Stability for quantitative photoacoustic tomography revisited

Abstract

This paper deals with the issue of stability in determining the absorption and the diffusion coefficients in quantitative photoacoustic imaging. We establish a global conditional Hölder stability inequality from the knowledge of two internal data obtained from optical waves, generated by two point sources in a region where the optical coefficients are known.

Appendix A: Proof of technical lemmas

Proof of Lemma 2

In this proof, $C=C(n,\mu ,\nu )>1$ is a generic constant.

It is well known that $G_{1,\nu }$, $\nu >0$, the fundamental solution of the operator $-\Delta +\nu$, is given by $G_{1,\nu }(x,\xi )={\mathcal {G}}_{1,\nu }(x-\xi )$, $x,\xi \in {\mathbb {R}}^n$, with

$$\begin{aligned} {\mathcal {G}}_{1,\nu }(x)= (2\pi )^{-n/2}(\sqrt{\nu }/|x|)^{n/2-1}K_{n/2-1}(\sqrt{\nu }|x|). \end{aligned}$$

In the particular case $n=3$, we have $K_{1/2}(z)=\sqrt{\pi /(2z)}e^{-z}$ and therefore

$$\begin{aligned} {\mathcal {G}}_{1,\nu }(x)=\frac{e^{-\sqrt{\nu }|x|}}{4\pi |x|}. \end{aligned}$$

Let $f\in C_0^\infty ({\mathbb {R}}^n)$, $\mu >0$ and $\nu >0$ be two constants, and denote by u the solution of the equation

$$\begin{aligned} (-\mu \Delta +\nu )u=f\quad \text{ in }\; {\mathbb {R}}^n. \end{aligned}$$

Then,

$$\begin{aligned} u(x)=\int _{{\mathbb {R}}^n}G_{\mu ,\nu }(x,\xi ) f(\xi )d\xi ,\quad x\in {\mathbb {R}}^n. \end{aligned}$$

(A.1)

We remark that $v(x)=u(\sqrt{\mu }x)$, $x\in {\mathbb {R}}^n$ satisfies $(-\Delta +\nu )v=f(\sqrt{\mu }\; \cdot )$. Whence

$$\begin{aligned} u(\sqrt{\mu }x)=v(x)&= \int _{{\mathbb {R}}^n}{\mathcal {G}}_{1,\kappa }(x-\xi ) f(\sqrt{\mu }\xi )d\xi \\&= \mu ^{-n/2}\int _{{\mathbb {R}}^n}{\mathcal {G}}_{1,\nu }(x-\xi /\sqrt{\mu }) f(\xi )d\xi ,\quad x\in {\mathbb {R}}^n. \end{aligned}$$

Hence,

$$\begin{aligned} u(x)=\mu ^{-n/2}\int _{{\mathbb {R}}^n}{\mathcal {G}}_{1,\nu }((x-\xi )/\sqrt{\mu }) f(\xi )d\xi ,\quad x\in {\mathbb {R}}^n. \end{aligned}$$

(A.2)

Comparing (A.1) and (A.2), we find

$$\begin{aligned} G_{\mu ,\nu }(x,\xi )=\mu ^{-n/2}{\mathcal {G}}_{1,\nu }((x-\xi )/\sqrt{\mu }),\quad x,\xi \in {\mathbb {R}}^n. \end{aligned}$$

Consequently, $G_{\mu ,\nu }(x,\xi )={\mathcal {G}}_{\mu ,\nu }(x-\xi )$ with

$$\begin{aligned} {\mathcal {G}}_{\mu ,\nu }(x)= (2\pi \mu )^{-n/2}(\sqrt{\nu \mu }/|x|)^{n/2-1}K_{n/2-1}(\sqrt{\nu }|x|/\sqrt{\mu }),\quad x\in {\mathbb {R}}^n. \end{aligned}$$

(A.3)

By the usual asymptotic formula for modified Bessel functions of the second kind (see for instance [5, 9.7.2, page 378]), we have, when $|x|\rightarrow \infty$,

$$\begin{aligned} K_{n/2-1}(\sqrt{\nu }|x|/\sqrt{\mu })=\left( \frac{\pi \sqrt{\mu }}{2\sqrt{\nu }|x|}\right) ^{1/2}e^{-\sqrt{\nu }|x|/\sqrt{\mu }}\left( 1+O(1/|x|)\right) , \end{aligned}$$

where O(1/|x|) only depends on n, $\mu$ and $\nu$.

Consequently, there exists $R=R(n,\mu ,\nu )>0$ so that

$$\begin{aligned} C^{-1}\frac{e^{-\sqrt{\nu }|x|/\sqrt{\mu }}}{|x|^{1/2}}\le K_{n/2-1}(\sqrt{\nu }|x|/\sqrt{\mu })\le C\frac{e^{-\sqrt{\nu }|x|/\sqrt{\mu }}}{|x|^{1/2}},\quad |x|\ge R. \end{aligned}$$

(A.4)

Substituting if necessary R by $\max (R,1)$, we have

$$\begin{aligned} \frac{1}{|x|^{n/2-1}}\le \frac{1}{|x|^{1/2}},\quad |x|\ge R. \end{aligned}$$

(A.5)

Moreover, we have

$$\begin{aligned} \frac{e^{-\sqrt{\nu }|x|/\sqrt{\mu }}}{|x|^{1/2}}=\left[ |x|^{(n-3)/2}e^{-\sqrt{\nu }|x|/(2\sqrt{\mu })}\right] \frac{e^{-\sqrt{\nu }|x|/(2\sqrt{\mu })}}{|x|^{n/2-1}},\quad |x|\ge R. \end{aligned}$$

Since the function $x\rightarrow |x|^{(n-3)/2}e^{-\sqrt{\nu }|x|/(2\sqrt{\mu })}$ is bounded in ${\mathbb {R}}^n$, we deduce

$$\begin{aligned} \frac{e^{-\sqrt{\nu }|x|/\sqrt{\mu }}}{|x|^{1/2}}\le C\frac{e^{-\sqrt{\nu }|x|/(2\sqrt{\mu })}}{|x|^{n/2-1}},\quad |x|\ge R. \end{aligned}$$

(A.6)

Using (A.5) and (A.6) in (A.4) in order to obtain

$$\begin{aligned} C^{-1}\frac{e^{-\sqrt{\nu }|x|/\sqrt{\mu }}}{|x|^{n/2-1}}\le K_{n/2-1}(\sqrt{\nu }|x|/\sqrt{\mu })\le C\frac{e^{-\sqrt{\nu }|x|/(2\sqrt{\mu })}}{|x|^{n/2-1}},\quad |x|\ge R. \end{aligned}$$

(A.7)

We now establish a similar estimate when $|x|\rightarrow 0$. To this end, we recall that according to formula [5, 9.6.9, p. 375] we have

$$\begin{aligned} K_{n/2-1}(\rho )\sim \frac{1}{2}\Gamma (n/2-1)\left( \frac{2}{\rho }\right) ^{n/2-1}\quad \text{ as }\; \rho \rightarrow 0, \end{aligned}$$

from which we deduce in a straightforward manner that there exists $0<r\le R$ so that

$$\begin{aligned} C^{-1}\frac{e^{-\sqrt{\nu }|x|/\sqrt{\mu }}}{|x|^{n/2-1}}\le K_{n/2-1}(\sqrt{\nu }|x|/\sqrt{\mu })\le C\frac{e^{-\sqrt{\nu }|x|/(2\sqrt{\nu })}}{|x|^{n/2-1}},\quad |x|\le r. \end{aligned}$$

(A.8)

The expected two-sided inequality (2.10) follows by combining (A.4), (A.7) and (A.8). $\square$

Proof of Lemma 3

Let ${\mathcal {Q}}$ be an open subset of ${\mathbb {R}}^n$, set $d=\text{ diam }({\mathcal {Q}})$, $d_x=\text{ dist }(x,\partial {\mathcal {Q}})$ and $d_{x,y}=\min (d_x,d_y)$.

We introduce the following weighted Hölder semi-norms and Hölder norms, where $\sigma \in {\mathbb {R}}$, $0<\gamma \le 1$, and k is nonnegative integer,

$$\begin{aligned}&[w]_{k,0;{\mathcal {Q}}}^{(\sigma )}=[w]_{k,{\mathcal {Q}}}^{(\sigma )}=\sup _{x\in {\mathcal {Q}},\; |\beta |=k}d_x^{k+\sigma }|\partial ^\beta w(x)|, \\&[w]_{k,\gamma ;{\mathcal {Q}}}^{(\sigma )}=\sup _{x,y\in {\mathcal {Q}},\; |\beta |=k}d_{x,y}^{k+\gamma +\sigma }\frac{|\partial ^\beta w(y)-\partial ^\beta w(x)|}{|y-x|^\gamma }, \\&|w|_{k;{\mathcal {Q}}}^{(\sigma )}=\sum _{j=0}^k[w]_{j;{\mathcal {Q}}}^{(\sigma )}, \\&|w|_{k,\gamma ;{\mathcal {Q}}}^{(\sigma )}=|w|_{k;{\mathcal {Q}}}^{(\sigma )}+[w]_{k,\gamma ;{\mathcal {Q}}}^{(\sigma )}. \end{aligned}$$

In terms of these notations, we have

$$\begin{aligned}&|a|_{0,\alpha ;{\mathcal {Q}}}^{(0)}=\sup _{x\in {\mathcal {Q}}}|a(x)|+\sup _{x,y\in {\mathcal {Q}}}d_{x,y}^{\alpha }\frac{|a(y)- a(x)|}{|y-x|^\alpha }\le (1+{\mathbf {d}})\lambda , \\&|\partial _ja|_{0,\alpha ;{\mathcal {Q}}}^{(1)}=\sup _{x\in {\mathcal {Q}}}d_x|\partial _j a(x)|+\sup _{x,y\in {\mathcal {O}}}d_{x,y}^{1+\alpha }\frac{|\partial _ja(y)- \partial _ja(x)|}{|y-x|^\alpha }\le ({\mathbf {d}}+{\mathbf {d}}^2)\lambda , \\&|b|_{0,\alpha ;{\mathcal {Q}}}^{(2)}=\sup _{x\in {\mathcal {O}}}d_x^2|b(x)|+\sup _{x,y\in {\mathcal {Q}}}d_{x,y}^{2+\alpha }\frac{|b(y)- b(x)|}{|y-x|^\alpha }\le ({\mathbf {d}}^2+{\mathbf {d}}^3)\lambda . \end{aligned}$$

In consequence,

$$\begin{aligned} |a|_{0,\alpha ;{\mathcal {Q}}}^{(0)}+|\partial _ja|_{0,\alpha ;{\mathcal {Q}}}^{(1)}+|b|_{0,\alpha ;{\mathcal {Q}}}^{(2)}\le \Lambda ({\mathbf {d}})=\left[ 1+2{\mathbf {d}}+2{\mathbf {d}}^2+{\mathbf {d}}^3\right] \lambda . \end{aligned}$$

(A.9)

Following [17], we define also

$$\begin{aligned}&[w]_{k,0;{\mathcal {Q}}}^*=[w]_{k,{\mathcal {O}}}^*=\sup _{x\in {\mathcal {Q}},\; |\beta |=k}d_x^{k}|\partial ^\beta w(x)|, \\&[w]_{k,\gamma ;{\mathcal {Q}}}^*=\sup _{x,y\in {\mathcal {Q}},\; |\beta |=k}d_{x,y}^{k+\alpha }\frac{|\partial ^\beta w(y)-\partial ^\beta w(x)|}{|y-x|^\gamma }, \\&|w|_{k;{\mathcal {Q}}}^*=\sum _{j=0}^k[w]_{j;{\mathcal {Q}}}^*, \\&|w|_{k,\gamma ;{\mathcal {Q}}}^*=|w|_{k;{\mathcal {Q}}}^*+[w]_{k,\gamma ;{\mathcal {O}}}^*. \end{aligned}$$

From [17, Lemma 6.32, page 130] and its proof, we have the following interpolation inequalities: Suppose that j and k, nonnegative integers, and $0\le \beta ,\gamma \le 1$ are so that $j+\beta <k+\gamma$. Then, there exist $C=C(n,\alpha ,\beta )>0$ and $\vartheta =\vartheta (\alpha ,\beta )$ so that, for any $w\in C^{k,\alpha }({\mathcal {Q}})$ and $\epsilon >0$, we have

$$\begin{aligned}&[w]_{j,\beta ;{\mathcal {Q}}}^*\le C\epsilon ^{-\vartheta }|w|_{0;{\mathcal {Q}}}+\epsilon [w]_{k,\gamma ;{\mathcal {Q}}}^*, \end{aligned}$$

(A.10)

$$\begin{aligned}&|w|_{j,\beta ;{\mathcal {Q}}}^*\le C\epsilon ^{-\vartheta }|w|_{0;{\mathcal {Q}}}+\epsilon [w]_{k,\gamma ;{\mathcal {Q}}}^*. \end{aligned}$$

(A.11)

Here, $|w|_{0;{\mathcal {Q}}}=\sup _{x\in {\mathcal {Q}}}|w(x)|$.

Checking carefully the proof of interior Schauder estimates in [17, Theorem 6.2, page 90], we get, taking into account inequalities (A.9)-(A.11), the following result: There exist a constant $C=C(n)>0$ and $\tau =\tau (\alpha )$ so that, for any $0<\mu \le 1/2$ and $w\in C^{k,\alpha }({\mathcal {Q}})$ satisfying $L_{a,b}w=0$ in ${\mathcal {Q}}$, we have

$$\begin{aligned}{}[w]_{2,\alpha ,{\mathcal {Q}}}^*\le C\Lambda ({\mathbf {d}})\left( \mu ^{-\tau }|w|_{0;{\mathcal {Q}}}+\mu ^\alpha [w]_{2,\alpha ,{\mathcal {Q}}}^*\right) . \end{aligned}$$

(A.12)

Substituting in (A.12) C by $\max (C,2^{\alpha -1})$, we may assume in (A.12) that $C=C(n,\alpha )\ge 2^{\alpha -1}$. Bearing in mind that $\Lambda ({\mathbf {d}})>1$, we can take in (A.12), $\mu =(2C\Lambda ({\mathbf {d}}))^{-1/\alpha }$. We find

$$\begin{aligned}{}[w]_{2,\alpha ,{\mathcal {Q}}}^*\le C\Lambda ({\mathbf {d}})^\varkappa |w|_{0;{\mathcal {Q}}}, \end{aligned}$$

(A.13)

for some constants $C=C(n,\alpha )>0$ and $\varkappa =\varkappa (\alpha )>1$.

Using again interpolation inequalities (A.10) and (A.11), we deduce that

$$\begin{aligned} |w|_{2,\alpha ,{\mathcal {Q}}}^*\le C\Lambda ({\mathbf {d}})^\varkappa |w|_{0;{\mathcal {Q}}}. \end{aligned}$$

(A.14)

Let $\delta >0$ be so that ${\mathcal {Q}}_\delta =\{x\in {\mathcal {Q}};\; \text{ dist }(x,\partial {\mathcal {Q}})>\delta \}$ is non-empty. If ${\mathcal {Q}}'$ is an open subset of ${\mathcal {Q}}_\delta$, then (A.14) yields in a straightforward manner

$$\begin{aligned} \Vert w\Vert _{C^{2,\alpha }\left( \overline{{\mathcal {Q}}'}\right) } \le C\max \left( \delta ^{-(2+\alpha )},1\right) \Lambda ({\mathbf {d}})^\varkappa |w|_{0;{\mathcal {Q}}}. \end{aligned}$$

This is the expected inequality. $\square$

Lemma 14

Let K be a compact subset of ${\mathbb {R}}^n$ and $f\in C^{2,\alpha }(K)$ satisfying $\min _K|f|\ge c_->0$. Then,

$$\begin{aligned} \Vert 1/f\Vert _{C^{2,\alpha }(K)}\le C c_+^4\left( 1+\Vert f\Vert _{C^{2,\alpha }(K)}\right) ^3, \end{aligned}$$

(A.15)

where $c_+=\max (1,c_-^{-1})$ and $C=C(\mathrm {diam}(K))$ is a constant.

Proof

Let $x,y\in K$. Using $|1/f|_{0;K}\le c_+$ and the following identities

$$\begin{aligned}&\frac{1}{f^2(y)}-\frac{1}{f^2(x)}=\left( \frac{1}{f(x)f^2(y)}+\frac{1}{f(x)^2f(y)}\right) (f(x)-f(y)), \\&\frac{1}{f^3(y)}-\frac{1}{f^3(x)}=\left( \frac{1}{f(x)f^3(y)}+\frac{1}{f^2(x)f^2(y)}+\frac{1}{f(x)^3f(y)}\right) (f(x)-f(y)), \end{aligned}$$

we easily get

$$\begin{aligned}{}[1/f^j]_{\alpha ;K}\le 3c_+^4[f]_{\alpha ;K},\quad j=2,3. \end{aligned}$$

(A.16)

Also, we have

$$\begin{aligned} \frac{\partial _if(y)\partial _j f(x)}{f^3(y)}-\frac{\partial _if(y)\partial _j f(x)}{f^3(x)}&=\frac{\partial _if(y)}{f^3(y)}(\partial _j f(y)-\partial _j f(x)) \\&\quad + \frac{\partial _jf(x)}{f^3(y)}(\partial _i f(y)-\partial _i f(x))\\&\quad +\left( \frac{1}{f^3(y)}-\frac{1}{f^3(x)}\right) (\partial _if(y)\partial _j f(x)). \end{aligned}$$

In light of (A.16), this identity yields

$$\begin{aligned} \left[ \partial _if\partial _jf/f^3\right] _{\alpha ;K}&\le c_+^4\left( [\partial _if]_{\alpha ;K}|\partial _jf|_{0;K}\right. \\&\quad \left. +[\partial _jf]_{\alpha ;K}|\partial _if|_{0;K}+[f]_{\alpha ;K}|\partial _if|_{0;K}|\partial _jf|_{0;K}\right) .\nonumber \end{aligned}$$

(A.17)

On the other hand, since

$$\begin{aligned} \frac{\partial _{ij}^2f(y)}{f^2(y)}-\frac{\partial _{ij}^2f(x)}{f^2(x)}=\frac{1}{f^2(y)}\left( \partial _{ij}^2f(y)-\partial _{ij}^2f(x)\right) +\left( \frac{1}{f^2(y)}-\frac{1}{f^2(y)}\right) \partial _{ij}^2f(x), \end{aligned}$$

we find, by using again (A.16),

$$\begin{aligned} \left[ \partial _{ij}^2f/f^2\right] _{\alpha ;K}\le 3c_+^4\left( \left[ \partial _{ij}^2f\right] _{\alpha ;K}+[f]_{\alpha ;K}\left| \partial _{ij}^2f\right| _{0,K}\right) . \end{aligned}$$

(A.18)

Inequalities (A.17), (A.18), the identity $\partial _{ij}^2(1/f)=2\partial _if\partial _jf/f^3-\partial _{ij}^2f/f^2$ and the interpolation inequality [17, Lemma 6.35, p. 135] (by proceeding as in Corollary 2) imply

$$\begin{aligned} {[}\partial _{ij}^2(1/f)]_{\alpha ,K}\le Cc_+^4\left( 1+\Vert f\Vert _{C^{2,\alpha }(K)}\right) ^3, \end{aligned}$$

(A.19)

where $C=C(\text{ diam }(K))$ is a constant.

The other terms for 1/f appearing in the norms $\Vert \cdot \Vert _{C^{2,\alpha }(K)}$ can be estimated similarly to the semi-norm in (A.19). Inequality (A.15) then follows. $\square$

Recall that $0<\theta<\alpha <1$.

Lemma 15

$C^{2,\alpha }(\overline{{\mathcal {O}}})$ is continuously embedded in $H^{2+\theta }({\mathcal {O}})$. Furthermore, there exists $C=C(n,\alpha -\theta )$ so that, for any $w\in C^{2,\alpha }(\overline{{\mathcal {O}}})$, we have

$$\begin{aligned} \Vert w\Vert _{H^{2+\theta }({\mathcal {O}})} \le C\max \left( {\mathbf {d}}^{n/2},{\mathbf {d}}^{n/2+\alpha -\theta }\right) \Vert w\Vert _{C^{2,\alpha }\left( \overline{{\mathcal {O}}}\right) }, \end{aligned}$$

(A.20)

where ${\mathbf {d}}=\mathrm {diam}({\mathcal {O}})$.

Proof

Let $w\in C^{2,\alpha }(\overline{{\mathcal {O}}})$ and, for fixed $1\le i,j\le n$, set $g=\partial _{ij}^2w$. Then,

$$\begin{aligned} \int _{{\mathcal {O}}} \int _{{\mathcal {O}}} \frac{|g(x)-g(y)|^2}{|x-y|^{n+2\theta }}dxdy\le [g]_{\alpha ;{\mathcal {O}}}^2 \int _{{\mathcal {O}}} \int _{{\mathcal {O}}} \frac{1}{|x-y|^{n-2(\alpha -\theta )}}dxdy. \end{aligned}$$

In light of [10, Lemma A3, p. 246], this inequality yields

$$\begin{aligned} \int _{{\mathcal {O}}} \int _{{\mathcal {O}}} \frac{|g(x)-g(y)|^2}{|x-y|^{n+2\theta }}dxdy\le \frac{|{\mathbb {S}}^{n-1}||{\mathcal {O}}|{\mathbf {d}}^{2(\alpha -\theta )}}{2(\alpha -\theta )}[g]_{\alpha ;{\mathcal {O}}}^2, \end{aligned}$$

But $|{\mathcal {O}}|\le |B(0,{\mathbf {d}})|$. Hence,

$$\begin{aligned} \int _{{\mathcal {O}}} \int _{{\mathcal {O}}} \frac{|g(x)-g(y)|^2}{|x-y|^{n+2\theta }}dxdy\le \frac{|{\mathbb {S}}^{n-1}|^2{\mathbf {d}}^{n+2(\alpha -\theta )}}{2(\alpha -\theta )}[g]_{\alpha ;{\mathcal {O}}}^2. \end{aligned}$$

(A.21)

Using (A.21) and the inequality

$$\begin{aligned} \Vert h\Vert _{L^2({\mathcal {O}})}^2\le |{\mathbb {S}}^{n-1}|{\mathbf {d}}^n|h|_{0,{\mathcal {O}}},\quad h\in C(\overline{{\mathcal {O}}}), \end{aligned}$$

we get from the definition of the norm of $H^s$-spaces in [18, formula (1.3.2.2), page 17]

$$\begin{aligned} \Vert w\Vert _{H^{2+\theta }({\mathcal {O}})} \le C\max \left( {\mathbf {d}}^{n/2},{\mathbf {d}}^{n/2+\alpha -\theta }\right) \Vert w\Vert _{C^{2,\alpha }(\overline{{\mathcal {O}}})}, \end{aligned}$$

for some constant $C=C(n,\alpha -\theta )>0$. This is the expected inequality $\square$