April  2017, 11(2): 277-304. doi: 10.3934/ipi.2017014

Applications of CGO solutions to coupled-physics inverse problems

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA

2. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

3. 

Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, USA

4. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

5. 

Department of Mathematics, Northeastern University, 360 Huntington Ave., Boston MA 02115, USA

Received  December 2015 Published  March 2017

Fund Project: R.-Y. L. was partly supported by the AMS-Simons Travel Grants. TZ. was supported by NSF grant DMS-1501049 and Alfred P. Sloan Research Fellowship FR-2015-65641

This paper surveys inverse problems arising in several coupled-physics imaging modalities for both medical and geophysical purposes. These include Photo-acoustic Tomography (PAT), Thermo-acoustic Tomography (TAT), Electro-Seismic Conversion, Transient Elastrography (TE) and Acousto-Electric Tomography (AET). These inverse problems typically consists of multiple inverse steps, each of which corresponds to one of the wave propagations involved. The review focuses on those steps known as the inverse problems with internal data, in which the complex geometrical optics (CGO) solutions to the underlying equations turn out to be useful in showing the uniqueness and stability in determining the desired information.

Citation: Ilker Kocyigit, Ru-Yu Lai, Lingyun Qiu, Yang Yang, Ting Zhou. Applications of CGO solutions to coupled-physics inverse problems. Inverse Problems & Imaging, 2017, 11 (2) : 277-304. doi: 10.3934/ipi.2017014
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[12]

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[13]

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Modality Equation and Data CGO solution Results
Section 2: Quantitative PAT (second step of PAT) $-\nabla\cdot\gamma\nabla u+\sigma u=0$
data: $u|_{\partial\Omega}\mapsto \sigma u|_{\Omega}$
(full boundary illuminations).
$u = e^{i \zeta\cdot x}(1+\psi_{\zeta})$
to the reduced equation $(\Delta+q)u=0$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [20]).
data: $u|_{\Gamma}\mapsto\sigma u|_{\Omega}$
(partial boundary illuminations).
$u=e^{\frac{1}{h}(\varphi+i\psi)}\big(a+r\big)+z$
with $\text{supp}~u|_{\partial\Omega}\subset\Gamma$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [34]).
Section 3: Electro Seismic Conversion Maxwell's equations:
$\begin{array}{l}\nabla\times E = i\omega\mu_{0} H,\\\nabla\times H = (\sigma - i\varepsilon\omega)E.\end{array}$
data:
$\nu\times E|_{\partial\Omega}\mapsto LE|_{\Omega}$
$E= e^{i\zeta\cdot x}(\eta + R_\zeta)$ Uniqueness and Stability in determining $(L,\sigma)$ (see [35]).
Section 4: Transient Elastography Elasticity system:
$\nabla\cdot (\lambda(\nabla\cdot u)I+2S(\nabla u)\mu)+k^2 u=0$
data:
$u|_{\partial\Omega}\mapsto u|_{\Omega}$.
$U = e^{i\zeta\cdot x} (C_0(x,\theta) p(\theta\cdot x)+O(\tau^{-1}))$
to the Schrödinger equation with external Yang-Mills potentials.
Uniqueness and Stability in determining the Lamé parameters $(\lambda,\mu)$(see [60]).
Section 5: Acousto Electric Tomography Step 1
Conductivity equation
$\nabla \cdot (\gamma \nabla u) = 0$.
data:
$m\mapsto(\Lambda_{\gamma_m}-\Lambda_{\gamma})(u|_{\partial\Omega})$
where $\Lambda_\gamma$ is the Dirichlet to Neumann map for $\gamma$ and $\gamma_m=(1+m)\gamma$.
$u={\gamma}^{-1/2}e^{i\zeta \cdot x} (1 + \psi_{\zeta})$
to the conductivity equation.
Reconstruction of $\sqrt{\gamma}\nabla u|_{\Omega}$ using CGOs (see [54]) or $\gamma |\nabla u|^2|_{\Omega}$ (see [9])
Step 2
data: $u|_{\partial\Omega}\mapsto \sqrt{\gamma}\nabla u|_{\Omega}$ or $u|_{\partial\Omega}\mapsto \gamma |\nabla u|^2 |_{\Omega}$
same as step 1 above Uniqueness and stability in determining $\gamma$(see [9,54]).
Section 6: Quantitative TAT Scalar Schrödinger
$(\Delta+q)u=0$
where $q=k^2+ik\sigma(x)$.
data: $u|_{\partial\Omega}\mapsto \sigma|u|^2_{\Omega}$
$u = e^{ i \zeta \cdot x}( 1 + \psi_\zeta)$ Uniqueness and Stability in determining $\sigma$(see [19]).
Maxwell system:
$-\nabla\times\nabla\times E+qE=0$
where $q=k^2n+ik\sigma$.
data: $\nu\times E|_{\partial\Omega}\mapsto \sigma|E|^2|_{\Omega}$
$E=\gamma_0^{-1/2} e^{i\zeta\cdot x}\big(\eta_\zeta+R_\zeta\big)$
where $\gamma_0=q/\kappa^2$.
Stability in determining $q$(see [22]).
Modality Equation and Data CGO solution Results
Section 2: Quantitative PAT (second step of PAT) $-\nabla\cdot\gamma\nabla u+\sigma u=0$
data: $u|_{\partial\Omega}\mapsto \sigma u|_{\Omega}$
(full boundary illuminations).
$u = e^{i \zeta\cdot x}(1+\psi_{\zeta})$
to the reduced equation $(\Delta+q)u=0$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [20]).
data: $u|_{\Gamma}\mapsto\sigma u|_{\Omega}$
(partial boundary illuminations).
$u=e^{\frac{1}{h}(\varphi+i\psi)}\big(a+r\big)+z$
with $\text{supp}~u|_{\partial\Omega}\subset\Gamma$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [34]).
Section 3: Electro Seismic Conversion Maxwell's equations:
$\begin{array}{l}\nabla\times E = i\omega\mu_{0} H,\\\nabla\times H = (\sigma - i\varepsilon\omega)E.\end{array}$
data:
$\nu\times E|_{\partial\Omega}\mapsto LE|_{\Omega}$
$E= e^{i\zeta\cdot x}(\eta + R_\zeta)$ Uniqueness and Stability in determining $(L,\sigma)$ (see [35]).
Section 4: Transient Elastography Elasticity system:
$\nabla\cdot (\lambda(\nabla\cdot u)I+2S(\nabla u)\mu)+k^2 u=0$
data:
$u|_{\partial\Omega}\mapsto u|_{\Omega}$.
$U = e^{i\zeta\cdot x} (C_0(x,\theta) p(\theta\cdot x)+O(\tau^{-1}))$
to the Schrödinger equation with external Yang-Mills potentials.
Uniqueness and Stability in determining the Lamé parameters $(\lambda,\mu)$(see [60]).
Section 5: Acousto Electric Tomography Step 1
Conductivity equation
$\nabla \cdot (\gamma \nabla u) = 0$.
data:
$m\mapsto(\Lambda_{\gamma_m}-\Lambda_{\gamma})(u|_{\partial\Omega})$
where $\Lambda_\gamma$ is the Dirichlet to Neumann map for $\gamma$ and $\gamma_m=(1+m)\gamma$.
$u={\gamma}^{-1/2}e^{i\zeta \cdot x} (1 + \psi_{\zeta})$
to the conductivity equation.
Reconstruction of $\sqrt{\gamma}\nabla u|_{\Omega}$ using CGOs (see [54]) or $\gamma |\nabla u|^2|_{\Omega}$ (see [9])
Step 2
data: $u|_{\partial\Omega}\mapsto \sqrt{\gamma}\nabla u|_{\Omega}$ or $u|_{\partial\Omega}\mapsto \gamma |\nabla u|^2 |_{\Omega}$
same as step 1 above Uniqueness and stability in determining $\gamma$(see [9,54]).
Section 6: Quantitative TAT Scalar Schrödinger
$(\Delta+q)u=0$
where $q=k^2+ik\sigma(x)$.
data: $u|_{\partial\Omega}\mapsto \sigma|u|^2_{\Omega}$
$u = e^{ i \zeta \cdot x}( 1 + \psi_\zeta)$ Uniqueness and Stability in determining $\sigma$(see [19]).
Maxwell system:
$-\nabla\times\nabla\times E+qE=0$
where $q=k^2n+ik\sigma$.
data: $\nu\times E|_{\partial\Omega}\mapsto \sigma|E|^2|_{\Omega}$
$E=\gamma_0^{-1/2} e^{i\zeta\cdot x}\big(\eta_\zeta+R_\zeta\big)$
where $\gamma_0=q/\kappa^2$.
Stability in determining $q$(see [22]).
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