# Applications of Microlocal Analysis in Inverse Problems

## Abstract

**:**

## 1. Introduction

- a kind of “variable coefficient Fourier analysis” for solving variable coefficient PDEs; or
- as a theory of pseudodifferential operators ($\Psi $DOs) and Fourier integral operators (FIOs); or
- as a phase space (or time-frequency) approach to studying functions, operators and their singularities (wave front sets).

- 1.
**Computed tomography/X-ray transform—**the X-ray transform is an FIO, and under certain conditions its normal operator is an elliptic $\Psi $DO. Microlocal analysis can be used to predict which sharp features (singularities) of the image can be reconstructed in a stable way from limited data measurements. Microlocal analysis is also a powerful tool in the study of geodesic X-ray transforms related to seismic imaging applications.- 2.
**Calderón problem/Electrical Impedance Tomography—**the boundary measurement map (Dirichlet-to-Neumann map) is a $\Psi $DO, and the boundary values of the conductivity as well as its derivatives can be computed from the symbol of this $\Psi $DO.- 3.
**Gel’fand problem/seismic imaging—**the boundary measurement operator (hyperbolic Dirichlet-to-Neumann map) is an FIO, and the scattering relation of the sound speed as well as certain X-ray transforms of the coefficients can be computed from the canonical relation and the symbol of this FIO.

#### Notation

## 2. Pseudodifferential Operators

- (1)
- a kind of “variable coefficient Fourier analysis” for solving variable coefficient PDEs; or
- (2)
- a theory of $\Psi $DOs and FIOs; or
- (3)
- a phase space (or time-frequency) approach to studying functions, operators and their singularities (wave front sets).

#### 2.1. Constant Coefficient PDEs

- If u is a function in ${\mathbb{R}}^{n}$, its Fourier transform$\widehat{u}=\mathcal{F}u$ is the function$$\widehat{u}(\xi ):={\int}_{{\mathbb{R}}^{n}}{e}^{-ix\xb7\xi}u(x)\phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{2.em}{0ex}}\xi \in {\mathbb{R}}^{n}.$$
- The Fourier transform converts derivatives to polynomials (this is why it is useful for solving PDEs):$$({D}_{j}u)\phantom{\rule{0.166667em}{0ex}}^\phantom{\rule{0.166667em}{0ex}}(\xi )={\xi}_{j}\widehat{u}(\xi ).$$
- A function u can be recovered from $\widehat{u}$ by the Fourier inversion formula $u={\mathcal{F}}^{-1}\widehat{u}$, where$${\mathcal{F}}^{-1}v(x):={(2\pi )}^{-n}{\int}_{{\mathbb{R}}^{n}}{e}^{ix\xb7\xi}v(\xi )\phantom{\rule{0.166667em}{0ex}}d\xi $$

#### 2.2. Variable Coefficient PDEs

**Definition**

**1.**

#### 2.3. Pseudodifferential Operators

**Definition**

**2.**

**Remark**

**1**

**.**We saw in Section 2.1 that the elliptic operator $-\Delta $ has the inverse

**Theorem**

**1**

**.**$\phantom{a}$

- (a)
- (Principal symbol) There is a one-to-one correspondence between operators in ${\Psi}^{m}$ and (full) symbols in ${S}^{m}$, and each operator $A\in {\Psi}^{m}$ has a well defined principal symbol ${\sigma}_{\mathrm{pr}}(A)$. The principal symbol may be computed by testing A against highly oscillatory functions (this is valid if A is a classical ΨDO):$${\sigma}_{\mathrm{pr}}(A)({x}_{0},{\xi}_{0})=\underset{\lambda \to \infty}{lim}{\lambda}^{-m}{e}^{-i\lambda x\xb7{\xi}_{0}}A({e}^{i\lambda x\xb7{\xi}_{0}}){|}_{x={x}_{0}};$$
- (b)
- (Composition) If $A\in {\Psi}^{m}$ and $B\in {\Psi}^{{m}^{\prime}}$, then $AB\in {\Psi}^{m+{m}^{\prime}}$ and ${\sigma}_{\mathrm{pr}}(AB)={\sigma}_{\mathrm{pr}}(A){\sigma}_{\mathrm{pr}}(B)$;
- (c)
- (Sobolev mapping properties) Each $A\in {\Psi}^{m}$ is a bounded operator ${H}^{s}({\mathbb{R}}^{n})\to {H}^{s-m}({\mathbb{R}}^{n})$ for any $s\in \mathbb{R}$;
- (d)
- (Elliptic operators have approximate inverses) If $A\in {\Psi}^{m}$ is elliptic, there is $B\in {\Psi}^{-m}$ so that $AB=\mathrm{Id}+K$ and $BA=\mathrm{Id}+L$ where $K,L\in {\Psi}^{-\infty}$, i.e., $K,L$ are smoothing (they map any ${H}^{-s}$ function to ${H}^{t}$ for any t, hence also to ${C}^{\infty}$ by Sobolev embedding).

- open sets in ${\mathbb{R}}^{n}$ (local setting) ([3], Section 18.1);
- compact manifolds without boundary, possibly acting on sections of vector bundles ([3], Section 18.1);
- compact manifolds with boundary (transmission condition/Boutet de Monvel calculus) [13];
- non-compact manifolds (e.g., Melrose scattering calculus) [14];
- operators with a small or large parameter (semiclassical calculus) [15]; and

## 3. Wave Front Sets and Fourier Integral Operators

#### 3.1. The Role of Singularities

**Definition**

**3**

**.**We say that a function or distribution u is${C}^{\infty}$ (resp. ${H}^{\alpha}$) near ${x}_{0}$if there is $\phi \in {C}_{c}^{\infty}({\mathbb{R}}^{n})$ with $\phi =1$ near ${x}_{0}$ such that $\phi u$ is in ${C}^{\infty}({\mathbb{R}}^{n})$ (resp. in ${H}^{\alpha}({\mathbb{R}}^{n})$). We define

**Example**

**1.**

**Definition**

**4**

**.**Let u be a distribution in ${\mathbb{R}}^{n}$. We say that u is (microlocally) ${C}^{\infty}$ (resp. ${H}^{\alpha}$) near $({x}_{0},{\xi}_{0})$ if there exist $\phi \in {C}_{c}^{\infty}({\mathbb{R}}^{n})$ with $\phi =1$ near ${x}_{0}$ and $\psi \in {C}^{\infty}({\mathbb{R}}^{n}\backslash \left\{0\right\})$ so that $\psi =1$ near ${\xi}_{0}$ and ψ is homogeneous of degree 0, such that

**Example**

**2.**

**Theorem**

**2**

**.**Any $A\in {\Psi}^{m}$ has the pseudolocal property

**Theorem**

**3**

**.**Let $A\in {\Psi}^{m}$ be elliptic. Then, for any u,

**Proof.**

#### 3.2. Fourier Integral Operators

**Motivation**

**1.**

#### 3.3. Propagation of Singularities

**Example**

**3.**

**Remark**

**2.**

**Theorem**

**4**

**.**Let $P\in {\Psi}^{m}$ have real principal symbol ${p}_{m}$ that is homogeneous of degree m in ξ. If

**Example**

**4.**

## 4. The Radon Transform in the Plane

#### 4.1. Basic Properties of the Radon Transform

**Definition**

**5.**

**Theorem**

**5**

**.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**6**

**.**One has

**Remark**

**3.**

**Proof.**

#### 4.2. Visible Singularities

**Example**

**5.**

**Theorem**

**7.**

- If $\mathcal{A}=\left\{\mathrm{lines}\phantom{\rule{4.pt}{0ex}}\mathrm{not}\phantom{\rule{4.pt}{0ex}}\mathrm{meeting}\phantom{\rule{4.pt}{0ex}}\overline{\mathbb{D}}\right\}$, then ${Rf|}_{\mathcal{A}}$ is called exterior data.
- If $0<a<\pi /2$ and $\mathcal{A}=\left\{\mathrm{lines}\phantom{\rule{4.pt}{0ex}}\mathrm{whose}\phantom{\rule{4.pt}{0ex}}\mathrm{angle}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}x-\mathrm{axis}\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}<a\right\}$, then ${Rf|}_{\mathcal{A}}$ is called limited angle data.

**Definition**

**6.**

- If $({x}_{0},{\xi}_{0})$ is visible from $\mathcal{A}$, then from the singularities of ${Rf|}_{\mathcal{A}}$ one can determine for any $\alpha $ whether or not $({x}_{0},{\xi}_{0})\in W{F}^{\alpha}(f)$. If ${Rf|}_{\mathcal{A}}$ uniquely determines f, one expects the reconstruction of visible singularities to be stable.
- If $({x}_{0},{\xi}_{0})$ is not visible from $\mathcal{A}$, then this singularity is smoothed out in the measurement ${Rf|}_{\mathcal{A}}$. Even if ${Rf|}_{\mathcal{A}}$ would determine f uniquely, the inversion is not Lipschitz stable in any Sobolev norms.

## 5. Gel’fand Problem

Gel’fand problem:Is it possible to determine the interior structure of Earth by controlling acoustic waves and measuring vibrations at the surface?

**Theorem**

**8**

**.**Let $T>0$ and assume that ${q}_{1},{q}_{2}\in {C}_{c}^{\infty}(\Omega )$. If

**Corollary**

**2.**

**Proof.**

- The map ${\Lambda}_{q}$ is an FIO of order 1 on $\partial \Omega \times (0,T)$.
- The X-ray transform of q can be read off from the symbol of ${\Lambda}_{q}$ (more precisely, from the principal symbol of ${\Lambda}_{q}-{\Lambda}_{0}$).

**Lemma**

**1**

**.**Assume that ${q}_{1},{q}_{2}\in {C}_{c}^{\infty}(\Omega )$. For any ${f}_{1},{f}_{2}\in {C}_{c}^{\infty}(\partial \Omega \times (0,T))$, one has

**Proof.**

**Proposition**

**1**

**.**Assume that $q\in {C}_{c}^{\infty}(\Omega )$, and let $\gamma :[\delta ,L]\to \overline{\Omega}$ be a maximal line segment in $\overline{\Omega}$ with $0<\delta <L<T$. For any $\lambda \ge 1$ there is a solution $u={u}_{\lambda}$ of $(\square +q)u=0$ in $\Omega \times (0,T)$ with $u={\partial}_{t}u=0$ on $\{t=0\}$, such that for any $\psi \in {C}_{c}^{\infty}(\Omega \times [0,T])$ one has

**Proof**

**(Proof of Theorem 8).**

**.**Using the assumption ${\Lambda}_{{q}_{1}}={\Lambda}_{{q}_{2}}$ and Lemma 1, we have

**Proof of Proposition**

**1.**

## 6. Calderón Problem: Boundary Determination

Calderón problem:Is it possible to determine the electrical conductivity of a medium by making voltage and current measurements on its boundary?

**Theorem**

**9**

**.**Let ${\gamma}_{1},{\gamma}_{2}\in {C}^{\infty}(\overline{\Omega})$ be positive. If

- The DN map ${\Lambda}_{\gamma}$ is an elliptic $\Psi $DO of order 1 on $\partial \Omega $.
- The Taylor series of $\gamma $ at a boundary point can be read off from the symbol of ${\Lambda}_{\gamma}$ computed in suitable coordinates. The symbol of ${\Lambda}_{\gamma}$ can be computed by testing against highly oscillatory boundary data (compare with (8)).

**Remark**

**4.**

**Example**

**6**

**.**Let $\Omega ={\mathbb{R}}_{+}^{n}=\{{x}_{n}>0\}$, so $\partial \Omega ={\mathbb{R}}^{n-1}=\{{x}_{n}=0\}$. We wish to compute the DN map for the Laplace equation (i.e., $\gamma \equiv 1$) in Ω. Consider

**Lemma**

**2**

**.**Let ${\gamma}_{1},{\gamma}_{2}\in {C}^{\infty}(\overline{\Omega})$. If ${f}_{1},{f}_{2}\in {C}^{\infty}(\partial \Omega )$, then

**Proof.**

**Proposition**

**2.**

**Proof of Theorem**

**9.**

**Lemma**

**3**

**.**Let ${f}_{j}\in {C}_{c}^{\infty}({\mathbb{R}}^{n-1})$ for $j=0,1,2,\dots $. There exists $f\in {C}_{c}^{\infty}({\mathbb{R}}^{n})$ such that

**Proof of Proposition**

**2.**

## Funding

## Conflicts of Interest

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Salo, M.
Applications of Microlocal Analysis in Inverse Problems. *Mathematics* **2020**, *8*, 1184.
https://doi.org/10.3390/math8071184

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Salo M.
Applications of Microlocal Analysis in Inverse Problems. *Mathematics*. 2020; 8(7):1184.
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Salo, Mikko.
2020. "Applications of Microlocal Analysis in Inverse Problems" *Mathematics* 8, no. 7: 1184.
https://doi.org/10.3390/math8071184