ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Finite limited-aperture Born samples miss a nonzero contrast

Example

Assume Countable Choice. Let $D = [0, 1]^{3}$ , $k > 0$ , and let $r, s \geq 1$ . Take nonempty finite lists $θ_{1}, \dots, θ_{r} \in Γ_{o b s}$ and $d_{1}, \dots, d_{s} \in Γ_{i n c}$ , where $Γ_{o b s}, Γ_{i n c} ⊊ S^{2}$ are proper direction sets. Put $m : = r s$ and $N : = m + 1$ . Use the partition $E_{n} = [(n - 1) / N, n / N) \times [0, 1]^{2} (1 \leq n < N), E_{N} = [(N - 1) / N, 1] \times [0, 1]^{2}$ and give the regularized Born algorithm these direction lists and this partition. The finite linearized Born data map has a nonzero piecewise-constant contrast in its kernel. Consequently, for zero sampled Born data, any admissible filter, any discrepancy threshold $δ \geq 0$ , and any positive budget $B \geq 1$ make the one-pass reconstruction return the zero estimate with discrepancy status, even though a nonzero contrast gives the same finite Born samples. This is a statement about the finite linearized model only; it does not assert equality of exact nonlinear scattering data.

Example references

The Born sample at $(θ, d)$ is the Fourier value at $k (θ - d) / (2 π)$ divided by $4 π$ .

The declared finite matrix maps piecewise-constant coefficients to sampled Born data and reports a residual status.

The sampled linear model is the Fréchet derivative at zero contrast.

The kernel of a linear map from $C^{m + 1}$ to $C^{m}$ is nonzero.

The image of a map into $C^{m}$ has dimension at most $m$ .

Facts & Assumptions

Given: Countable Choice; $D = [0, 1]^{3}$ ; $k > 0$ ; integers $r, s \geq 1$ ; observation directions $θ_{b} \in Γ_{o b s}$ for $1 \leq b \leq r$ ; incident directions $d_{a} \in Γ_{i n c}$ for $1 \leq a \leq s$ ; proper sets $Γ_{o b s}, Γ_{i n c} ⊊ S^{2}$ ; zero data $y = 0$ ; an admissible filter; a threshold $δ \geq 0$ ; and a budget $B \geq 1$ . Write $m = r s$ .

[F1]

For each acquisition pair $(θ_{b}, d_{a})$ , the Born sample is $B_{k} h (θ_{b}, d_{a}) = \frac{1}{4 π} \hat{h_{0}} (k (θ_{b} - d_{a}) / (2 π))$ (Fixed-frequency far-field data on the small-norm volume branch, Fourier transform on complex L1 classes).

[F2]

With a partition $D = ⨆_{n = 1}^{N} E_{n}$ , the discrete matrix entries are $F_{j n} = \frac{1}{4 π} \int_{E_{n}} e^{- 2 π i x \cdot ξ_{j}} d x$ and coefficients $c$ reconstruct $\sum_{n} c_{n} 1_{E_{n}}$ (One-pass filtered reconstruction from finite Born data).

[F3]

Every cell $E_{n}$ below has Lebesgue measure $1 / (m + 1) > 0$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F4]

For a linear map $A : V \to W$ with $V$ finite-dimensional, $\dim V = \dim \ker A + \dim im A$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F5]

The image of a linear map into the $m$ -dimensional space $C^{m}$ is a subspace of dimension at most $m$ (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

[F6]

The operator bound $ρ_{k} (q) \leq C_{D} ∥ q ∥_{\infty}$ implies $t h \in U_{D, k}$ for every sufficiently small $t$ (The volume integral operator is well defined and bounded on L2, Fixed-frequency far-field data on the small-norm volume branch).

[F7]

For zero data, any declared linear spectral filter returns the zero coefficient vector; its residual is zero and the status is discrepancy when the threshold is nonnegative (One-pass filtered reconstruction from finite Born data).

[A1]

Countable Choice is inherited from the volume-model branch (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

Given: The data in the Statement.

1.1F1F2F3givenconstruct

Put $N = m + 1$ and partition $D$ into the $N$ measurable slabs $E_{n} = [(n - 1) / N, n / N) \times [0, 1]^{2} (1 \leq n < N), E_{N} = [(N - 1) / N, 1] \times [0, 1]^{2} .$ They are pairwise disjoint, cover $D$ , and have measure $1 / N$ by [F3]. For each pair $(θ_{b}, d_{a})$ set $ξ_{b a} = k (θ_{b} - d_{a}) / (2 π)$ and define $F \in C^{m \times N}$ by $F_{(b, a), n} = \frac{1}{4 π} \int_{E_{n}} e^{- 2 π i x \cdot ξ_{b a}} d x$ .

1.2F4F5givenchoosealgebra

Regard $F$ as the linear map $C^{m + 1} \to C^{m}$ . Its image is a subspace of dimension at most $m$ by [F5], so [F4] gives $\dim \ker F = (m + 1) - \dim im F \geq 1$ . Choose a nonzero $c = (c_{n})_{n = 1}^{m + 1} \in \ker F$ .

2.1F1F2step 1.2algebra

Define $h : = \sum_{n = 1}^{m + 1} c_{n} 1_{E_{n}}$ . Since at least one $c_{n} \neq 0$ and each $E_{n}$ has positive measure, $h$ is a nonzero element of $L^{\infty} (D; C)$ . By [F1] and [F2], $B_{k} h (θ_{b}, d_{a}) = (F c)_{(b, a)} = 0$ for every $1 \leq b \leq r$ , $1 \leq a \leq s$ . Thus the finite Born sample map has a nontrivial kernel.

3.1A1F1F6step 2.1algebra

For any scalar $t$ , [F6] gives $ρ_{k} (t h) \leq C_{D} ∥ t h ∥_{\infty} = C_{D} ∣ t ∣ ∥ h ∥_{\infty}$ . The right-hand side tends to zero as $∣ t ∣ \to 0$ , so choose $t \neq 0$ small enough that $t h \in U_{D, k}$ . Set $q : = t h$ . Linearity of $B_{k}$ gives $B_{k} q (θ_{b}, d_{a}) = 0$ for every acquisition pair while $q \neq 0$ . The vector $c$ is a nonzero coefficient representation of a contrast whose sampled first Born data are exactly those of zero.

4.1F7step 3.1algebra∎

For input $y = 0$ , any filter application in the algorithm gives $\hat{c} = 0$ by [F7]; the reconstruction is zero and $R = ∥ F (0) - 0 ∥_{2} = 0 \leq δ$ , so the status is discrepancy for every $δ \geq 0$ and positive budget. This status cannot distinguish the zero contrast from the nonzero contrast in step 3.1 within the finite Born model, and no statement about the exact nonlinear data was used.

Depends on

Used by

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Sources

Fioralba Cakoni, David Colton, and Houssem Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS 88 (standard reference, not scraped)
Otmar Scherzer and Leonidas Mindrinos, Inverse Problems: Lecture Notes, University of Vienna, Sommersemester 2015 (standard reference, not scraped)
Semyon Dyatlov, MIT 18.155 Lecture Notes: Distributions, Elliptic Regularity, and Applications to PDEs (standard reference, not scraped)