reduced problem∆u + κ2u = f in BR ,

∂νu = Tu on ΓR .

Define boundary data at discrete frequency

‖u(·, κn)‖2ΓR

=

∫ΓR

(|Tu(x , κn)|2 + κ2

n|u(x , κn)|2)dγ(x),

whereκn = n

(πR

), n = |n|, n ∈ Z2 \ 0.

ISP 2. Let f be a complex function with a compact support Ω ⊂ BR . TheISP is to determine f from the datau(x , κ), x ∈ ΓR , κ ∈ (0, πR ] ∪ ∪Nn=1κn, where 1 < N ∈ N.

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Main result - ISP 2

Define

FM = f ∈ FM :

∫Ωf (x)dx = 0.

Theorem (L-Yuan)

Let f ∈ FM and u be the solution of the scattering problem correspondingto f . Then

‖f ‖2L2(Ω) . ε2

1 +M2(

N58 | ln ε2|

19

(6m−15)3

)2m−5,

where

ε1 =

∑n≤N‖u(·, κn)‖2

ΓR

12

, ε2 = supκ∈(0,π

R]‖u(·, κ)‖ΓR

.

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Sketch of proof

Step 1: energy estimate

|fn|2 . ‖u(·, κn)‖2ΓR, n ∈ Z2 \ 0.

Step 2: low frequency estimate

I (s) =

∣∣∣∣∫BR

f (x)e−isx ·ddx

∣∣∣∣2 , s ∈ (0,π

R].

Step 3: high frequency tail estimate

∞∑n=N0

|fn|2 ≤ N−(2m−5)0 ‖f ‖2

Hm(BR).

Step 4: link between low and high frequencies.

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References

(with G. Bao and C. Chen) Inverse random source scatteringproblems in several dimensions, SIAM/ASA J. UncertaintyQuantitication, 2016.

(with G. Bao and C. Chen) Inverse random source scattering forelastic waves, preprint.

(with G. Bao and C. Chen) Inverse random source scattering for theHelmholtz equation in inhomogeneous media, preprint.

(with G. Yuan) Stability on the inverse random source scatteringproblem for the one-dimensional Helmholtz equation, J. Math. Anal.Appl., to appear.

(with G. Yuan) Increasing stability for the inverse source scatteringproblem with multi-frequencies, preprint.

(with G. Bao and Y. Zhao) Stability in the inverse source problem forelastic and electromagnetic waves with multi-frequencies, preprint.

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Ongoing and future work

Partial data

Inhomogeneous media

Inverse random medium problem

Time-domain inverse problems

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Thank You !