Fixed-frequency inverse scattering and Born linearization

Assume Countable Choice. Let $D\subset {\mathbb{R}}^3$ be a bounded Lebesgue-measurable set with $0<M:=\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$, and let $k>0$. Write $S^2:=\{v\in {\mathbb{R}}^3:\mathrm{\mid }v\mathrm{\mid }=1\}$ and $C_D:=\sqrt{M\mathrm{diam}(D)\mathrm{/}(4\pi )}$. For $q\in L^{\mathrm{\infty }}(D;\mathbb{C})$, extended by zero off $D$, set $G_k(z):=e^{ik\mathrm{\mid }z\mathrm{\mid }}\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$ for $z\mathrm{\ne }0$ and $G_k(0):=0$, and let $K_{q,k}v(x):={\int }_D{G}_k(x-y)q(y)v(y)dy,{\rho }_k(q):=\mathrm{\parallel }{K}_{q,k}{\mathrm{\parallel }}_{L^2(D)\to L^2(D)}\mathrm{.}$ The small-norm contrast class is ${\mathcal{U}}_{D,k}:=\{q\in L^{\mathrm{\infty }}(D;\mathbb{C}):{\rho }_k(q)<1\}\mathrm{.}$ For $d\in S^2$, put $f_{k,d}(x):=e^{ikd\cdot x}$ on $D$. For $q\in {\mathcal{U}}_{D,k}$, let $u_{q,k,d}\in L^2(D;\mathbb{C})$ denote the unique solution of $u=f_{k,d}+K_{q,k}u$. For $q\in {\mathcal{U}}_{D,k}$, define the exact far-field data family by ${\mathcal{S}}_k(q)(\theta ,d):=\frac{1}{4\pi }{\int }_D{e}^{-ik\theta \cdot y}q(y)u_{q,k,d}(y)dy,(\theta ,d)\in S^2\times S^2\mathrm{.}$

For $h\in L^{\mathrm{\infty }}(D;\mathbb{C})$, define its first Born data by ${\mathcal{B}}_{k}h(\theta ,d):=\frac{1}{4\pi }{\int }_D{e}^{-ik\theta \cdot y}h(y)e^{ikd\cdot y}dy=\frac{1}{4\pi }\widehat{h_0}\text{\hspace{0.17em}⁣}(\frac{k}{2\pi }(\theta -d)),$ where $h_0$ is the zero extension of $h$ to ${\mathbb{R}}^3$ and $\widehat{g}(\xi ):={\int }_{{\mathbb{R}}^3}g(y)e^{-2\pi iy\cdot \xi }dy$ for $g\in L^1({\mathbb{R}}^3;\mathbb{C})$. No uniqueness or stability outside ${\mathcal{U}}_{D,k}$ is asserted.

The diameter is $\mathrm{diam}(D)=\sup \{\mathrm{\mid }x-y\mathrm{\mid }:x,y\in D\}$.

The norm and dot product on ${\mathbb{R}}^3$ are the Euclidean ones.

The spaces $L^2$ and $L^{\mathrm{\infty }}$ consist of measurable functions modulo equality almost everywhere.

The $L^{\mathrm{\infty }}$ norm is the essential supremum with respect to Lebesgue measure.

The operator norm is the least bound on a bounded linear map between normed spaces.

The outgoing volume model fixes $G_k(z)=e^{ik\mathrm{\mid }z\mathrm{\mid }}\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$ and the far-field convention $e^{-ik\theta \cdot y}q(y)v(y)\mathrm{/}(4\pi )$.

The displayed volume kernel defines a bounded operator on complex $L^2(D)$.

The far-field map is continuous and is the uniform outgoing limit of the exterior volume potential.

The library Fourier transform uses $\widehat{g}(\xi )=\int g(y)e^{-2\pi iy\cdot \xi }dy$, giving the frequency $k(\theta -d)\mathrm{/}(2\pi )$.

For the strict operator norm bound, the volume equation has a unique L2 solution, which supplies the field used here.

Countable Choice is the declared axiom branch for the L2 volume model and its Neumann result.

Assume Countable Choice and the hypotheses and notation of the definition of fixed-frequency far-field data. For each $q\in {\mathcal{U}}_{D,k}$ there is an $\epsilon >0$ such that $\mathrm{\parallel }{q}^{\mathrm{\prime }}-q{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$ implies $q^{\mathrm{\prime }}\in {\mathcal{U}}_{D,k}$; in particular this metric neighborhood condition holds at zero. For every $q\in {\mathcal{U}}_{D,k}$ and every $d,\theta \in S^2$, the solution $u_{q,k,d}$ exists uniquely in $L^2(D;\mathbb{C})$, the integrals defining ${\mathcal{S}}_k(q)(\theta ,d)$ and ${\mathcal{B}}_{k}q(\theta ,d)$ are absolutely convergent and independent of representatives, and ${\mathcal{S}}_k(q)(\cdot ,d)$ is continuous. If $R_0>{\sup }_{y\in D}\mathrm{\mid }y\mathrm{\mid }$ and $r\ge 2R_0$, then ${\sup }_{\theta \in S^2}\mid r{e}^{-ikr}{V}_k{u}_{q,k,d}(r\theta )-{\mathcal{S}}_k(q)(\theta ,d)\mid \le \frac{2R_0+\mathrm{\mid }k\mathrm{\mid }{R}_0^2}{4\pi r}\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}\sqrt{M}\mathrm{\parallel }{u}_{q,k,d}{\mathrm{\parallel }}_2\mathrm{.}$ Thus ${\mathcal{S}}_k(q)(\cdot ,d)$ is the uniform outgoing far-field amplitude in the stated convention.

The small-norm domain, volume operator, incident plane wave, scattering data, and first Born data family.

The volume model uses equivalence classes in complex L2 and the stated outgoing Green kernel.

The far-field map is continuous and has the uniform outgoing-wave error bound.

The volume operator is linear and satisfies the Hilbert–Schmidt norm bound used to show openness.

For operator norm strictly below one, the Lippmann–Schwinger equation has a unique L2 solution.

Complex L2 Cauchy–Schwarz and representative-independent pairings.

An L-infinity representative is bounded almost everywhere by its essential norm.

The L1 Fourier transform is defined at every frequency and is representative-independent.

Countable Choice is the declared axiom branch of the volume and L2 suppliers.

The proof establishes only the small-norm branch selected by ${\rho }_k(q)<1$. It does not import global scattering uniqueness from smoother coefficient models.

Assume Countable Choice and the hypotheses and notation of the fixed-frequency far-field definition. For $q\in {\mathcal{U}}_{D,k}$ put $\rho ={\rho }_k(q)$. Uniformly over all incident and observation directions, ${\sup }_{\theta ,d\in S^2}\mid {\mathcal{S}}_k(q)(\theta ,d)-{\mathcal{B}}_{k}q(\theta ,d)\mid \le \frac{M}{4\pi }\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}\frac{\rho }{1-\rho }\mathrm{.}$ Whenever $C_D\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}<1$, this implies ${\sup }_{\theta ,d\in S^2}\mid {\mathcal{S}}_k(q)(\theta ,d)-{\mathcal{B}}_{k}q(\theta ,d)\mid \le \frac{M{C}_D}{4\pi }\frac{\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}^2}{1-C_D\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}}\mathrm{.}$ For each fixed $(\theta ,d)$, the scalar map $q\mapsto {\mathcal{S}}_k(q)(\theta ,d)$ on the open domain ${\mathcal{U}}_{D,k}$, with $L^{\mathrm{\infty }}(D;\mathbb{C})$ and $\mathbb{C}$ viewed as real Banach spaces, is Fréchet differentiable at $q=0$ and has derivative $h\mapsto {\mathcal{B}}_{k}h(\theta ,d)$. The remainder bound is uniform in $(\theta ,d)$; it asserts no nonlinear uniqueness or stability result.

The exact scattering family, small-norm domain, and first Born data formula.

The small-norm solution, data integrals, and open contrast domain are well-defined.

The operator-norm estimate ${\rho }_k(q)\le C_D\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$.

For $\rho <1$, the Lippmann–Schwinger solution satisfies the first partial-sum tail bound.

Fréchet differentiability means a bounded real-linear first-order approximation with remainder $o(\mathrm{\parallel }h\mathrm{\parallel })$.

Countable Choice makes complex $L^{\mathrm{\infty }}$ complete.

The declared axiom for L-infinity completeness and volume-scattering suppliers.

At zero contrast the first Born data are ${\mathcal{B}}_{k}h(\theta ,d)=\frac{1}{4\pi }\widehat{h_0}\text{\hspace{0.17em}⁣}(\frac{k}{2\pi }(\theta -d))\mathrm{.}$ Thus each fixed-frequency direction pair samples one Fourier frequency on the corresponding Ewald surface. The result is local at $q=0$ and does not extend a derivative or inverse-uniqueness assertion to arbitrary contrasts.

For fixed $k>0$ and unit directions $d,\theta$, the wavevectors $kd$ and $k\theta$ lie on the radius-$k$ sphere in their span. The arrow from $kd$ to $k\theta$ is exactly $k(\theta -d)$; the Fourier variable in the library's $2\pi$ convention is that chord divided by $2\pi$. The diagram is only a geometric description of the frequency label; the Fréchet estimate above is proved by the integral and norm bounds, not by the drawing.

Inputs and admissible conditions. Fix the domain $D$, wavenumber $k>0$, and library Fourier convention from the fixed-frequency data definition. Give nonempty finite lists of incident directions $d_1,\dots ,d_s\in S^2$ and observation directions ${\theta }_1,\dots ,{\theta }_r\in S^2$. Give a finite measurable partition $D={⨆}_{n=1}^N{E}_n$ with $\mathrm{\mid }{E}_n\mathrm{\mid }>0$, complex Born-data samples $y=(y_{ba})\in {\mathbb{C}}^{rs}$, a discrepancy threshold $\delta \ge 0$, and an iteration budget $B\in {\mathbb{N}}_0:=\{0,1,2,\dots \}$. The sample frequency for pair $({\theta }_b,d_a)$ is ${\xi }_{ba}:=k({\theta }_b-d_a)\mathrm{/}(2\pi )$. The declared finite discrete Born Fourier matrix $F\in {\mathbb{C}}^{rs\times N}$ has entries $F_{(b,a),n}:=\frac{1}{4\pi }{\int }_{E_n}{e}^{-2\pi ix\cdot {\xi }_{ba}}dx\mathrm{.}$ For a piecewise-constant contrast $q_c:={\sum }_{n=1}^N{c}_n{1}_{E_n}$, its sampled first Born data vector is exactly $Fc$. The input also declares a reduced singular-value decomposition $Fc={\sum }_{j=1}^p{\sigma }_j⟨c,v_j⟩u_j(c\in {\mathbb{C}}^N),$ where $p=\mathrm{rank}F$, the positive singular values ${\sigma }_j$ have orthonormal right vectors $v_j$ and orthonormal left vectors $u_j$, and the empty sum is used when $F=0$. Give a scalar filter $g:[0,\mathrm{\infty })\to \mathbb{C}$ whose values at the ${\sigma }_j$ are finite and with $g(0)=0$. The threshold filter $g_{\tau }(\sigma )=1\mathrm{/}\sigma$ for $\sigma >\tau$ and $g_{\tau }(\sigma )=0$ for $0\le \sigma \le \tau$ is the truncated singular-value pseudoinverse at threshold $\tau \ge 0$.

Instructions and outputs.

1. If $B=0$, return no estimate, no residual, and status budget-exhausted, with zero filter applications.
2. If $B\ge 1$, form the coefficient estimate $\widehat{c}:={\sum }_{j:{\sigma }_j>0}g({\sigma }_j)⟨y,u_j⟩v_j,$ where $⟨z,w⟩={\sum }_{\mathrm{\ell }}{z}_{\mathrm{\ell }}\overline{w_{\mathrm{\ell }}}$ is linear in its first argument. Return the piecewise-constant reconstruction $\widehat{q}:={\sum }_{n=1}^N{\widehat{c}}_n{1}_{E_n}$ after exactly one filter application.
3. Compute the finite data residual $R:=\mathrm{\parallel }F\widehat{c}-y{\mathrm{\parallel }}_2$, where $\mathrm{\parallel }z{\mathrm{\parallel }}_2=({\sum }_{\mathrm{\ell }}\mathrm{\mid }{z}_{\mathrm{\ell }}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Return status discrepancy when $R\le \delta$, and residual-above-threshold when $R>\delta$; in either case also return $\widehat{q}$, $R$, and the application count $1$.

This procedure only defines one filtered finite-dimensional Born reconstruction and a residual status. The residual is not an exact-range test for an infinite-dimensional compact forward operator and does not establish a nonlinear solution, identifiability, stability, convergence, or correctness of the reconstructed contrast.

The linearized far-field samples are $\widehat{q}(k(\theta -d)\mathrm{/}(2\pi ))\mathrm{/}(4\pi )$.

The first Born family is the local Fréchet derivative of the nonlinear scattering data at zero contrast.

The special filter $g_{\tau }$ is the finite-matrix truncated singular-value pseudoinverse.

For a compact Hilbert-space map, exact range membership also requires orthogonality to the adjoint kernel and a finite Picard sum.

Countable Choice is inherited from the fixed-frequency data, Born-linearization, and contextual Picard suppliers.

The library Fourier transform uses the stated $2\pi$ normalization.