Consistency of the Bayes method for the inverse scattering problem

In this work, we study the Bayes method for solving the inverse medium scattering problem. Our aim is to prove the consistency property of the posterior distribution. Let the refractive index $n\unicode{x2A7E} 0$ and $1-n$ be a compactly supported function in $\mathbb{R}^3$ with $\mathrm{supp}(1-n)\subset D$, where D is an open bounded smooth domain, and having suitable regularity, which will be specified later. Let $u = u^\textrm{inc}+u_n^\textrm{sca}$ satisfy

$$\begin{equation} \Delta u + \kappa^{2} nu = 0\quad\mbox{in}\quad\mathbb{R}^3, \end{equation} \tag{ 1.1a }$$

and

$$\begin{equation} \lim_{|x|\to\infty}|x|\left(\frac{\partial u_n^{\mathrm{sca}}}{\partial |x|} - \mathbf{i} \kappa u_n^{\mathrm{sca}}\right) = 0. \end{equation} \tag{ 1.1b }$$

Assume that $u^\textrm{inc}$ is the plane incident field, i.e. $u^\textrm{inc} = e^{\mathbf{i} \kappa x\cdot\theta}$ with $\theta\in\mathcal{S}^{2}$, where $\mathcal{S}^{2} = \left\{\theta \in \mathbb{R}^{3} : \lvert \theta \rvert = 1 \right\}$. Then the scattered field $u_n^\textrm{sca}$ satisfies

$$\begin{equation} u_n^{\mathrm{sca}}\left(x,\theta\right) = \frac{e^{\mathbf{i} \kappa \lvert x \rvert}}{\lvert x \rvert}u_n^\infty\left(\theta^{^{\prime}},\theta\right)+o\left(|x|^{-1}\right)\quad{\mathrm{as}}\quad|x|\to\infty, \end{equation} \tag{ 1.1c }$$

where $\theta^{^{\prime}} = x/|x|$, see for example, [Ser17, p 232]. The inverse scattering problem is to determine the medium perturbation $1-n$ from the knowledge of the scattering amplitude $u_n^\infty(\theta^{^{\prime}},\theta)$ for all $\theta^{^{\prime}},\theta\in\mathcal{S}^{2}$ at one fixed energy κ².

It was known that the scattering amplitude $u_n^\infty(\theta^{^{\prime}},\theta)$ uniquely determines the near-field data of (1.1a ) on $\partial D$, which in turn, determines the Dirichlet-to-Neumann map of (1.1a ) provided κ² is not an Dirichlet eigenvalue of $-\Delta$ on D, for example, see [Nac88]. Combining this fact and the uniqueness results proved in [SU87], one can show that the scattering amplitude $u_n^\infty(\theta^{^{\prime}},\theta)$ for all $\theta^{^{\prime}},\theta\in\mathcal{S}^{2}$ uniquely determines n, at least when n is essentially bounded. For the stability, a log-type estimate was derived in [HH01].

In this paper, we would like to apply the Bayes approach to the inverse scattering problem. The study of this inverse method is motivated by the practical consideration. In practice, it is impossible to obtain the full knowledge of the scattering amplitude. Instead, the following experiment is more realistic. We first randomly choose an incident direction and send out the corresponding incident field towards the probe region. Then we measure the scattering amplitude at another randomly chosen direction. The experiment can be repeated as many times as we wish. The goal now is to make an inference of the refractive index based on such observation data. To describe the method more precisely, we first introduce the measurement model. Let µ be the uniform distribution on $\mathcal{S}^{2}\times\mathcal{S}^{2}$, i.e. $\mu = \mathrm{d}\omega/|\mathcal{S}^{2}|^2$, where $\mathrm{d}\omega$ is the product measure on $\mathcal{S}^{2}\times\mathcal{S}^{2}$, that is, $\int_{\mathcal{S}^{2}\times\mathcal{S}^{2}}\mathrm{d}\omega = |\mathcal{S}^{2}|^2$. We also write $\mu = \mathrm{d}\xi$ and hence $\int_{\mathcal{S}^2\times\mathcal{S}^2}\mathrm{d}\xi = 1$. Consider the iid random variables $X_i\sim\mu$, $i = 1,2,\ldots,N$ with $N\in\mathbb{N}$. In other words, $\{X_i\}_{i = 1}^N$ is a sequence of independent samples of µ on $\mathcal{S}^{2}\times\mathcal{S}^{2}$. Denote

$$\begin{equation} G\left(n\right)\left(X_i\right) = u_n^\infty\left(\theta^{^{\prime}},\theta\right), \end{equation} \tag{ 1.2 }$$

where $(\theta^{^{\prime}},\theta)$ is a realization of X_i . The observation of the scattering amplitude $G(n)(X_i)$ is polluted by the measurement noise which is assumed to be a Gaussian random variable. Since $G(n)(X_i)$ is a complex-valued function, we treat it as a $\mathbb{R}^2$-valued function. For convenience, we slightly abuse the notation by writing

$$\begin{align*} G\left(n\right)\left(X_i\right) = \begin{pmatrix}{\mathrm{Re}}\{G\left(n\right)\left(X_i\right)\}\\{\mathrm{Im}}\{G\left(n\right)\left(X_i\right)\}\end{pmatrix}. \end{align*}$$

Consequently, the statistical model of the scattering problem is given as

$$\begin{equation} Y_i = G\left(n\right)\left(X_i\right)+\sigma W_i,\;\; W_i\overset{\scriptsize iid}{\sim} {\mathcal N}\left(0,I_2\right),\;\; i = 1,\ldots,N, \end{equation} \tag{ 1.3 }$$

where σ > 0 is the noise level, I₂ is the $2\times 2$ unit matrix. We also assume that $W^{(N)}: = \{W_i\}_{i = 1}^N$ and $X^{(N)}: = \{X_i\}_{i = 1}^N$ are independent.

The theme of this paper is to consider the inference of n from the observational data $(Y^{(N)},X^{(N)})$ with $Y^{(N)} = \{Y_i\}_{i = 1}^N$ by the Bayes method. In particular, we are interested in the asymptotic behavior of the posterior distribution induced from a large class of Gaussian process priors on n as $N\to\infty$. The aim is to establish the statistical consistency theory of recovering n in (1.1c ) with an explicit convergence rate as the number of measurements N increases, i.e. the contraction rate of the posterior distribution to the 'ground truth' n₀ when the observation data is indeed generated by n₀. Gaussian process priors are often used in applications in which efficient numerical simulations can be carried out based on modern MCMC algorithms such as the pCN (preconditioned Crank–Nicholson) method [CRSW13].

The study of inverse problems in the Bayesian inversion framework has recently attracted much attention since Stuart's seminal article [Stu10] (also see [DS17]). The setting of the problem considered in this paper is closely related to the ones studied in [GN20, Kek22]. In [GN20], Gaussian process priors were used in the Bayesian approach to study the recovery of the diffusion coefficient in the elliptic equation by measuring the solution at randomly chosen interior points with uniform distribution. It was shown that the posterior distribution concentrates around the true parameter at a rate N^−λ for some λ > 0 as $N\to\infty$, where N is the number of measurements (or sample size). Previously, (frequentist) consistency of Bayesian inversion in the elliptic PDE considered in [GN20] was derived in [Vol13]. However, the contraction rates obtained in [Vol13] were only implicitly given. Based on the method in [GN20], similar results were proved in [Kek22] for the parabolic equation where the aim is to recover the absorption coefficient by the interior measurements of the solution. For further results on the Bayesian inverse problems in the non-linear settings, we refer the reader to other interesting papers [Abr19, NS17, NS19, MNP21, Nic20, NP21, NW20]. On the other hand, for linear inverse problems, the statistical guarantees of nonparametric Bayesian methods with Gaussian priors have been extensively studied and well understood, see for example [ALS13, KLS16, KvdVvZ11, MNP19, Ray13] and references therein.

The ideas used in proving the consistency of the Bayesian inversion for the inverse scattering problem studied here are similar to those used in [GN20, Kek22] in which main ideas are from [MNP21]. Unlike the polynomial contraction rates derived in [GN20, Kek22], the posterior distribution $\Pi(\cdot|Y^{(N)},X^{(N)})$ of $n|(Y^{(N)},X^{(N)})$ contracts at the true refractive index n₀ as $N\to\infty$ with a logarithmic rate. The logarithmic rate is due to the ill-posedness of the inverse scattering medium problem by the knowledge of the scattering amplitude at one fixed energy, see [HH01].

This paper is organized as follows. In section 2, we describe the statistical model arising from the scattering problem. We state main consistency theorems with contraction rates assuming re-scaled Gaussian processes priors and re-scaled Gaussian sieve priors. The proofs of theorems are given in section 3. In appendix A, we derive some estimates for the forward scattering problem, and in appendix B, we prove the optimality of the logarithmic stability estimate in the inverse medium scattering problem based on Mandache's idea. Similar instability estimate was also derived in [Isa13]. To make the paper self contained, we present our own proof and slightly refine the estimate obtained in [Isa13].

2.1. Some function spaces and notations

Throughout this paper, we shall use the symbol $\lesssim$ and $\gtrsim$ for inequalities holding up to a universal constant. For two real sequences $(a_{N})$ and $(b_{N})$, we say that $\simeq$ if both $a_{N} \lesssim b_{N}$ and $b_{N}\lesssim a_{N}$ for all sufficiently large N. For a sequence of random variables Z_N and a real sequence $(a_{N})$, we write $Z_N = O_{\mathbb{P}}(a_N)$ if for all ε > 0 there exists $1 \unicode{x2A7D} M_\varepsilon\lt\infty$ such that for all N large enough, $\mathbb{P}(|Z_N|\unicode{x2A7E} M_\varepsilon a_N)\lt\varepsilon$. Denote ${\mathcal L}(Z)$ the law of a random variable Z. Let $C_c^t({\mathcal O})$ with $t\unicode{x2A7E} 0$ denote the Hölder space of order t with compact supports in the bounded smooth domain ${\mathcal O}$.

Let D be a bounded smooth domain in $\mathbb{R}^{3}$, let $s \unicode{x2A7E} 0$ be an integer and we consider the Hilbert space

$$\begin{align*} \begin{aligned} &H^s\left(D\right) = \left\{f\in L^2\left(D\right) : \text{$D ^jf\in L^2(D)$ for all $\lvert j \rvert\unicode{x2A7D} s$} \right\}, \\ & \text{with scalar product} \quad \langle f,g\rangle_{H^s\left(D\right)} = \sum_{|j|\unicode{x2A7D} s}\langle D\,^jf, D^jg\rangle_{L^2\left(D\right)}. \end{aligned} \end{align*}$$

For non-integer s, $H^s(D)$ is defined in terms of the interpolation, see [LM72]. It is known that the restriction operator to D is a continuous linear map of $H^s(\mathbb{R}^3)$ to $H^s(D)$ [LM72, (8.6)]. The space $H_{0}^{s}(D)$ is the completion of $C_c^\infty(D)$ with respect to $H^s(D)$. Also, for $s\gt1/2$ with $s\ne \mathbb{Z} +1/2$, the zero extension of $f\in H_{0}^s(D)$ (extension of f by 0 outside of D) is a continuous map $H_{0}^s(D)\to H^s(\mathbb{R}^3)$ [LM72, theorem 11.4]. Let ${\mathcal K}$ be a compact subset, define $H^s_{\mathcal K} = \{f\in H^s(\mathbb{R}^3): \mathrm{supp}(f)\subseteq {\mathcal K}\}$. In fact, we have

$$\begin{equation*} H_{\overline{D}}^{s} \subset H_{0}^{s}\, \mathrm{for}\, \mathrm{all}\, s\unicode{x2A7E} 0\, \textrm{and equality (up to equivalent norms) holds when } s\notin \mathbb{Z}+\frac{1}{2}, \end{equation*}$$

see [McL00, theorems 3.29 and 3.33].

We now define the space of parameters. For ${M}_{0}\gt1$ and $s \gt \frac{3}{2}$, let

$$\begin{equation*} \mathcal{F}^s_{M_{0}} = \left\{n\in H^s\left(D\right) : 0 < n < M_0,\, n|_{\partial D} = 1,\, \displaystyle{\left. \frac{\partial^j n}{\partial\nu^j} \right|_{\partial D} = 0}\,\, \textrm{for all}\, 1\unicode{x2A7D} j\unicode{x2A7D} s-1 \right\}, \end{equation*}$$

where the traces are defined in the sense of [LM72, theorem 8.3]. For each $n\in\mathcal{F}^s_{M_0}$ we extend $n\equiv 1$ in $\mathbb{R}^3\setminus D$, still denoted by n. Then it is clear that $\mathrm{supp}(1-n)\, \subset D$. Note that for $n\in\mathcal{F}^s_{M_0}$, we only put the restriction on the size of n, but not on the $H^s(D)$-norm of n. As in [NvdGW20, GN20, AN19, Kek22], we will consider a re-parametrization of $\mathcal{F}^s_{M_0}$. We consider the link function Φ satisfying

• (i)  
  $\Phi:(-\infty,\infty)\to(0,M_0)$, $\Phi(0) = 1$, $\Phi^{^{\prime}}(z)\gt0$ for all z;
• (ii)  
  for any $k\in{\mathbb{N}}$

  $$\begin{align*} \sup_{-\infty < z < \infty}|\Phi^{\left(k\right)}\left(z\right)| < \infty. \end{align*}$$

One example to satisfy (i) and (ii) is the logistic function

$$\begin{align*} \Phi\left(z\right) = \frac{M_0}{M_0+\left(M_0-1\right)\left(e^{-z}-1\right)},\quad -\infty < z < \infty. \end{align*}$$

As pointed out in [NvdGW20, section 3], by utilizing a characterization of the space $H_{0}^{s}(D)$ (see e.g. [LM72, theorem 11.5]), one can show that the parameter space can be realized as

$$\begin{equation} \mathcal{F}^s_{M_0} = \left\{n = \Phi\left(F\right) : F\in H_{0}^s\left(D\right) \right\}. \end{equation} \tag{ 2.1 }$$

We end this subsection by emphasizing that our link function is different to those in [NvdGW20, GN20, AN19, Kek22], see (i).

2.2. An abstract statistical model

For each forward map $G : {\mathcal F}_{M_0}^s\rightarrow L^2({\mathcal S}^2\times{\mathcal S}^2)$, we define the reparametrized forward map by

$$\begin{equation} \mathcal{G}\left(F\right) = G\left(\Phi\left(F\right)\right) \quad \textrm{for all}\, F \in H_{0}^{s}\left(D\right), \end{equation} \tag{ 2.2a }$$

and consider the following random design regression model

$$\begin{equation} Y_i = {\mathcal G}\left(F\right)\left(X_i\right)+\sigma W_i,\;\; W_i\overset{\scriptsize iid}{\sim} N\left(0,I_2\right),\;\; i = 1,\ldots,N. \end{equation} \tag{ 2.2b }$$

Assume that $\mathcal{G}$ satisfies

$$\begin{equation} S_{1} : = \sup_{F\in H_0^s\left(D\right)} \|{\mathcal G}\left(F\right)\|_{L^\infty\left(\mathcal{S}^2\times\mathcal{S}^2\right)} < \infty, \end{equation} \tag{ 2.2c }$$

and for each $F_{1},F_{2} \in (H^{1}(D))^{*}$ one has

$$\begin{equation} \|{\mathcal G}\left(F_1\right)-{\mathcal G}\left(F_2\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}\unicode{x2A7D} S_2\left(1+\|F_1\|_{C^t\left(D\right)}^2\vee\|F_2\|_{C^t\left(D\right)}^2\right)\|F_1-F_2\|_{\left(H^{1}\left(D\right)\right)^{*}}, \end{equation} \tag{ 2.2d }$$

for some constant $S_2\gt0$ and $t\unicode{x2A7E} 1$.

Remark 2.1. We implicitly assume that taking pointwise value of ${\mathcal G}(F)$ on ${\mathcal S}^2\times{\mathcal S}^2$ is legitimate. For instance, in the inverse scattering problem here, ${\mathcal G}(F)$ (corresponding to the scattering amplitude $u^\infty_n$) is, in fact, analytic on ${\mathcal S}^2\times{\mathcal S}^2$.

The statistical model (2.2b ) with conditions (2.2c ) and (2.2d ) falls into the general framework described in [GN20]. We want to remark that the uniform boundedness of the forward map ${\mathcal G}$ (condition (2.2c )) in [GN20, NvdGW20] (elliptic boundary value problem) or in [Kek22] (parabolic initial-boundary value problem) is ensured by the positivity assumption of the coefficient and the bound S₁ is determined by the fixed boundary value or the fixed initial and boundary values. Due to these facts, the ranges of the link functions used in [GN20, NvdGW20] and [Kek22] do not required to have finite upper bounds. In the scattering problem, the boundedness requirement of the forward map (2.2c ) cannot be guaranteed by the sign restriction of the potential. In this case, we choose a link with finite range (like Φ given above) to ensure (2.2c ).

Remark 2.2. In view of (1.2) and (2.1), one notes that the statistical model (1.3) fits into the framework of (2.2b ). For the inverse scattering problem studied here, if the forward map G(n) is defined by the far-field pattern (1.2) with refractive index $n \in \mathcal{F}^s_{M_{0}}$ with $s \gt \frac{3}{2}$, from (A.9) we see that (2.2c ) satisfies with $S_{1} = S_{1}(D,\kappa,M_{0})$. From (A.14) we have

$$\begin{equation*} \begin{aligned} & \|\mathcal{G}\left(F_{1}\right)-\mathcal{G}\left(F_{2}\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)} \\ & \quad \unicode{x2A7D} C\left(1+\|\Phi\left(F_{1}\right)\|_{C^1\left(D\right)}\vee\|\Phi\left(F_{2}\right)\|_{C^1\left(D\right)}\right)\|\Phi\left(F_{1}\right)-\Phi\left(F_{2}\right)\|_{\left(H^1\left(D\right)\right)^\ast}, \end{aligned} \end{equation*}$$

where $C = C(D,k,M_0)$. By using (ii) and [NvdGW20, lemma 29], we have

$$\begin{equation*} \|\Phi\left(F_{j}\right)\|_{C^1\left(D\right)} \lesssim 1 + \lVert F_{j} \rVert_{C^{1}\left(D\right)} \quad \textrm{for all }\, j = 1,2, \end{equation*}$$

and

$$\begin{equation*} \|\Phi\left(F_{1}\right)-\Phi\left(F_{2}\right)\|_{\left(H^1\left(D\right)\right)^\ast} \lesssim \left(1 + \lVert F_{1} \rVert_{C^{1}\left(D\right)}\vee \lVert F_{2} \rVert_{C^{1}\left(D\right)}\right) \lVert F_{1}-F_{2} \rVert_{\left(H^{1}\left(D\right)\right)^{\ast}}, \end{equation*}$$

therefore we see that (2.2d ) satisfies with $S_{2} = S_{2}(D,\kappa,M_{0})$ with t = 1.

Observe that the random vectors $(Y_i,X_i)$ are iid with laws $\mathbb{P}_{F}^{i}$. It turns out the Radon–Nikodym derivative of $\mathbb{P}_{F}^{i}$ is given by

$$\begin{equation} p_F\left(y,\xi\right) = \frac{\mathrm{d} \mathbb{P}_{F}^{i}}{\mathrm{d} y \times \mathrm{d}\xi} = \frac{1}{2\pi\sigma^2}e^{-\frac{\left(y-\mathcal{G}\left(F\right)\left(\xi\right)\right)^T\left(y-\mathcal{G}\left(F\right)\left(\xi\right)\right)}{2\sigma^2}}. \end{equation} \tag{ 2.3 }$$

By slightly abusing the notation, now we define $\mathbb{P}_F^N = \otimes_{i = 1}^N \mathbb{P}_F^i$ the joint law of the random vectors $(Y_i,X_i)_{i = 1}^N$. Moreover, $\mathbb{E}_F^i$, $\mathbb{E}_F^N$ denote the expectation operators in terms of the laws $\mathbb{P}_F^i, \mathbb{P}_F^N$, respectively.

In the Bayesian approach, let Π be a Borel probability measure on the parameter space $H_{0}^{s}(D)$ supported in the Banach space C(D). From the continuity property of $(F,(y,\xi))\to p_F(y,\xi)$, the posterior distribution $\Pi(\cdot|Y^{(N)},X^{(N)})$ of $F|(Y^{(N)},X^{(N)})$ is given by

$$\begin{equation} \Pi\left(B|Y^{\left(N\right)},X^{\left(N\right)}\right) = \frac{\int_{B} e^{\ell^{\left(N\right)}\left(F\right)} \, \mathrm{d} \Pi\left(F\right)}{\int_{C\left(D\right)} e^{\ell^{\left(N\right)}\left(F\right)} \, \mathrm{d} \Pi\left(F\right)} \quad \textrm{for all Borel set}\, B \subset C\left(D\right), \end{equation} \tag{ 2.4a }$$

where the log-likelihood function is written as

$$\begin{equation} \ell^{\left(N\right)}\left(F\right) = -\frac{1}{2\sigma^2}\sum_{i = 1}^N\left(Y_i-\mathcal{G}\left(F\right)\left(X_i\right)\right)^T\left(Y_i-\mathcal{G}\left(F\right)\left(X_i\right)\right). \end{equation} \tag{ 2.4b }$$

Finally, we end this subsection by referring to the monograph [Nic23] for a nice introduction on the above preliminaries.

2.3. Statistical convergence rates

In this work we would like to show that the posterior distribution arising from certain priors concentrates near sufficiently regular ground truth $\Phi(F_{0})$, and derive a bound on the rate of contraction, assuming that the observation data $(Y^{(N)},X^{(N)})$ are generated through the model (2.2a )–(2.2d ) of law $\mathbb{P}_{F_{0}}^N$.

2.3.1. Rescaled Gaussian priors.

We now describe explicitly Gaussian priors introduced in [GN20] (also see [Kek22]).

Assumption 2.3. Let $s\gt t+3/2$, $t\unicode{x2A7E} 1$, and ${\mathcal H}$ be a Hilbert space continuously embedded into $H_{0}^s(D)$, and let $\Pi^{^{\prime}}$ be a centered Gaussian Borel probability measure on the Banach space $C^0_c(D)$ that is supported on a separable measurable linear subspace of $C_c^t(D)$. Assume that the reproducing kernel Hilbert space (RKHS) of $\Pi^{^{\prime}}$ equals to ${\mathcal H}$.

Here, we refer to [GN21, definition 2.6.4] for the definition of the RKHS, and we refer to [GN20, example 25] for an example satisfies assumption 2.3. For each s given in assumption 2.3, let $\Pi^{^{\prime}}$ be given in assumption 2.3 and $F^{^{\prime}}\sim\Pi^{^{\prime}}$, we consider the rescaled prior

$$\begin{equation} \Pi_N\equiv \Pi_N\left[s\right] = {\mathcal L}\left(F_N\right)\quad\mbox{with}\quad F_N = \frac{1}{N^{3/\left(4s+10\right)}}F^{^{\prime}}. \end{equation} \tag{ 2.5 }$$

Again, $\Pi_N$ defines a centered Gaussian prior on C(D) and its RKHS ${\mathcal H}_N$ is still ${\mathcal H}$ but with the norm

$$\begin{equation} \|F\|_{{\mathcal H}_N} = N^{3/\left(4s+10\right)}\|F\|_{{\mathcal H}}, \end{equation} \tag{ 2.6 }$$

for all $F\in{\mathcal H}$. We now introduce the main device in our proof, concerning the posterior contraction with an explicit rate around F₀, whose proof can be found in [GN20, theorem 14].

Theorem 2.4. Let $(\mathcal{H},\Pi^{^{\prime}})$ satisfies assumption 2.3 with integer s, let $\Pi_{N}\equiv\Pi_{N}[s]$ be the rescaled prior given in (2.5), let $\Pi_N(\cdot|Y^{(N)},X^{(N)})$ be the posterior distribution given in (2.4a ) with $\Pi = \Pi_{N}$. Assume that $F_0\in{\mathcal H}$ and the observation $(Y^{(N)},X^{(N)})$ to be generated through model (2.2b )–(2.2d ) of law $\mathbb{P}_{F_{0}}^{N}$. If we denote $\delta_N = N^{-(s+1)/(2s+5)}$, then for any K > 0, there exists a large L > 0, depending on $\sigma, F_0, K, s, t, D, S_1,S_2$, such that

$$\begin{align} \begin{aligned} & \Pi_N \left( F:\|{\mathcal G}\left(F\right)-{\mathcal G}\left(F_0\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}>L\delta_N|Y^{\left(N\right)},X^{\left(N\right)} \right) \\ & \quad = O_{\mathbb{P}^N_{F_0}}\left(e^{-KN\delta_N^2}\right)\;\;\mbox{as}\;\; N\to\infty. \end{aligned} \end{align} \tag{ 2.7a }$$

In addition, there exists a large $L^{^{\prime}}\gt0$, depending on $\sigma, K, s, t$, such that

$$\begin{equation} \Pi_N \left( F:\|F\|_{C^t}>L^{^{\prime}}|Y^{\left(N\right)},X^{\left(N\right)} \right) = O_{\mathbb{P}^N_{F_0}}\left(e^{-KN\delta_N^2}\right)\;\;\mbox{as}\;\; N\to\infty. \end{equation} \tag{ 2.7b }$$

We now apply theorem 2.4 to the inverse scattering problem considered here. Relying on the contraction rate (2.7a ) and the regularization property (2.7b ) and taking account of the stability estimate of G⁻¹ (see theorem A.1), we can show that the posterior distribution arising from the statistical inverse scattering model (1.3) contracts around n₀ in the $L^\infty$-risk using ideas from [MNP21]. In light of the link function, we define the push-forward posterior on the refractive index n by

$$\begin{align*} \tilde\Pi_N\left(\cdot|Y^{\left(N\right)}, X^{\left(N\right)}\right): = {\mathcal L}\left(n\right)\;\;\mbox{with}\;\; n = \Phi\circ F: F\sim\Pi_N\left(\cdot|Y^{\left(N\right)}, X^{\left(N\right)}\right). \end{align*}$$

By the push-forward map, we can rewrite (2.7a ) and (2.7b ) in terms of n. That is, we have that for $N\to\infty$ that

$$\begin{equation} \tilde\Pi_N \left( n:\|{G}\left(n\right)-{G}\left(n_0\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}>L\delta_N|Y^{\left(N\right)},X^{\left(N\right)} \right) = O_{\mathbb{P}^N_{n_0}}\left(e^{-KN\delta_N^2}\right), \end{equation} \tag{ 2.8a }$$

and

$$\begin{equation} \tilde\Pi_N \left( n:\|n\|_{C^t}>L^{^{\prime}}|Y^{\left(N\right)},X^{\left(N\right)} \right) = O_{\mathbb{P}^N_{n_0}}\left(e^{-KN\delta_N^2}\right), \end{equation} \tag{ 2.8b }$$

where the second estimate can be derived from the estimates proved in [NvdGW20, lemma 29] (see also [GN20, (27)]). Here L depends on $\sigma, n_0, K, s, t, D, \kappa, M_0$ and Lʹ depends on $\sigma, K, s, t, \kappa, M_0$.

Theorem 2.5. Let $t \unicode{x2A7E} 2$ and $s\gt t+3/2$ be integers, and fix a real parameter $M_0\gt1$. We further assume that is any constant satisfying $0\lt\epsilon\lt\frac{2t-3}{2t+3}$. Let $\Pi_N(\cdot|Y^{(N)},X^{(N)})$, $F_{0} \in \mathcal{H}$ and $\delta_N = N^{-(s+1)/(2s+5)}$ be given in theorem 2.4. The ground truth refractive index is $n_0 = \Phi\circ F_0\in{\mathcal F}^s_{M_0}$. Then for any K > 0, there exists a constant $C = C(\sigma, n_0,K,s,t,D,\kappa,\epsilon,M_0)\gt0$ such that

$$\begin{equation} \tilde\Pi_N \left( n:\|n-n_0\|_{L^\infty\left(D\right)}>C\left(\text{ln} N\right)^{-\frac{2t-3}{2t+3}+\epsilon}|Y^{\left(N\right)},X^{\left(N\right)} \right) = O_{\mathbb{P}^N_{n_0}}\left(e^{-KN\delta_N^2}\right), \end{equation} \tag{ 2.9 }$$

as $N\rightarrow \infty$.

It is clear that we can replace $\|n-n_0\|_{L^\infty(D)}$ by $\|n-n_0\|_{L^2(D)}$ in (2.9). Unlike the polynomial rate proved in [GN20, theorem 5], we obtain a logarithmic contraction rate in (2.9), which is due to a log-type stability estimate of G⁻¹. To obtain an estimator of the unknown coefficient n, in view of the link function Φ, it is often convenient to derive an estimator of F. The posterior mean $\bar F_N : = \mathbb{E}^{\Pi}[F|Y^{(N)},X^{(N)}]$ of $\Pi_N(\cdot|Y^{(N)},X^{(N)})$, which can be approximated numerically by an MCMC algorithm, is the most natural choice of estimator. In light of theorem 2.5, we can also prove a contraction rate for the convergence $\bar F_N$ to F₀.

Theorem 2.6. Assume that the hypotheses of theorem 2.5 hold. Then, there exists a $\left.\tilde{C} = \tilde{C}(\sigma, n_0,K,s,t,D,\kappa,\epsilon,M_0)\gt0\right.$ such that

$$\begin{equation} \mathbb{P}_{F_0}^N \left( \|\bar F_N-F_0\|_{L^\infty}>\tilde C\left(\text{ln} N\right)^{-\frac{2t-3}{2t+3}+\epsilon} \right) \to 0 \quad \textrm{as}\, N\to\infty. \end{equation} \tag{ 2.10 }$$

Corollary 2.7. Assume that the hypotheses of theorem 2.5 hold. Then there exists a sufficiently large $\tilde{C}^{^{\prime}} = \tilde{C}^{^{\prime}}(\sigma, n_0,K,s,t,D,\kappa,\epsilon,M_0)\gt0$ such that

$$\begin{equation} \mathbb{P}_{n_0}^N \left( \|\Phi\circ\bar F_N-n_0\|_{L^\infty}>\tilde{C}^{^{\prime}}\left(\text{ln} N\right)^{-\frac{2t-3}{2t+3}+\epsilon} \right) \to 0 \quad \textrm{as}\, N\to\infty. \end{equation} \tag{ 2.11 }$$

The logarithmic contraction rates obtained in theorems 2.5, 2.6, and corollary 2.7 inherit from the log-type estimate of the inverse scattering problem. Nonetheless, in the next theorem, we will show that this contraction rate for the estimator $\hat n: = \Phi\circ\bar F_N$ is optimal is the statistical minimax sense, at least up to the exponent of $\text{ln} N$. We first define a parameter space. Let $s\gt3/2$ be an integer, β > 0, and define

$$\begin{align*} \hat{\mathcal F}_{\beta}^s = \left\{n = 1+q:\; q\in H_{\overline{B_{1/2}}}^s\left(\mathbb{R}^3\right)\;\;\mbox{with}\;\; \|q\|_{H^s}\unicode{x2A7D}\beta\right\}. \end{align*}$$

Theorem 2.8. For integer $s\gt3/2$, there exists $\beta = \beta(s)\gt0$ such that for any $\delta\gt\frac{5s}{3}$ and $\varepsilon\in (0,1)$, we have

$$\begin{equation} \inf_{\tilde n}\sup_{n\in\hat{\mathcal F}^s_{\beta}}\mathbb{P}_{n}^N \left( \lVert \tilde n-n \rVert_{L^\infty}>\frac 12\left(\text{ln} N\right)^{-\delta}\right)>1-\varepsilon, \end{equation} \tag{ 2.12 }$$

for all N large enough, where the infimum is taken over all measurable functions $\tilde n = \tilde n(Y,X)$ of the data $(Y,X)\sim{\mathbb{P}}_n^N$.

2.3.2. High-dimensional Gaussian sieve priors.

From computational perspective, it is useful to consider sieve priors that are finite-dimensional approximations of the function space supporting the prior. Here we will use a randomly truncated Karhunen–Loéve type expansion in terms of Daubechies wavelets considered in [GN20, appendix B] or [GN21, chapter 4]. Let $\left\{\Psi_{\ell r} : \ell\unicode{x2A7E} -1, r\in\mathbb{Z}^3 \right\}$ be the (3-dimensional) compactly supported Daubechies wavelets, which forms an orthonormal basis of $L^{2}(\mathbb{R}^{3})$. Let $\mathcal{K}$ be a compact subset in D and let $R_{\ell} = \left\{r \in \mathbb{Z}^{d} : \mathrm{supp}\,(\Psi_{\ell r}) \cap \mathcal{K} \neq \emptyset \right\}$. Let $\mathcal{K}^{^{\prime}}$ be another compact subset in D such that $\mathcal{K}^{^{\prime}} \subsetneq \mathcal{K}$, and let $\chi\in C_c^\infty({D})$ be a cut-off function with χ = 1 on ${\mathcal K}^{^{\prime}}$. Let $s\gt1+3/2$ and consider the prior

$$\begin{equation} \Pi_J^{^{\prime}}\equiv \Pi_J^{^{\prime}}\left[s\right] = {\mathcal L}\left(\chi F_{J}\right),\quad F_{J} = \sum_{\begin{subarray}{c}-1\unicode{x2A7D}\ell\unicode{x2A7D} J \\ r\in{\mathcal R}_\ell\end{subarray}} 2^{-\ell s}F_{\ell r}\Psi_{\ell r},\;\; F_{\ell r}\overset{\mathrm{iid}}{\sim}{\mathcal N}\left(0,1\right), \end{equation} \tag{ 2.13 }$$

where $J\in\mathbb{N}$ is the truncation level. In fact $\Pi^{^{\prime}}_J$ defines a centered Gaussian prior that is supported on the finite-dimensional space

$$\begin{align*} {\mathcal H}_J = \mathrm{span}\left\{\chi\Psi_{\ell r}, : -1\unicode{x2A7D}\ell\unicode{x2A7D} J,\, r\in{\mathcal R}_\ell\right\}\subset C\left(D\right), \end{align*}$$

with RKHS norm satisfying [GN20, (B2)]. As above, we consider the 're-scaled' prior $\Pi_N$ defined in (2.5) with $F^{^{\prime}}\sim\Pi^{^{\prime}}_J$.

In analogy to theorem 2.4, we will derive a contraction rate for the statistical model (2.2b ) with the prior $\Pi_N$ defined above. As in theorem 2.4, we obtain the same contraction rate in the L²-prediction risk of the regression function.

Theorem 2.9. Let $t \unicode{x2A7E} 1$, $s\gt t+3/2$ be integers, and $\Pi_N\equiv \Pi_{N}[s]$ be the 're-scaled' prior defined in (2.5) with priors $F^{^{\prime}}\sim\Pi^{^{\prime}}_{J_{N}} \equiv \Pi^{^{\prime}}_{J_{N}}[s]$ where $2^{J_N}\simeq N^{1/(2s+5)}$. Denote $\Pi_N(\cdot|Y^{(N)},X^{(N)})$ the posterior distribution arising from the noisy discrete measurements $(Y^{(N)},X^{(N)})$ of (2.2b ). Let $F_0\in H^{s}_{\mathcal K}(D)$ and $\delta_N = N^{-(s+1)/(2s+5)}$. Then for any K > 0, there exists a large L > 0, depending on $\sigma, F_0, K, s, D, S_1,S_2$, such that

$$\begin{align} & \Pi_N\left(F:\|{\mathcal G}\left(F\right)-{\mathcal G}\left(F_0\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}>L\delta_N|Y^{\left(N\right)},X^{\left(N\right)}\right)\nonumber\\ & \quad = O_{P^N_{F_0}}\left(e^{-KN\delta_N^2}\right)\;\;\mbox{as}\;\; N\to\infty, \end{align} \tag{ 2.14 }$$

and for sufficiently large $L^{^{\prime}}\gt0$, depending on $\sigma, K, s, t$, we have

$$\begin{equation} \Pi_N\left(F:\|F\|_{C^t}>L^{^{\prime}}|Y^{\left(N\right)},X^{\left(N\right)}\right) = O_{P^N_{F_0}}\left(e^{-KN\delta_N^2}\right)\;\;\mbox{as}\;\; N\to\infty. \end{equation} \tag{ 2.15 }$$

The proof of theorem 2.9 only requires minor modification from the proof of theorem 2.4, and all necessary modifications are listed in [GN20, section 3.2]. Therefore we omit the details. Having established theorem 2.9, similar to [GN20, proposition 7], theorems 2.5 and 2.6 can be directly extended to the case of Gaussian sieve priors.

Remark 2.10. As in [GN20], it is likely to extend the results for the Gaussian sieve priors with deterministic truncation level to randomly truncated ones, where the truncation level J itself is an appropriate random number. However, due to the log-type stability estimate of the inverse scattering problem, such extension is highly nontrivial. We will discuss the Bayes method with randomly truncated sieve priors for the inverse scattering problem in another paper.

The main theme of this section is to prove theorem 2.5, 2.6 and 2.8.

Proof of theorem 2.5. For each M > 0 satisfies $\lVert 1-n \rVert_{H^{t}(D)} \vee \lVert 1-n_{0} \rVert_{H^{t}(D)} \unicode{x2A7D} M$, one has the stability estimate of G⁻¹ in theorem A.1:

$$\begin{equation} \|n-n_0\|_{L^\infty\left(D\right)}\unicode{x2A7D} C\left[-\text{ln}^-\left(\|G\left(n\right)-G\left(n_0\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}\right)\right]^{-\alpha}, \end{equation} \tag{ 3.1 }$$

where $\alpha = \frac{2t-3}{2t+3}-\epsilon\gt0$ and $C = C(D,t,\kappa,M,\epsilon)$.

If $\lVert n \rVert_{C^{t}(D)} \unicode{x2A7D} M^{^{\prime}}$ for some $M^{^{\prime}}\unicode{x2A7E} M_0$, since t is an integer, then $\|1-n\|_{H^t(D)}\unicode{x2A7D} CM^{^{\prime}}$ with $C = C(t,D)$. We now set the constant $M = (CM^{^{\prime}})\vee\|1-n_0\|_{H^t(D)}$. In view of (2.8a ), (2.8b ), and (3.1), for any K > 0, there exists large constants $L, M^{^{\prime}}$ and Lʹ (Lʹ is determined by L and Mʹ) such that

$$\begin{align*} \begin{aligned} &\tilde\Pi_N\left(n:\|n-n_0\|_{L^\infty\left(D\right)}>|\text{ln}\left(L\delta_N\right)|^{-\alpha}|Y^{\left(N\right)},X^{\left(N\right)}\right)\nonumber\\ & \quad \unicode{x2A7D}\tilde\Pi_N\left(n:\|G\left(n\right)-G\left(n_0\right)\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}>L^{^{\prime}}\delta_N|Y^{\left(N\right)},X^{\left(N\right)}\right)\nonumber\\ &\qquad +\tilde\Pi_N\left(n:\|n\|_{C^t\left(D\right)}>M^{^{\prime}}|Y^{\left(N\right)},X^{\left(N\right)}\right) = O_{\mathbb{P}^N_{n_0}}\left(e^{-KN\delta_N^2}\right), \end{aligned} \end{align*}$$

which conclude our result. □

Having proved theorem 2.5, we then establish theorem 2.6 using the contraction rate in theorem 2.5 and the link function Φ.

Proof of theorem 2.6. We proof the theorem by modifying some ideas in [GN20, theorem 6]. By the Jensen's inequality, it suffices to prove that there exists a large $\tilde{C}\gt0$ such that

$$\begin{align*} \mathbb{P}_{F_0}^N\left(\mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]>\tilde{C}\left(\text{ln} N\right)^{-\frac{2t-3}{2t+3}+\epsilon}\right)\to 0 \quad \text{as $N\rightarrow\infty$}. \end{align*}$$

For a large M > 0 to be chosen later, we write

$$\begin{align} \begin{aligned} \mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]& = \mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\unicode{x2A7D} M}|Y^{\left(N\right)},X^{\left(N\right)}\right]\\ & \quad +\mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}>M}|Y^{\left(N\right)},X^{\left(N\right)}\right]. \end{aligned} \end{align} \tag{ 3.2 }$$

Part I: Estimating $\mathbb{E}^\Pi[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\gt M}|Y^{(N)},X^{(N)}]$. By using Cauchy-Schwartz inequality, it is easy to see that

$$\begin{equation} \begin{aligned} & \mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}>M}|Y^{\left(N\right)},X^{\left(N\right)}\right] \\ & \quad \unicode{x2A7D} \sqrt{\mathbb{E}^\Pi\left[\|F-F_0\|^2_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]}\sqrt{\Pi_N\left(F: \|F\|_{C^t}>M|Y^{\left(N\right)},X^{\left(N\right)}\right)}. \end{aligned} \end{equation} \tag{ 3.3 }$$

Let B > 0 be a constant to be determined later. By using (2.7b ), one can choose a sufficiently large $M = M(\sigma,B,s,t)\gt0$ such that

$$\begin{align*} \begin{aligned} & \mathbb{P}_{F_0}^N \left(\mathbb{E}^\Pi\left[\|F-F_0\|^2_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]\Pi_N\left(F: \|F\|_{C^t}>M|Y^{\left(N\right)},X^{\left(N\right)}\right)>\left(\text{ln} N\right)^{-\frac{2\left(2t-3\right)}{2t+3}+2\epsilon} \right)\\ & \quad \unicode{x2A7D} \mathbb{P}_{F_0}^N \left( \mathbb{E}^\Pi\left[\|F-F_0\|^2_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]e^{-B N\delta_N^2}>\left(\text{ln} N\right)^{-\frac{2\left(2t-3\right)}{2t+3}+2\epsilon} \right)+o\left(1\right). \end{aligned} \end{align*}$$

We now define ${\mathcal B}_N$ by

$$\begin{equation} {\mathcal B}_N = \left\{F: \mathbb{E}^1_{F_0}\left[\text{log}\frac{p_{F_0}\left(Y_1,X_1\right)}{p_{F}\left(Y_1,X_1\right)}\right]\unicode{x2A7D}\delta_N^2, \mathbb{E}^1_{F_0}\left[\text{log}\frac{p_{F_0}\left(Y_1,X_1\right)}{p_{F}\left(Y_1,X_1\right)}\right]^2\unicode{x2A7D}\delta_N^2\right\}. \end{equation} \tag{ 3.4 }$$

By using [GN20, lemmas 16 and 23], we have

$$\begin{equation*} \Pi_N\left({\mathcal B}_N\right)\unicode{x2A7E} ae^{-AN\delta_N^2} \quad \textrm{for some}\, a,A>0. \end{equation*}$$

Let $\nu(\cdot) = \Pi_N\left(\cdot\cap{\mathcal B}_N\right)/\Pi_N\left({\mathcal B}_N\right)$ and set the event

$$\begin{equation} {\mathcal C}_N = \left\{\int_{{\mathcal B}_N}\prod_{i = 1}^N\frac{p_F}{p_{F_0}}\left(Y_i,X_i\right) \, \mathrm{d}\nu\left(F\right)\unicode{x2A7E} e^{-2N\delta_N^2}\right\}. \end{equation} \tag{ 3.5 }$$

By [GN21, lemma 7.3.2], we can show that

$$\begin{equation} \mathbb{P}^N_{F_0}\left({\mathcal C}_N\right)\to 1 \left(\mathrm{i.e.}\, \mathbb{P}^N_{F_0}\left({\mathcal C}^c_N\right)\to 0\right)\, \textrm{as}\, N\to\infty. \end{equation} \tag{ 3.6 }$$

By using the properties (3.6) of $\mathcal{C}_{N}$, we obtain

$$\begin{align*} \begin{aligned} &\mathbb{P}_{F_0}^N \left( \mathbb{E}^\Pi\left[\|F-F_0\|^2_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]e^{-BN\delta_N^2}>\left(\text{ln} N\right)^{-\frac{2\left(2t-3\right)}{2t+3}+2\epsilon} \right)\\ & \quad \unicode{x2A7D} \mathbb{P}_{F_0}^N\left(\frac{\int_{C\left(D\right)}\|F-F_0\|^2_{L^\infty}\prod_{i = 1}^Np_{F}/p_{F_0}\left(Y_i,X_i\right) \, \mathrm{d}\Pi_N\left(F\right)}{\Pi\left({\mathcal B}_N\right)\int_{{\mathcal B}_N}\prod_{i = 1}^Np_{F}/p_{F_0}\left(Y_i,X_i\right) \, \mathrm{d}\nu\left(F\right)}e^{- B N\delta_N^2}>\left(\text{ln} N\right)^{-\frac{2\left(2t-3\right)}{2t+3}+2\epsilon},{\mathcal C}_N\right) +o\left(1\right)\\ & \quad \unicode{x2A7D} \mathbb{P}_{F_0}^N\left(\int_{C\left(D\right)}\|F-F_0\|^2_{L^\infty}\prod_{i = 1}^{N} \frac{p_{F}}{p_{F_0}}\left(Y_i,X_i\right) \, \mathrm{d}\Pi_N\left(F\right)>\left(\text{ln} N\right)^{-\frac{2\left(2t-3\right)}{2t+3}+2\epsilon}ae^{\left(B-A-2\right)N\delta_N^2}\right) +o\left(1\right), \end{aligned} \end{align*}$$

which is bounded from above, using Markov's inequality and Fubini's theorem, by

$$\begin{equation} \begin{aligned} & \left(\text{ln} N\right)^{\frac{2\left(2t-3\right)}{2t+3}-2\epsilon}a^{-1}e^{-\left(B-A-2\right)N\delta_N^2} \int_{C\left(D\right)}\|F-F_0\|_{L^\infty}^2 \mathbb{E}_{F_0}^N\left(\prod_{i = 1}^N\frac{p_F}{p_{F_0}}\left(Y_i,X_i\right)\right) \,\mathrm{d} \Pi_N\left(F\right) \\ & \quad = \left(\text{ln} N\right)^{\frac{2\left(2t-3\right)}{2t+3}-2\epsilon}a^{-1}e^{-\left(B-A-2\right)N\delta_N^2} \int_{C\left(D\right)}\|F-F_0\|_{L^\infty}^2 \,\mathrm{d} \Pi_N\left(F\right). \end{aligned} \end{equation} \tag{ 3.7 }$$

By using Fernique's theorem (see e.g. [GN21, exercises 2.1.1, 2.1.2 and 2.1.5]) one has $\mathbb{E}^{\Pi_{N}}\lVert F \rVert_{L^{\infty}}^{2} \lt \infty$. Taking $B \gt A+2$, from (3.7) we conclude

$$\begin{equation} \mathbb{P}_{F_0}^N \left( \mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}>M}|Y^{\left(N\right)},X^{\left(N\right)}\right]>\left(\text{ln} N\right)^{-\frac{\left(2t-3\right)}{2t+3}+\epsilon} \right) \to 0. \end{equation} \tag{ 3.8 }$$

Part II: Estimating $\mathbb{E}^\Pi[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t} \unicode{x2A7D} M}|Y^{(N)},X^{(N)}]$. The above discussions still valid if we replace M by a (possibly larger) $M = M(\sigma,B,s,t,F_{0})\gt0$ with $\lVert F_{0} \rVert_{L^{\infty}} \unicode{x2A7D} M$. Since $n = \Phi\circ F$ and $n_0 = \Phi\circ F_0$, by (i), mean value theorem and inverse function theorems, there exists η lying between $n_{0}(x)$ and n(x) such that

$$\begin{equation*} \lvert F\left(x\right) - F_{0}\left(x\right) \rvert = \frac{1}{\lvert \Phi^{^{\prime}}\left(\Phi^{-1}\left(\eta\right)\right) \rvert}\lvert n\left(x\right)-n_{0}\left(x\right) \rvert, \end{equation*}$$

for all $x\in D$. Since $F,F_{0} \in \left[\Phi(-M),\Phi(M)\right]$, by (i) we reach

$$\begin{align*} |F\left(x\right)-F_0\left(x\right)|\unicode{x2A7D}\frac{1}{\min_{z\in\left[-M,M\right]}\Phi^{^{\prime}}\left(z\right)}|n\left(x\right)-n_0\left(x\right)|\lesssim|n\left(x\right)-n_0\left(x\right)| \quad \text{for all $x\in D$.} \end{align*}$$

Therefore we see that

$$\begin{align*} \|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\unicode{x2A7D} M}\lesssim\|n-n_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\unicode{x2A7D} M}, \end{align*}$$

and

$$\begin{equation} \begin{aligned} & \mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\unicode{x2A7D} M}|Y^{\left(N\right)},X^{\left(N\right)}\right]\lesssim \mathbb{E}^{\tilde\Pi}\left[\|n-n_0\|_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]. \end{aligned} \end{equation} \tag{ 3.9 }$$

Let C > 0 be a constant to be determined later. From (3.9), we see that

$$\begin{equation*} \begin{aligned} & \mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\unicode{x2A7D} M}|Y^{\left(N\right)},X^{\left(N\right)}\right]\lesssim \mathbb{E}^{\tilde\Pi}\left[\|n-n_0\|_{L^\infty}|Y^{\left(N\right)},X^{\left(N\right)}\right]\\ & \quad \unicode{x2A7D} C\left(\text{ln} N\right)^{-\frac{2t-3}{2t+3}+\epsilon} + \mathbb{E}^{\tilde\Pi}\left[\|n-n_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|n-n_0\|_{L^\infty}\unicode{x2A7E} C\left(\text{ln} N\right)^{-\frac{2t-3}{2t+3}+\epsilon}}|Y^{\left(N\right)},X^{\left(N\right)}\right]. \end{aligned} \end{equation*}$$

We can modify the arguments as in Part I above by replacing the event $\left\{F : \lVert F \rVert_{C^{t}}\gt M \right\}$ (resp. (2.7b )) by the event $\left\{n : \lVert n-n_{0} \rVert_{L^{\infty}}\gt C(\text{ln} N)^{-\frac{2t-3}{2t+3}+\epsilon} \right\}$ (resp. (2.9)) with $C = C(\sigma, n_0,K,s,t,D,\kappa,\epsilon,M_0)\gt0$ given in theorem 2.5 to show that

$$\begin{equation} \mathbb{P}_{F_0}^N\left(\mathbb{E}^\Pi\left[\|F-F_0\|_{L^\infty}{\unicode{x1D7D9}}_{\|F\|_{C^t}\unicode{x2A7D} M}|Y^{\left(N\right)},X^{\left(N\right)}\right]>C\left(\text{ln} N\right)^{-\frac{\left(2t-3\right)}{2t+3}+\epsilon}\right)\to 0. \end{equation} \tag{ 3.10 }$$

Finally, putting together (3.2), (3.8), and (3.10) yields theorem 2.6 with $\tilde{C} = \text{max}\{C,1\}$. □

Next, we prove the optimality of contraction rate in corollary 2.7 in the minimax sense.

Proof of theorem 2.8. We apply the method in the proof of the lower bound [AN19, theorem 2] to our case here. The idea is to find $n_0, n_1\in\hat{\mathcal{F}}^s_{\beta}$ (both are allowed to depend on N) such that, for some small ζ sufficiently small,

• (a)  
  $\|n_0-n_1\|_{L^{\infty}} \unicode{x2A7E} \theta_{N,\delta}: = (\text{ln} N)^{-\delta}$.
• (b)  
  $\textrm{KL}(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N})\unicode{x2A7D}\zeta$,

where $p_{n}^{\otimes N}$ is the Radon Nikodym derivative of the joint law $\mathbb{P}_{n}^{N}$ and the Kullback–Leibler divergence $\textrm{KL}(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N})$ is defined by

$$\begin{equation} {\mathrm{KL}}\left(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N}\right) = \mathbb{E}_{n_0}^N\left[\text{log}\frac{p_{n_0}^{\otimes N}}{p_{n_1}^{\otimes N}}\right]. \end{equation} \tag{ 3.11 }$$

By independence, we note that

$$\begin{align*} {\mathrm{KL}}\left(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N}\right) = N\mathbb{E}_{n_0}^1\left[\text{log}\frac{p_{n_0}\left(Y_1,X_1\right)}{p_{n_1}\left(Y_1,X_1\right)}\right], \end{align*}$$

and [GN20, lemma 23] implies

$$\begin{equation} {\mathrm{KL}}\left(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N}\right) = \frac{N}{2\sigma^2}\|{G}\left(n_0\right)-{G}\left(n_1\right)\|_{L^2\left({\mathcal S}^2\times{\mathcal S}^2\right)}^2, \end{equation} \tag{ 3.12 }$$

then using the standard arguments as in [GN21, section 6.3.1] (see also [Tsy09, chapter 2]), we conclude the theorem.

For the sake of completeness, here we present the details. From condition (a), we see that $\psi = {\unicode{x1D7D9}}_{\{\|\tilde n-n_1\|_{L^{\infty}}\lt\|\tilde n-n_0\|_{L^{\infty}}\}}$ yields a test of

$$\begin{equation} H_{0}:n = n_{0}\quad \mathrm{against}\, \quad H_{1}:n = n_{1}. \end{equation} \tag{ 3.13 }$$

It follows from a general reduction principle that

$$\begin{align*} \inf_{\tilde n}\sup_{n\in\hat{\mathcal F}_{\beta}^s}\mathbb{P}_n^N \left( \|\tilde n-n\|_{L^{\infty}}\unicode{x2A7E}\frac 12\theta_{N,\delta} \right) \unicode{x2A7E} \inf_\psi\text{max} \left\{\mathbb{P}_{n_0}^N\left(\psi\ne 0\right),\mathbb{P}_{n_1}^N\left(\psi\ne 1\right) \right\}, \end{align*}$$

where the second infimum is over all tests ψ of (3.13). Similar to the proof of [GN21, theorem 6.3.2], we introduce the event

$$\begin{align*} \Omega = \left\{\frac{p_{n_0}^{\otimes N}}{p_{n_1}^{\otimes N}}\unicode{x2A7E} \frac 12\right\}. \end{align*}$$

Note that

$$\begin{equation*} \mathbb{P}_{n_0}^N\left(\psi\ne 0\right) \unicode{x2A7E} \mathbb{E}_{n_{0}}^{N} \left[\unicode{x1D7D9}_{\Omega}\psi\right] = \mathbb{E}_{n_1}^N\left[\frac{p_{n_0}^{\otimes N}}{p_{n_1}^{\otimes N}}{\unicode{x1D7D9}}_{\Omega}\psi\right]\unicode{x2A7E}\frac 12\left[\mathbb{P}_{n_1}^N\left(\psi = 1\right)-\mathbb{P}_{n_1}^N\left(\Omega^c\right)\right]. \end{equation*}$$

Let $p_1 = \mathbb{P}_{n_1}^N(\psi = 1)$, then

$$\begin{align*} \begin{aligned} \text{max} \left\{\mathbb{P}_{n_0}^N\left(\psi\ne 0\right),\mathbb{P}_{n_1}^N\left(\psi\ne 1\right) \right\} &\unicode{x2A7E}\text{max} \left\{\frac 12\left(p_1-\mathbb{P}_{n_1}^N\left(\Omega^c\right)\right), 1-p_1 \right\}\\ & \unicode{x2A7E}\inf_{p\in\left[0,1\right]}\text{max} \left\{\frac 12\left(p-\mathbb{P}_{n_1}^N\left(\Omega^c\right)\right), 1-p \right\}. \end{aligned} \end{align*}$$

It is clear that the infimum above is attained when $\frac 12(p-\mathbb{P}_{n_1}^N(\Omega^c)) = 1-p$ and has the value $\frac 13\mathbb{P}_{n_1}^N(\Omega)$. Hence,

$$\begin{equation} \inf_{\tilde n}\sup_{n\in\hat{\mathcal F}_{\beta}^s}\mathbb{P}_n^N \left( \|\tilde n-n\|_\infty\unicode{x2A7E}\frac 12\theta_{N,\delta} \right) \unicode{x2A7E}\frac 13\mathbb{P}_{n_1}^N\left(\Omega\right). \end{equation} \tag{ 3.14 }$$

Next, let us estimate

$$\begin{align*} \begin{aligned} \mathbb{P}_{n_1}^N\left(\Omega\right)& = \mathbb{P}_{n_1}^N\left[{p_{n_0}^{\otimes N}}/{p_{n_1}^{\otimes N}}\unicode{x2A7D} 2\right] = 1-\mathbb{P}_{n_1}^N\left[\text{log}\left({p_{n_0}^{\otimes N}}/{p_{n_1}^{\otimes N}}\right)>\text{log} 2\right]\\ &\unicode{x2A7E} 1-\mathbb{P}_{n_1}^N\left[|\text{log}\left({p_{n_0}^{\otimes N}}/{p_{n_1}^{\otimes N}}\right)|>\text{log} 2\right]\\ &\unicode{x2A7E} 1-\left(\text{log} 2\right)^{-1}\mathbb{E}_{n_1}^N\left[|\text{log}\left({p_{n_0}^{\otimes N}}/{p_{n_1}^{\otimes N}}\right)|\right]\quad\left(\text{by Markov's inequality}\right). \end{aligned} \end{align*}$$

Using the second Pinsker inequality [GN21, proposition 6.1.7b] and condition (b), we have

$$\begin{align*} \begin{aligned} \frac 13\mathbb{P}_{n_1}^N\left(\Omega\right)&\unicode{x2A7E}\frac 13\left(1-\left(\text{log} 2\right)^{-1}\left[{\mathrm{KL}}\left(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N}\right)+\sqrt{2{\mathrm{KL}}\left(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N}\right)}\right]\right)\\ &\unicode{x2A7E} \frac 13\left(1-\left(\text{log} 2\right)^{-1}\left(\zeta+\sqrt{2\zeta}\right)\right). \end{aligned} \end{align*}$$

We now choose ζ sufficiently small such that the last term above is bounded below by $1-\varepsilon$. Then the estimate (2.12) follows in view of (3.14).

The remaining task is to find $n_0, n_1$ satisfying conditions (a) and (b). For $\delta\gt\frac{5s}{3}$, setting $\theta = \theta_{N,\delta} = (\text{ln} N)^{-\delta}$ in theorem B.1, there exist $n_0, n_1 \in\hat{\mathcal F}^s_{\beta}$ satisfying $\|n_0-n_1\|_\infty\gt\theta_{N,\delta}$ and

$$\begin{align*} \|u_{n_0}^\infty-u_{n_1}^\infty\|_{L^2\left({\mathcal S}^2\times{\mathcal S}^2\right)}\unicode{x2A7D} 2 \text{exp} \left(-\theta^{-\frac{3}{5s}}\right) = 2\text{exp}\left(-\left(\text{ln} N\right)^{\frac{3\delta}{5s}}\right). \end{align*}$$

To verify condition (b), we use (3.12) to conclude that

$$\begin{align*} \begin{aligned} {\mathrm{KL}}\left(p_{n_0}^{\otimes N},p_{n_1}^{\otimes N}\right)& = \frac{N}{2\sigma^2}\|u^\infty_{n_0}-u^\infty_{n_1}\|_{L^2\left({\mathcal S}^2\times{\mathcal S}^2\right)}^2 \unicode{x2A7D}\frac{4N}{2\sigma^2}\text{exp}\left(-2\left(\text{ln} N\right)^{\frac{3\delta}{5s}}\right)\\ & = \frac{2}{\sigma^2}\text{exp}\left(\text{ln} N-2\left(\text{ln} N\right)^{\frac{3\delta}{5s}}\right)\to 0\;\;\mbox{as}\;\; N\to\infty, \end{aligned} \end{align*}$$

since $\frac{3\delta}{5s}\gt1$. Therefore, we can make ζ as small as we wish by taking N sufficiently large. □

No new data were created or analysed in this study.

In what follows, assume that

$$\begin{equation} n\in C^1\left(\mathbb{R}^3\right),\;\;\mathrm{supp}\left(1-n\right)\subset D,\;\; 0\unicode{x2A7D} n\unicode{x2A7D} M_0\;\;\mbox{with}\;\; M_0\unicode{x2A7E} 1. \end{equation} \tag{ A.1 }$$

We first discuss the existence and uniqueness of the scattered field $u_n^\textrm{sca}$. It is known that $w: = u_n^\textrm{sca}$ satisfies the following boundary value problem [CCH23, (1.51)–(1.52)]:

$$\begin{equation} \left\{ \begin{aligned} &-\Delta w-\kappa^{2}nw = \kappa^2\left(n-1\right)u^{\mathrm{inc}}\quad\mbox{in}\;\; B_R,\\ &{\partial w}/{\partial r} = S_R\left(w|_{\partial B_R}\right)\quad\mbox{on}\;\;\partial B_R, \end{aligned} \right. \end{equation} \tag{ A.2 }$$

where B_R is a ball of radius R such that $\bar D\subset B_R$. Here $S_R: H^{1/2}(\partial B_R)\to H^{-1/2}(\partial B_R)$ is the Dirichlet-to-Neumann map, defined for $g\in H^{1/2}(\partial B_R)$ by $S_Rg = (\partial u_g/\partial r)|_{\partial B_R}$ , where u_g is the solution of the Helmholtz equation satisfying the Sommerfeld radiation condition in $R^3\setminus B_R$ and the Dirichlet condition $u_g = g$ on $\partial B_R$. It has been shown that

$$\begin{equation} \mathrm{Re}\langle S_R\left(v\right),v\rangle\unicode{x2A7D} 0\quad\mbox{and}\quad{\mathrm{Im}}\langle S_R\left(v\right),v\rangle\unicode{x2A7E} 0,\;\;\forall\;\, v\in H^{1/2}\left(\partial B_R\right), \end{equation} \tag{ A.3 }$$

where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between $H^{-1/2}(\partial B_R)$ and $H^{1/2}(\partial B_R)$, see, for example, [CCH23, definition 1.36 and (1.50)].

To proceed further, let us replace the right-hand side of the first equation in (A.2) by a general source term f with $\mathrm{supp}(f)\,\subset B_R$, i.e.

$$\begin{equation} \left\{ \begin{aligned} &-\Delta w-\kappa^2nw = f\quad\mbox{in}\;\; B_R,\\ &{\partial w}/{\partial r} = S_R\left(w|_{\partial B_R}\right)\quad\mbox{on}\;\;\partial B_R. \end{aligned} \right. \end{equation} \tag{ A.4 }$$

In view of the integration by parts, (A.4) is equivalent to the following variational formulation: find $w\in H^1(B_R)$ such that for all $v\in H^1(B_R)$,

$$\begin{align*} a_1\left(w,v\right)+a_2\left(w,v\right) = F\left(v\right), \end{align*}$$

where

$$\begin{align*} a_1\left(w,v\right) = \int_{B_R}\nabla w\cdot\nabla\bar v \, \mathrm{d} x-\kappa^2\int_{B_R}nw\hspace{1pt}\bar v \, \mathrm{d} x+\kappa^2\left(M_0+1\right)\int_{B_R}w\hspace{1pt}\bar v \, \mathrm{d} x-\langle S_R\left(w\right), \bar v\rangle, \end{align*}$$

$$\begin{align*} a_2\left(w,v\right) = -\kappa^2\left(M_0+1\right)\int_{B_R}w\hspace{1pt}\bar v \, \mathrm{d} x, \end{align*}$$

and

$$\begin{align*} F\left(v\right) = \int_{B_R}f\hspace{1pt}\bar v \, \mathrm{d} x. \end{align*}$$

We can see that

$$\begin{align*} {\mathrm{Re}}\hspace{2pt}a_1\left(w,w\right) = \int_{B_R}\left(|\nabla w|^2+k^2\left(M_0+1-n\right)|w|^2\right) \, \mathrm{d} x\unicode{x2A7E} C\left(\kappa\right)\|w\|^2_{H^1\left(B_R\right)}. \end{align*}$$

In other words, $a_1(\cdot,\cdot)$ is strictly coercive. Combining the Riesz representation theorem and the Lax–Milgram theorem, there exists an invertible operator ${\mathcal A}:H^1(B_R)\to (H^{1}(B_R))^\ast$, where $(H^1(B_R))^\ast$ is the dual space of $H^1(B_R)$, such that

$$\begin{align*} a_1\left(w,v\right) = \left({\mathcal A}w,v\right)_{H^1\left(B_R\right)}\quad\forall\;\, v\in H^1\left(B_R\right). \end{align*}$$

Similarly, define the bounded linear operator ${\mathcal B}:H^1(B_R)\to (H^1(B_R))^\ast$ by $a_2(w,v) = ({\mathcal B}w,v)_{H^1(B_R)}$. It is not difficult to see that ${\mathcal B}$ is compact. Consequently, ${\mathcal A}+{\mathcal B}$ is a Fredholm operator. By the Fredholm alternative, ${\mathcal A}+{\mathcal B}:H^1(B_R)\to (H^1(B_R))^\ast$ is bounded invertible provided the kernel of ${\mathcal A}+{\mathcal B}$ is trivial, which follows the uniqueness of the scattered solution (by the combination of the Rellich lemma and the unique continuation property). Furthermore, we have the following estimate:

$$\begin{equation} \|w\|_{H^1\left(B_R\right)}\unicode{x2A7D} C\|f\|_{\left(H^1\left(B_R\right)\right)^\ast}, \end{equation} \tag{ A.5 }$$

where $C = C(D,\kappa,M_0)$. Let $f = \kappa^2(n-1)u^\textrm{inc}$ and $w = u_n^\textrm{sca}$, then (A.5) implies

$$\begin{equation} \|u_n^{\mathrm{sca}}\|_{H^1\left(B_R\right)}\unicode{x2A7D} C\|1-n\|_{L^{2}\left(D\right)}, \end{equation} \tag{ A.6 }$$

uniformly in $\theta\in\mathcal{S}^2$. We now choose $R^{^{\prime}}\lt R$ such that $\overline{D} \subset B_{R^{^{\prime}}}$. By the interior estimate [GT01, theorem 8.8], we further have

$$\begin{align*} \|u_n^{\mathrm{sca}}\|_{H^2\left(B_{R^{^{\prime}}}\right)}\unicode{x2A7D} C\|1-n\|_{L^{2}\left(D\right)}, \end{align*}$$

which, by the Sobolev imbedding theorem, implies

$$\begin{equation} \|u_n^{\mathrm{sca}}\|_{C\left(D\right)}\unicode{x2A7D} C\|1-n\|_{L^{2}\left(D\right)}. \end{equation} \tag{ A.7 }$$

The scattering amplitude $u_n^\infty(\theta^{^{\prime}},\theta)$ can be expressed explicitly by

$$\begin{equation} \begin{aligned} u_n^\infty\left(\theta^{^{\prime}},\theta\right)& = -\frac{\kappa^2}{4\pi}\int_{\mathbb{R}^3}e^{-\mathbf{i}\kappa\theta^{^{\prime}}\cdot y}\left(1-n\right)\left(y\right)u\left(y,\theta\right)\,\mathrm{d} y\\ & = -\frac{\kappa^2}{4\pi}\int_{D}e^{-\mathbf{i}\kappa\theta^{^{\prime}}\cdot y}\left(1-n\right)\left(y\right)u\left(y,\theta\right)\,\mathrm{d} y, \end{aligned} \end{equation} \tag{ A.8 }$$

where $u(y,\theta) = u^\textrm{inc}(y,\theta)+u_n^\textrm{sca}(y,\theta)$ is the total field with $u^\textrm{inc}(y,\theta) = e^{\mathbf{i}\kappa y\cdot\theta}$, see [CCH23, (1.22)] or [CK19, (8.28)] or [Ser17, p 232]. From (A.7) and (A.8), we have

$$\begin{equation} \|u_n^\infty\|_{L^\infty\left(\mathcal{S}^2\times\mathcal{S}^2\right)}\unicode{x2A7D} C\|1-n\|_{L^{2}\left(D\right)}\unicode{x2A7D} S<\infty, \end{equation} \tag{ A.9 }$$

with $S = S(D,k,M_0) = C(1+M_0)$. Since

$$\begin{equation*} -\Delta \nabla u_{n}^{\mathrm{sca}} - \kappa^{2} n \nabla u_{n}^{\mathrm{sca}} = \kappa^{2} \nabla n u_{n}^{\mathrm{sca}} + \kappa^{2} \nabla n u^{\mathrm{inc}} + \kappa^2\left(n-1\right) \nabla u^{\mathrm{inc}} \quad \textrm{in }\, B_{R}, \end{equation*}$$

applying the interior estimate and the Sobolev imbedding theorem again, we have that

$$\begin{equation} \|u_n^{\mathrm{sca}}\|_{C^1\left(D\right)}\unicode{x2A7D} C\left(1+\|n\|_{C^1\left(D\right)}\right)\|1-n\|_{L^\infty\left(D\right)} \unicode{x2A7D} C\left(1+\|n\|_{C^1\left(D\right)}\right), \end{equation} \tag{ A.10 }$$

with $C = C(D,k,M_0)$.

Next, assume that $n_1, n_2$ satisfy (A.1). For each open set Ω in Euclidean space, we observe that

$$\begin{align} \begin{aligned} \lVert f_{1}f_{2} \rVert_{\left(H^{1}\left(\Omega\right)\right)^{*}}& = \sup_{\lVert \varphi \rVert_{H^{1}\left(\Omega\right)} \unicode{x2A7D} 1} \left| \int_{\Omega} f_{1}f_{2}\varphi \, \mathrm{d} x \right| \\ & \unicode{x2A7D} \lVert f_{1} \rVert_{\left(H^{1}\left(\Omega\right)\right)^{*}} \sup_{\lVert \varphi \rVert_{H^{1}\left(\Omega\right)} \unicode{x2A7D} 1} \lVert f_{2}\varphi \rVert_{H^{1}\left(\Omega\right)} \unicode{x2A7D} C \lVert f_{1} \rVert_{\left(H^{1}\left(\Omega\right)\right)^{*}} \lVert f_{2} \rVert_{C^{1}\left(\Omega\right)}. \end{aligned} \end{align} \tag{ A.11 }$$

Let $w = u^\textrm{sca}_{n_2}-u^\textrm{sca}_{n_1}$ and $f = \kappa^2(n_2-n_1)u^\textrm{sca}_{n_1}+\kappa^2(n_2-n_1)u^\textrm{inc}$, then combining (A.5), (A.7), (A.10) and (A.11), yields

$$\begin{equation} \|u_{n_1}^{\mathrm{sca}}-u_{n_2}^{\mathrm{sca}}\|_{H^1\left(B_R\right)}\unicode{x2A7D} C\left(1+\|n_1\|_{C^1\left(D\right)}\right)\|n_1-n_2\|_{\left(H^1\left(D\right)\right)^\ast}, \end{equation} \tag{ A.12 }$$

uniformly in θ and $C = C(D,k,M_0)$. Then it yields from (A.8) that

$$\begin{align} \begin{aligned} &\left(\int_{\mathcal{S}^2\times\mathcal{S}^2}|\left(u_{n_1}^\infty-u_{n_2}^\infty\right)\left(\theta^{^{\prime}},\theta\right)|^2 \,\mathrm{d}\omega\right)^{\frac{1}{2}}\\ & \quad = \frac{\kappa^2}{4\pi}\left(\int_{\mathcal{S}^2\times\mathcal{S}^2}\left|\int_{D} e^{-\mathbf{i}\kappa\theta^{^{\prime}}\cdot y}\left[\left(1-n_1\right)\left(e^{\mathbf{i}\kappa\theta\cdot y}+u_{n_1}^{\mathrm{sca}}\right)-\left(1-n_2\right)\left(e^{\mathbf{i}\kappa\theta\cdot y}+u_{n_2}^{\mathrm{sca}}\right)\right] \, \mathrm{d} y\right|^2 \,\mathrm{d}\omega\right)^{\frac{1}{2}}\\ & \quad \unicode{x2A7D}\frac{\kappa^2}{4\pi}\int_D\left(\int_{\mathcal{S}^2\times\mathcal{S}^2}\left|e^{-\mathbf{i}\kappa\theta^{^{\prime}}\cdot y}\left[\left(1-n_1\right)\left(e^{\mathbf{i}\kappa\theta\cdot y}+u_{n_1}^{\mathrm{sca}}\right)-\left(1-n_2\right)\left(e^{\mathbf{i}\kappa\theta\cdot y}+u_{n_2}^{\mathrm{sca}}\right)\right]\right|^2 \,\mathrm{d}\omega\right)^{\frac{1}{2}} \, \mathrm{d} y\\ & \quad = \frac{\kappa^2}{4\pi}\int_D\left(\int_{\mathcal{S}^2\times\mathcal{S}^2}\left|\left(n_2-n_1\right)e^{\mathbf{i}\kappa\theta\cdot y} - \left(n_1-n_2\right)u_{n_1}^{\mathrm{sca}}+\left(1-n_2\right)\left(u_{n_1}^{\mathrm{sca}}-u_{n_2}^{\mathrm{sca}}\right)\right|^2 \,\mathrm{d}\omega\right)^{\frac{1}{2}} \,\mathrm{d} y, \end{aligned} \end{align} \tag{ A.13 }$$

where the inequality above is due to the integral form of Minkowski's inequality. Plugging (A.6), (A.10) and (A.12) into (A.13) gives

$$\begin{equation} \|u_{n_1}^\infty-u_{n_2}^\infty\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}\unicode{x2A7D} C\left(1+\|n_1\|_{C^1\left(D\right)}\vee\|n_2\|_{C^1\left(D\right)}\right)\|n_1-n_2\|_{\left(H^1\left(D\right)\right)^\ast}, \end{equation} \tag{ A.14 }$$

where $C = C(D,k,M_0)$.

Next, we recall the following stability estimate for the determination of the potential from the scattering amplitude.

Theorem A.1  (HH01, theorem 1.2).  Let $t\gt3/2$, M > 0, and $0\lt\epsilon\lt\frac{2t-3}{2t+3}$ be given constants. Assume that $1-n_j\in H^t(\mathbb{R}^3)$ satisfying $\|1-n_j\|_{H^t(\mathbb{R}^3)}\unicode{x2A7D} M$ and $\mathrm{supp}(1-n_j)\subset D$, $j = 1,2$. Then

$$\begin{equation} \|n_1-n_2\|_{L^\infty\left(D\right)}\unicode{x2A7D} C\left[-\text{ln}^-\left(\|u^\infty_{n_1}-u^\infty_{n_2}\|_{L^2\left(\mathcal{S}^2\times\mathcal{S}^2\right)}\right)\right]^{-\left(\frac{2t-3}{2t+3}-\epsilon\right)}, \end{equation} \tag{ A.15 }$$

where $C = C(D,t,k,M,\epsilon)$ and

$$\begin{align*} \text{ln}^-\left(z\right) = \left\{ \begin{aligned} &\text{ln}\left(z\right),\;\;\mbox{if}\;\; t\unicode{x2A7D} e^{-1},\\ &-1,\;\;\mbox{otherwise}. \end{aligned} \right. \end{align*}$$

Remark A.2 Here the constant M may be different from M₀ given above.

The purpose of this section is to show that the logarithmic estimate obtained in theorem A.1 is optimal by deriving an instability estimate. Similar instability estimate was already proved in [Isa13]. To make the paper self contained, we present our own proof here and also slightly refine the estimate obtained in [Isa13]. Throughout this section, we denote $q(x) = n(x)-1$.

Theorem B.1. Consider the inverse scattering problem (1.1a )–(1.1c ) with frequency κ > 0. Let integer s > 0 be a given regularizing parameter. Then there exists constants $\beta = \beta(s,\kappa) \gt 0$ and $\vartheta_{0} = \vartheta_{0}(s,\kappa)\gt0$ such that: for each $0 \lt \vartheta \lt \vartheta_{0}$ there exists non-negative $q_{1},q_{2} \in C^{\infty}(\mathbb{R}^{3})$ with $\mathrm{supp}\,(q_{j}) \subset B_{\frac{1}{2}}$ satisfying the apriori bound $\lVert q_{j} \rVert_{C^{s}(\mathbb{R}^{3})} \unicode{x2A7D} \beta$ and

$$\begin{equation*} \lVert q_{1}-q_{2} \rVert_{L^{\infty}\left(\mathbb{R}^{3}\right)} \unicode{x2A7E} \vartheta ,\quad \lVert u_{1+q_{1}}^{\infty} - u_{1+q_{2}}^{\infty} \rVert_{L^{2}\left(\mathcal{S}^{2} \times \mathcal{S}^{2}\right)} \unicode{x2A7D} 2 \text{exp} \left(-\vartheta^{-\frac{3}{5s}}\right). \end{equation*}$$

Remark B.2. From the properties of the Hilbert-Schmidt norm [Con90, exercise IX.2.19(h)], one has

$$\begin{equation} \begin{aligned} & \lVert G\left(1+q\right) \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)} = \lVert u_{1+q}^{\infty} \rVert_{L^{2}\left(\mathcal{S}^{2} \times \mathcal{S}^{2}\right)}, \\ & \lVert G\left(1+q_{1}\right)-G\left(1+q_{2}\right) \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)} = \lVert u_{1+q_{1}}^{\infty} - u_{1+q_{2}}^{\infty} \rVert_{L^{2}\left(\mathcal{S}^{2} \times \mathcal{S}^{2}\right)}. \end{aligned} \end{equation} \tag{ B.1 }$$

Recall $G(1+q)$ is the far-field operator given in (1.2).

Our main strategy is to modify the ideas in [DCR03] (see also [KRS21] for more details about the mechanism). Given any ϑ > 0, $s \unicode{x2A7E} 0$ and β > 0, we consider the following set:

$$\begin{equation*} \mathcal{N}_{s \beta}^{\vartheta} : = \left\{q \unicode{x2A7E} 0 : \mathrm{supp}\,\left(q\right)\subset B_{1/2} ,\quad \lVert q \rVert_{L^{\infty}\left(\mathbb{R}^{d}\right)} \unicode{x2A7D} \vartheta ,\quad \lVert q \rVert_{C^{s}\left(\mathbb{R}^{d}\right)} \unicode{x2A7D} \beta \right\}, \end{equation*}$$

where B_r denotes the ball or radius r centered at the origin. The following lemma verifies the assumption (a) of [DCR03, theorem 2.2], can be proved as in [Man01, lemma 2] (we omit the proof), see also [Isa13, KUW21, KW22, ZZ19]. We refer [KT61] for a version in a more abstract form.

Lemma B.3. Fix $d \in \mathbb{N}$ and $s \unicode{x2A7E} 0$. There exists a constant ⁶ $\mu = \mu(d,s) \gt 0$ such that the following statement holds for all β > 0 and for all $\vartheta \in (0,\mu\beta)$: there exists a ϑ-discrete subset $\mathcal{Z}_{\vartheta}$ of $\left( \mathcal{N}_{s \beta}^{\vartheta} , \lVert \cdot \rVert_{L^{\infty}(\mathbb{R}^{d})} \right)$ ⁷ such that

$$\begin{equation} \lvert \mathcal{Z}_{\vartheta} \rvert \unicode{x2A7E} \text{exp} \left( \left( \frac{\text{log} 2}{2}\right)^{d} \left( \frac{\mu\beta}{\vartheta} \right)^{\frac{d}{s}} \right). \end{equation} \tag{ B.2 }$$

In addition, all elements in $\mathcal{Z}_{\vartheta}$ are in $C^{\infty}(\mathbb{R}^{d})$.

Similar to [DCR03], the proof of theorem B.1 is quite delicate, which is not an obvious consequence of the abstract theorem in [DCR03, theorem 2.2]. From the asymptotic expansion (1.1c ), it is easy to see that

$$\begin{equation*} \lvert x \rvert \lvert u_{1+q}^{\mathrm{sca}}\left(x,\theta\right) \rvert = O\left(1\right) \textrm{as}\, \lvert x \rvert\rightarrow\infty\, \mathrm{uniformly}\, \mathrm{for}\, \textrm{all}\, \theta^{^{\prime}} = x/\lvert x \rvert\in\mathcal{S}^{2}. \end{equation*}$$

A crucial point is to bound $\lvert x \rvert \lvert u_{1+q}^\textrm{sca}(x,\theta) \rvert$ for all $\lvert x \rvert \unicode{x2A7E} 2$, by some constant which is independent of θ and q, like [DCR03, (4.21)]. From now on, for simplicity, we restrict ourselves for the case when d = 3. We now prove the following lemma.

Lemma B.4. Let κ > 0 and let $\lVert q \rVert_{L^{\infty}(\mathbb{R}^{3})} \unicode{x2A7D} 1$ with $\mathrm{supp}\,(q) \subset B_{1/2}$. Then there exists a constant $C = C(\kappa)\gt0$ such that the following uniform decay estimate holds:

$$\begin{equation*} \sup_{\theta \in \mathcal{S}^{2}} \lvert u_{1+q}^{\mathrm{sca}}\left(x,\theta\right) \rvert \unicode{x2A7D} C \lvert x \rvert^{-1} \quad \textrm{for all}\, x \in \mathbb{R}^{3} \setminus \overline{B_{2}}. \end{equation*}$$

Proof. This lemma is an easy consequence of (A.6) and [Ron03, lemma 3.2] (with R = 1). □

As in [HH01], we introduce the following index set

$$\begin{equation*} \mathcal{M} : = \left\{\left(m,j,n,k\right) \in \mathbb{Z}_{\unicode{x2A7E} 0} \times \mathbb{Z} \times \mathbb{Z}_{\unicode{x2A7E} 0} \times \mathbb{Z} : \lvert j \rvert\unicode{x2A7D} m \text{ and } \lvert k \rvert\unicode{x2A7D} n \right\}. \end{equation*}$$

Let $\{Y\,{_{m}^{j}}: m \in \mathbb{Z}_{\unicode{x2A7E} 0},\;\lvert j \rvert \unicode{x2A7D} m\}$ be the set of spherical harmonics. For each $F \in \textrm{HS}(L^{2}(\mathcal{S}^{2}),L^{2}(\mathcal{S}^{2}))$, we denote $F_{mn}^{jk} : = \left( F Y\,{_{m}^{j}} , Y_{n}^{k} \right)_{L^{2}(\mathcal{S}^{2}),L^{2}(\mathcal{S}^{2})}$. Accordingly, we write

$$\begin{equation*} G\left(1+q\right)_{mn}^{jk} : = \left( G\left(1+q\right) Y\,{_{m}^{j}} , Y_{n}^{k} \right)_{L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)}. \end{equation*}$$

Lemma B.5. Let κ > 0 and let $\lVert q \rVert_{L^{\infty}(\mathbb{R}^{3})} \unicode{x2A7D} 1$ with $\mathrm{supp}\,(q) \subset B_{1/2}$. Then there exist constants $c_\textrm{abs} = c_\textrm{abs}(\kappa)\gt0$ and $C_\textrm{abs} = C_\textrm{abs}(\kappa)\gt e$ such that

$$\begin{equation*} \lvert G\left(1+q\right)_{mn}^{jk} \rvert \unicode{x2A7D} C_{\mathrm{abs}} e^{-c_{\mathrm{abs}} \text{max}\left\{m,n\right\}} \quad \textrm{for all}\, \left(m,j,n,k\right)\in\mathcal{M}. \end{equation*}$$

Proof. By [CK19, theorem 2.15], we can write

$$\begin{equation*} u_{1+q}^{\mathrm{sc}}\left(x,\theta\right) = \sum_{m = 0}^{\infty} \sum_{j = -m}^{m} \left( u_{1+q}^{\mathrm{sca}} \right)_{m}^{j} \left(\theta\right) h_{m}^{\left(1\right)}\left(\kappa\lvert x \rvert\right) Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right), \end{equation*}$$

where $h_{m}^{(1)}$ is the spherical Hankel function of the first kind or order m. In fact,

$$\begin{equation*} \left( u_{1+q}^{\mathrm{sca}} \right)_{m}^{j}\left(\theta\right) h_{m}^{\left(1\right)}\left(\kappa r\right) = \int_{\mathcal{S}^{2}} u_{1+q}^{\mathrm{sca}}\left(r \theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)} \, \mathrm{d} \theta^{^{\prime}} \quad \textrm{for all}\, r>1, \end{equation*}$$

see the proof of [CK19, theorem 2.16]. For each $\theta \in \mathcal{S}^{2}$, also from [CK19, theorem 2.16], we have the following expansion

$$\begin{equation*} u_{1+q}^{\infty}\left(\theta^{^{\prime}},\theta\right) = \sum_{m = 0}^{\infty} \sum_{j = -m}^{m} \frac{1}{\mathbf{i}^{m+1}} \left(u_{1+q}^{\mathrm{sca}}\right)_{m}^{j}\left(\theta\right) Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right), \end{equation*}$$

which converges uniformly in $\theta^{^{\prime}} \in \mathcal{S}^{2}$, and hence

$$\begin{equation} \int_{\mathcal{S}^{2}} u_{1+q}^{\infty}\left(\theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)} \, \mathrm{d} \theta^{^{\prime}} = \frac{1}{\mathbf{i}^{m+1}} \left(u_{1+q}^{\mathrm{sc}}\right)_{m}^{j}\left(\theta\right) = \lvert h_{m}^{\left(1\right)}\left(\kappa r\right) \rvert^{-1} \int_{\mathcal{S}^{2}} u_{1+q}^{\mathrm{sca}}\left(r \theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)} \, \mathrm{d} \theta^{^{\prime}}, \end{equation} \tag{ B.3 }$$

for all r > 1. We now combine (B.3) with lemma B.4 to obtain

$$\begin{equation*} \left| \int_{\mathcal{S}^{2}} u_{1+q}^{\infty}\left(\theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)} \, \mathrm{d} \theta^{^{\prime}} \right| \unicode{x2A7D} C r^{-1} \lvert h_{m}^{\left(1\right)}\left(\kappa r\right) \rvert^{-1}, \end{equation*}$$

uniformly in $\theta^{^{\prime}}\in\mathcal{S}^2$. We now choose $\kappa r = 2$ and reach

$$\begin{equation} \left| \int_{\mathcal{S}^{2}} u_{1+q}^{\infty}\left(\theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)} \, \mathrm{d} \theta^{^{\prime}} \right| \unicode{x2A7D} C \lvert h_{m}^{\left(1\right)}\left(2\right) \rvert^{-1}. \end{equation} \tag{ B.4 }$$

From [KW22, (48)], we see that

$$\begin{equation} \lvert h_{m}^{\left(1\right)}\left(2\right) \rvert^{-1} \unicode{x2A7D} 2\sqrt{\pi} \frac{1}{\Gamma\left(m + \frac{1}{2}\right)} \quad \textrm{for all }\, m = 0,1,2,\cdots. \end{equation} \tag{ B.5 }$$

In view of the quantitative Stirling's formula [Rob55], we obtain

$$\begin{equation} \begin{aligned} \lvert h_{m}^{\left(1\right)}\left(2\right) \rvert^{-1} &\unicode{x2A7D} 2\sqrt{\pi} \frac{1}{\left(m-1\right)!} \unicode{x2A7D} \sqrt{2} \left(m-1\right)^{-\left(m-\frac{1}{2}\right)} e^{m-1} e^{-\frac{1}{12m-11}} \\ & \unicode{x2A7D} C \left(m-1\right)^{-\left(m-1\right)} e^{m-1} \quad \textrm{for all}\, m = 6,7,8,\cdots. \end{aligned} \end{equation} \tag{ B.6 }$$

From (B.5), it is clearly that

$$\begin{equation*} \lvert h_{m}^{\left(1\right)}\left(2\right) \rvert^{-1} \unicode{x2A7D} C \quad \textrm{for}\, m = 0,1,2,\ldots,5. \end{equation*}$$

Combining (B.4) and (B.6) implies

$$\begin{equation*} \begin{aligned} \lvert G\left(1+q\right)_{nm}^{kj} \rvert& = \left| \int_{\mathcal{S}^{2}} \int_{\mathcal{S}^{2}} u_{1+q}^{\infty}\left(\theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)} Y_{n}^{k}\left(\theta\right) \, \mathrm{d} \theta^{^{\prime}} \, \mathrm{d} \theta \right| \\ & \unicode{x2A7D} \lvert \mathcal{S}^{2} \rvert^{\frac{1}{2}} \left| \int_{\mathcal{S}^{2}} u_{1+q}^{\infty}\left(\theta^{^{\prime}},\theta\right) \overline{Y\,{_{m}^{j}}\left(\theta^{^{\prime}}\right)}\, \mathrm{d} \theta^{^{\prime}} \right| \unicode{x2A7D} C e^{-c m} \quad \textrm{for all}\, \left(m,j,n,k\right)\in\mathcal{M}. \end{aligned} \end{equation*}$$

Finally, by the first reciprocity relation [CK19, theorem 8.8], we have

$$\begin{equation*} \lvert G\left(1+q\right)_{mn}^{jk} \rvert = \lvert G\left(1+q\right)_{nm}^{kj} \rvert \quad \textrm{for all}\, \left(m,j,n,k\right)\in\mathcal{M}, \end{equation*}$$

and the lemma is proved. □

We also need a technical lemma.

Lemma B.6. Consider the normed space

$$\begin{equation*} {\mathrm{HS}}^{^{\prime}} = \left\{F \in {\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right) : \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} < \infty \right\}, \end{equation*}$$

where

$$\begin{equation*} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} : = \sup_{\left(m,j,n,k\right)\in\mathcal{M}} \lvert F_{mn}^{jk} \rvert\left(1 + \text{max}\left\{m,n\right\}\right)^{3}. \end{equation*}$$

Then

$$\begin{equation*} \lVert F \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)} \unicode{x2A7D} C_{0} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} ,\quad C_{0} = 4 \left( \sum_{n = 0}^{\infty} \frac{1}{\left(1+n\right)^{3}} \right)^{\frac{1}{2}} < \infty. \end{equation*}$$

Proof. Since

$$\begin{equation} \lVert F \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)}^{2} = \sum_{\left(m,j,n,k\right)\in\mathcal{M}} \lvert F_{mn}^{jk} \rvert^{2}, \end{equation} \tag{ B.7 }$$

we can estimate

$$\begin{align*} \begin{aligned} \lVert F \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)} &= \left( \sum_{\left(m,j,n,k\right)\in\mathcal{M}} \lvert F_{mn}^{jk} \rvert^{2} \frac{\left(1 + \text{max}\left\{m,n\right\}\right)^{6}}{\left(1 + \text{max}\left\{m,n\right\}\right)^{6}} \right)^{\frac{1}{2}} \\ & \unicode{x2A7D} \left( \sum_{\left(m,j,n,k\right)\in\mathcal{M}} \frac{1}{\left(1 + \text{max}\left\{m,n\right\}\right)^{6}} \right)^{\frac{1}{2}} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} \\ & \unicode{x2A7D} 2 \left( \sum_{m,n = 0,1,2,\cdots} \frac{1}{\left(1 + \text{max}\left\{m,n\right\}\right)^{4}} \right)^{\frac{1}{2}} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} \\ & \unicode{x2A7D} 2 \left( \left( \sum_{n\unicode{x2A7E} m \unicode{x2A7E} 0} + \sum_{m\unicode{x2A7E} n \unicode{x2A7E} 0} \right) \frac{1}{\left(1 + \text{max}\left\{m,n\right\}\right)^{4}} \right)^{\frac{1}{2}} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} \\ & = 4 \left( \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} \frac{1}{\left(1 + n\right)^{4}} \right)^{\frac{1}{2}} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}} = 4 \left( \sum_{n = 0}^{\infty} \frac{1}{\left(1 + n\right)^{3}} \right)^{\frac{1}{2}} \lVert F \rVert_{{\mathrm{HS}}^{^{\prime}}}, \end{aligned} \end{align*}$$

which implies the lemma. □

We now construct a δ-net by following the procedures in [DCR03, lemma 2.3].

Lemma B.7. Let $\mathcal{N}^{1} : = \left\{q\unicode{x2A7E} 0 : \mathrm{supp}\,(q) \subset B_{1/2}, \quad \lVert q \rVert_{L^{\infty}(\mathbb{R}^{3})} \unicode{x2A7D} 1 \right\}$. Then there exists a constant $C_\textrm{abs}^{^{\prime}} = C_\textrm{abs}^{^{\prime}}(\kappa) \gt e$ such that: for each $0 \lt \delta \lt 1/e$ we can find a δ-net ⁸ $\mathcal{Y}_{\delta}$ for $\left( G(1+\mathcal{N}^{1}) , \lVert \cdot \rVert_{\textrm{HS}(L^{2}(\mathcal{S}^{2}),L^{2}(\mathcal{S}^{2}))} \right)$ such that

$$\begin{equation} \lvert \mathcal{Y}_{\delta} \rvert \unicode{x2A7D} C_{\mathrm{abs}}^{^{\prime}} \text{exp}\left(- \text{log} \delta\right)^{5}. \end{equation} \tag{ B.8 }$$

Proof. Fix $\delta \in (0 , 1/e)$. Let $\tilde{\ell}$ be the smallest positive integer such that

$$\begin{equation} C_{\mathrm{abs}} e^{-c_{\mathrm{abs}}\left(t-1\right)} \left(1+t\right)^{3} \unicode{x2A7D} \frac{\delta}{2C_{0}} \quad \textrm{for all}\, t \unicode{x2A7E} \tilde{\ell}, \end{equation} \tag{ B.9 }$$

where $C_\textrm{abs} \gt e$ and $c_\textrm{abs} \gt 0$ are the absolute constants appeared in lemma B.5. One sees that there exists an absolute constant C > 0 such that

$$\begin{equation} \tilde{\ell} \unicode{x2A7D} C \text{log} \delta^{-1}. \end{equation} \tag{ B.10 }$$

We define a finite subset of the complex plane $\mathbb{C}$ by

$$\begin{equation*} \Psi_{\delta} : = \left( \delta^{^{\prime}} \mathbb{Z} \cap \left[-C_{\mathrm{abs}},C_{\mathrm{abs}}\right] \right) + \mathbf{i} \left( \delta^{^{\prime}} \mathbb{Z} \cap \left[-C_{\mathrm{abs}},C_{\mathrm{abs}}\right] \right) ,\quad \delta^{^{\prime}} = \frac{\left(1+\tilde{\ell}\right)^{-3}}{2C_{0}} \delta. \end{equation*}$$

It is easy to see that

$$\begin{equation} \lvert \Psi_{\delta} \rvert \unicode{x2A7D} \left( \frac{2 C_{\mathrm{abs}} + 1}{\delta^{^{\prime}}} \right)^{2} = \left( \frac{2C_{0}\left(2 C_{\mathrm{abs}} + 1\right)\left(1+\tilde{\ell}\right)^{3}}{\delta} \right)^{2}. \end{equation} \tag{ B.11 }$$

We now define the set

$$\begin{equation*} \mathcal{Y}_{\delta} : = \left\{ F \in {\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right) : \begin{array}{l} F_{mn}^{jk} \in \Psi_{\delta} \,\mathrm{if}\, \mathrm{max}\,\left\{m,n\right\}\unicode{x2A7D}\tilde{\ell} \\ \textrm{and}\, F_{mn}^{jk} = 0\,\,\, \mathrm{otherwise} \end{array}\right\}. \end{equation*}$$

Let $\ell_{*}$ be the number of $(m,j,n,k)\in\mathcal{M}$ such that $\text{max}\{m,n\} \unicode{x2A7D} \tilde{\ell}$, then we have

$$\begin{equation} \lvert \mathcal{Y}_{\delta} \rvert = \lvert \Psi_{\delta} \rvert^{\ell_{*}} \unicode{x2A7D} \lvert \Psi_{\delta} \rvert^{16\left(1+\tilde{\ell}\right)^{4}}. \end{equation} \tag{ B.12 }$$

Plugging (B.10) and (B.11) into (B.12) yields

$$\begin{equation*} \lvert \mathcal{Y}_{\delta} \rvert \unicode{x2A7D} \left( 2C_{0}\left(2 C_{\mathrm{abs}} + 1\right) \frac{\left(C \text{log} \delta^{-1}\right)^{3}}{\delta} \right)^{16\left(C\text{log} \delta^{-1}\right)^{4}}, \end{equation*}$$

and, furthermore, from the trivial estimate $\text{log}(1+t)\unicode{x2A7D} t$ for all $t \unicode{x2A7E} 0$, we see that

$$\begin{equation*} \begin{aligned} \text{log} \lvert \mathcal{Y}_{\delta} \rvert& \unicode{x2A7D} 16\left(C\text{log} \delta^{-1}\right)^{4} \text{log} \left( 2C_{0}\left(2 C_{\mathrm{abs}} + 1\right) \frac{\left(C\text{log} \delta^{-1}\right)^{3}}{\delta} \right) \\ & = 16\left(C\text{log} \delta^{-1}\right)^{4} \left( \text{log} \left(2C_{0}\left(2C_{\mathrm{abs}}+1\right)\right) + 3\text{log} \left(C\text{log} \delta^{-1}\right) + \text{log} \delta^{-1} \right) \\ & \unicode{x2A7D} 16\left(C\text{log} \delta^{-1}\right)^{4} \left( \text{log} \left(2C_{0}\left(2C_{\mathrm{abs}}+1\right)\right) + 3C\text{log} \delta^{-1} + \text{log} \delta^{-1} \right) \\ & \unicode{x2A7D} C \left(\text{log} \delta^{-1}\right)^{5}, \end{aligned} \end{equation*}$$

which gives (B.8).

The remaining task now is to verify that the set $\mathcal{Y}_{\delta}$ constructed above is a δ-net for $\left( G(1+\mathcal{N}^{1}) , \lVert \cdot \rVert_{\textrm{HS}(L^{2}(\mathcal{S}^{2}),L^{2}(\mathcal{S}^{2}))} \right)$. Fix $q \in \mathcal{N}^{1}$ and consider the far-field pattern $G(1+q)$. For each $(m,j,n,k)\in\mathcal{M}$ with $\text{max}\{m,n\}\unicode{x2A7D}\tilde{\ell}$, we choose $F_{mn}^{jk} \in \Psi_{\delta}$ be the closest element to $G(1+q)_{mn}^{jk}$; otherwise for those $(m,j,n,k)\in\mathcal{M}$ with $\text{max}\{m,n\} \gt \tilde{\ell}$, we simply choose $F_{mn}^{jk} = 0$. We define the operator $F \in \mathcal{Y}_{\delta}$ by

$$\begin{equation*} \left( F Y\,{_{m}^{j}} , Y_{m}^{k} \right)_{L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)} : = F_{mn}^{jk} \quad \textrm{for all}\, \left(m,j,n,k\right)\in\mathcal{M}. \end{equation*}$$

By lemma B.5, we see that

$$\begin{equation*} G\left(1+\mathcal{N}^{1}\right) \subset B_{C_{\mathrm{abs}}} \subset \mathbb{C}. \end{equation*}$$

Therefore, if $\text{max}\{m,n\} \unicode{x2A7D} \tilde{\ell}$, we have that

$$\begin{equation*} \lvert G\left(1+q\right)_{mn}^{jk} - F_{mn}^{jk} \rvert\left(1 + \text{max}\left\{m,n\right\}\right)^{3} \unicode{x2A7D} \sqrt{2} \delta^{^{\prime}} \left(1 + \tilde{\ell}\right)^{3} = \frac{\delta}{\sqrt{2} C_{0}}. \end{equation*}$$

Otherwise, when $t : = \text{max}\{m,n\} \gt \tilde{\ell}$, from lemma B.5 and (B.9), we estimate

$$\begin{equation*} \lvert G\left(1+q\right)_{mn}^{jk} - F_{mn}^{jk} \rvert\left(1 + \text{max}\left\{m,n\right\}\right)^{3} = \lvert G\left(1+q\right)_{mn}^{jk} \rvert\left(1 + t\right)^{3} \unicode{x2A7D} C_{\mathrm{abs}} e^{-c_{\mathrm{abs}} t} \left(1+t^{3}\right) \unicode{x2A7D} \frac{\delta}{2C_{0}}. \end{equation*}$$

Consequently, by lemma B.6, we finally obtain

$$\begin{equation*} \lVert G\left(1+q\right)-F \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)} \unicode{x2A7D} C_{0} \sup_{\left(m,j,n,k\right)\in\mathcal{M}} \lvert G\left(1+q\right)_{mn}^{jk} - F_{mn}^{jk} \rvert\left(1 + \text{max}\left\{m,n\right\}\right)^{3} \unicode{x2A7D} \delta. \end{equation*}$$

 □

We are now ready to prove the main instability estimate by combining lemmas B.3 and B.7.

Proof of theorem B.1. Let $s \gt \frac{3}{2}$ and $C_\textrm{abs}^{^{\prime}} \gt e$ be the constant obtained in lemma B.7. We choose $\beta = \beta(s) \gt 0$ such that

$$\begin{equation} 1 \unicode{x2A7D} \frac{1}{2} \left( \frac{\text{log} 2}{2} \right)^{d} \left(\mu\beta\right)^{\frac{3}{s}}, \end{equation} \tag{ B.13 }$$

where $\mu = \mu(s)$ is the constant given in lemma B.3 (with d = 3). Let us pick ϑ satisfying

$$\begin{equation*} 0 < \vartheta < \text{min} \left\{\mu\beta, \left( \left( \frac{\text{log} 2}{2} \right)^{d} \frac{1}{2 \text{log} C_{\mathrm{abs}}^{^{\prime}}} \right)^{\frac{s}{3}}\left(\mu\beta\right) ,1 \right\}. \end{equation*}$$

Assume that the set $\mathcal{Z}_{\vartheta}$ is given in lemma B.3 (also take d = 3). Next we choose

$$\begin{equation*} \delta : = \text{exp} \left(-\vartheta^{-\frac{3}{5s}}\right) \in \left(0,1/e\right), \end{equation*}$$

and construct the $\mathcal{Y}_{\delta}$ described in lemma B.7. Since $\mathcal{Z}_{\vartheta} \subset \mathcal{N}_{s \beta}^{\vartheta} \subset \mathcal{N}_{s \beta}^{1}$, it is clear that $\mathcal{Y}_{\delta}$ is also a δ-net for $\left( G(1+\mathcal{Z}_{\vartheta}) , \lVert \cdot \rVert_{\textrm{HS}(L^{2}(\mathcal{S}^{2}),L^{2}(\mathcal{S}^{2}))} \right)$. Moreover, we can check that

$$\begin{equation*} \vartheta \unicode{x2A7D} \left( \left( \frac{\text{log} 2}{2} \right)^{d} \frac{1}{2 \text{log} C_{\mathrm{abs}}^{^{\prime}}} \right)^{\frac{s}{3}} \mu\beta \iff \text{log} C_{\mathrm{abs}}^{^{\prime}} \unicode{x2A7D} \frac{1}{2} \left( \frac{\text{log} 2}{2} \right)^{d} \left(\mu\beta\right)^{\frac{3}{s}} \vartheta^{-\frac{3}{s}}. \end{equation*}$$

Combining (B.2), (B.8) and (B.13) implies

$$\begin{equation*} \lvert \mathcal{Y}_{\delta} \rvert \unicode{x2A7D} C_{\mathrm{abs}}^{^{\prime}} \text{exp}\left( \vartheta^{-\frac{3}{s}}\right) = \text{exp}\left( \vartheta^{-\frac{3}{s}} + \text{log} C_{\mathrm{abs}}^{^{\prime}}\right) < \text{exp} \left( \left( \frac{\text{log} 2}{2} \right)^{d} \left(\mu\beta\right)^{\frac{3}{s}} \vartheta^{-\frac{3}{s}} \right) \unicode{x2A7D} \lvert \mathcal{Z}_{\vartheta} \rvert. \end{equation*}$$

This enables us to choose two different $q_{1},q_{2} \in \mathcal{Z}_{\vartheta}$ (by definition of $\mathcal{Z}_{\vartheta}$ it holds that $\lVert q_{1}-q_{2} \rVert_{L^{\infty}(\mathbb{R}^{3})} \unicode{x2A7E} \vartheta$) such that there exists $F \in \mathcal{Y}_{\delta}$ such that

$$\begin{equation*} \lVert G\left(1+q_{j}\right) - F \rVert_{{\mathrm{HS}}\left(L^{2}\left(\mathcal{S}^{2}\right),L^{2}\left(\mathcal{S}^{2}\right)\right)} \unicode{x2A7D} \delta = \text{exp} \left(-\vartheta^{-\frac{3}{5s}}\right), \end{equation*}$$

which proves the theorem by using (B.1). □