Introduction
Passive gamma emission tomography (PGET) is a non-invasive technique for verifying the internal configuration of spent nuclear fuel assemblies [2 ] . Gamma rays emitted by the fuel rods are attenuated as they pass through the rods and the moderator before reaching a row of collimated gamma-ray detectors. Mathematically, these measurements are modeled by the attenuated X-ray transform (AtRT) R a f \mathcal{R}_a f R a f f f f a a a Ω \Omega Ω
In standard emission tomography, the attenuation is often treated as known, at least approximately, and only the source is recovered. Reconstruction methods include iterative approaches [25 , 10 ] and, for the attenuated X-ray transform, Novikov’s inversion formula [19 ] . In PGET, both a a a f f f [3 ] and has been studied since the late 1970s [5 , 15 ] . The problem is not uniquely solvable in general, so unique recovery requires additional structure.
We impose geometric and material assumptions motivated by the design of nuclear fuel assemblies. The domain is a fixed disk containing finitely many disjoint smaller disks of equal radius at known locations. The smaller disks represent cross-sections of fuel rods. We model a a a f f f [12 ] analyzed the singularities associated with tangential rays and recovered attenuation jumps under a non-cancellation condition.
Our contribution has two parts. First, we give a condition involving a common tangent and a second emitting rod that ensures non-cancellation on an open family of tangencies of a rod with a nonzero attenuation jump (Lemma 3.3 ). This gives a sufficient condition for recovery from full data (Theorem 3.5 ). Second, for rods on sufficiently separated parallel rows with strictly positive rod sources, we select tangencies at which non-cancellation holds uniformly over the parameter space. These tangencies give explicit reconstruction formulas and stability estimates for both coefficients in a specific singular data seminorm (Theorem 4.7 ). The analysis does not include noise or scattering. Stability remains open when the assembly contains a non-emitting rod, such as fresh nuclear fuel or a special absorber rod.
Related literature
When a = 0 a = 0 a = 0 [16 ] , for which explicit inversion formulas are available [26 ] . Hertle [9 ] studied the identification of an unknown constant attenuation parameter when the source is also unknown. Solmon [23 ] studied the corresponding identification problem, including the role of radial symmetry. Novikov established an explicit inversion formula [19 ] and a range characterization [18 ] for the two-dimensional attenuated X-ray transform. Natterer [17 ] derived a closely related exact inversion formula using a different approach. Boman and Strömberg [4 ] gave an alternative proof of Novikov’s inversion formula and extended it to a broader class of weighted Radon transforms. Kunyansky [14 ] developed a noniterative SPECT reconstruction algorithm based on Novikov’s inversion formula for known, spatially varying attenuation. An earlier approach used the theory of A A A [8 , 20 Zho24, 21 ] .
Dicken [6 ] studied joint recovery using the stability and mapping properties of the AtRT in L p L^p L p [24 ] studied uniqueness, stability, and the linearized problem, and also demonstrated non-uniqueness when both coefficients are radial. Holman and Richardson [11 ] established unique recovery under a piecewise constant assumption on the attenuation and with a sufficiently regular source. Their subsequent work [12 ] treated piecewise smooth sources and real-analytic interfaces under a non-cancellation condition.
PGET has been developed for nuclear safeguards and the verification of spent nuclear fuel [30 , 28 ] , and has been approved by the International Atomic Energy Agency (IAEA) since 2017 [30 ] . A reconstruction method used in practice combines an iterative Levenberg–Marquardt trust-region algorithm with physical constraints and variational regularization [2 , 27 , 28 ] . Other methods have been developed, for example in [SHG15, WMD+17, FADL+21]. The fixed rod geometry of PGET motivates our analysis of uniqueness and stability.
Organization of the article
This article addresses two questions in the PGET disk model.
When can non-cancellation be established from the geometry and the coefficient assumptions?
How stable are the recovered attenuation and source with respect to perturbations of the data?
§ 3 addresses the first question, and § 4 addresses the second. We present related numerical experiments in § 5 to help illustrate some of the presented concepts and theoretical results. These codes are also available in https://github.com/tommheik/PGET_stability.
The following diagram summarizes the structure of the proofs of the two main results.
Lemma 2.12 \hspace{4em} [19 ]
Lemma 2.8 \hspace{2em} → Corollary 2.14 → Lemma 3.3 → Theorem 3.5 \hspace{2em} Lemma 2.3
Remark 2.9 \hspace{2em} Proposition 2.19 ← Lemma 2.5
Corollary 2.14 → Proposition 4.4 → Theorem 4.7 ← Remark 2.18
Lemma 2.7
We prove Theorem 3.5 in § 3 . The main step is Lemma 3.3 , which combines the chord formulas of Lemma 2.8 with the tangential trace representation in Corollary 2.14 to show that the non-cancellation coefficient does not vanish identically along a family of simple tangencies near a common tangent. The singular identities of Proposition 2.19 , based on [12 ] , then determine the attenuation. Once the attenuation is known, Novikov’s inversion formula [19 ] recovers the source.
The proof of Theorem 4.7 in § 4 uses the admissible tangencies constructed in Proposition 4.4 . Under the row-separation assumption, this proposition provides, for every rod, a tangency at which the tangential trace vanishes. The positive lower bound on the rod sources therefore gives a uniform non-cancellation bound. Combining the identities of Proposition 2.19 with the geometric bounds of Lemma 2.7 and the seminorm estimates of Remark 2.18 yields the stability estimates.
Measurement geometry and mathematical model
We introduce the forward model and the fixed geometry, then describe tangencies and backward-ray chords used in reconstruction. We also specify the coefficient space and the singular data functionals.
The attenuated X-ray transform
We work in the planar cross-section R 2 \mathbb{R}^{2} R 2 ω ∈ [ 0 , 2 π ) \omega\in[0,2\pi) ω ∈ [ 0 , 2 π )
θ ( ω ) = ( cos ω sin ω ) ∈ S 1 , θ ⊥ ( ω ) = ∂ ω θ ( ω ) = ( − sin ω cos ω ) . \theta(\omega)=\begin{pmatrix}\cos\omega\\ \sin\omega\end{pmatrix}\in S^{1}, \qquad\theta^{\perp}(\omega)=\partial_{\omega}\theta(\omega)=\begin{pmatrix}-\sin\omega\\ \cos\omega\end{pmatrix}. θ ( ω ) = ( cos ω sin ω ) ∈ S 1 , θ ⊥ ( ω ) = ∂ ω θ ( ω ) = ( − sin ω cos ω ) . Angles are understood modulo 2 π 2\pi 2 π ( R / 2 π Z ) 2 (\mathbb{R}/2\pi\mathbb{Z})^{2} ( R /2 π Z ) 2 [ 0 , 2 π ) 2 [0,2\pi)^{2} [ 0 , 2 π ) 2 x ∈ R 2 x \in\mathbb{R}^{2} x ∈ R 2 θ ( ω ) \theta(\omega) θ ( ω )
L ( x , θ ( ω ) ) : = { x + t θ ( ω ) : t ∈ R } . L(x,\theta(\omega)):=\{x+t\theta(\omega):t\in\mathbb{R}\}. L ( x , θ ( ω )) := { x + tθ ( ω ) : t ∈ R } . The attenuation a a a f f f Ω \Omega Ω beam transform of a a a
D a ( x , θ ( ω ) ) = ∫ 0 ∞ a ( x + ρ θ ( ω ) ) d ρ . (1) Da(x,\theta(\omega))=\int_{0}^{\infty}a(x+\rho\theta(\omega))\,d\rho. \tag*{(1)} D a ( x , θ ( ω )) = ∫ 0 ∞ a ( x + ρθ ( ω )) d ρ . (1) The beam transform measures the attenuation accumulated along the forward ray from x x x θ ( ω ) \theta(\omega) θ ( ω ) attenuated X-ray transform (AtRT) of f f f a a a
R a f ( x , θ ( ω ) ) = ∫ − ∞ ∞ f ( x + t θ ( ω ) ) e − D a ( x + t θ ( ω ) , θ ( ω ) ) d t . R_a f(x,\theta(\omega))=\int_{-\infty}^{\infty}f(x+t\theta(\omega))e^{-D_a(x+t\theta(\omega),\theta(\omega))}\,dt. R a f ( x , θ ( ω )) = ∫ − ∞ ∞ f ( x + tθ ( ω )) e − D a ( x + tθ ( ω ) , θ ( ω )) d t . A change of variable shows that
R a f ( x + s θ , θ ) = R a f ( x , θ ) , s ∈ R . R_a f(x+s\theta,\theta)=R_a f(x,\theta),\qquad s\in\mathbb{R}. R a f ( x + s θ , θ ) = R a f ( x , θ ) , s ∈ R . Thus, the data depend only on the oriented line, independently of the point chosen to represent it.
For a bounded convex Lipschitz domain Ω ⊂ R 2 \Omega\subset\mathbb{R}^2 Ω ⊂ R 2 ν \nu ν
Γ − = { ( x , θ ) ∈ ∂ Ω × S 1 : θ ⋅ ν ( x ) ≤ 0 } . \Gamma^-=\{(x,\theta)\in\partial\Omega\times\mathbb{S}^1:\theta\cdot\nu(x)\leq0\}. Γ − = {( x , θ ) ∈ ∂ Ω × S 1 : θ ⋅ ν ( x ) ≤ 0 } . The transport equation
θ ⋅ ∇ x u ( x , θ ) + a ( x ) u ( x , θ ) = f ( x ) , ( x , θ ) ∈ Ω × S 1 , u ∣ Γ − = 0 , (2) \theta\cdot\nabla_xu(x,\theta)+a(x)u(x,\theta)=f(x),\qquad(x,\theta)\in\Omega\times\mathbb{S}^1,\qquad u|_{\Gamma^-}=0, \tag*{(2)} θ ⋅ ∇ x u ( x , θ ) + a ( x ) u ( x , θ ) = f ( x ) , ( x , θ ) ∈ Ω × S 1 , u ∣ Γ − = 0 , (2) has a solution u u u
u ( x , θ ( ω ) ) = ∫ − ∞ 0 f ( x + t θ ( ω ) ) exp ( − ∫ t 0 a ( x + ( t − ρ ) θ ( ω ) ) d ρ ) d t . (3) u(x,\theta(\omega))=\int_{-\infty}^{0}f(x+t\theta(\omega))\exp\left(-\int_t^0a(x+(t-\rho)\theta(\omega))\,d\rho\right)\,dt. \tag*{(3)} u ( x , θ ( ω )) = ∫ − ∞ 0 f ( x + tθ ( ω )) exp ( − ∫ t 0 a ( x + ( t − ρ ) θ ( ω )) d ρ ) d t . (3) see [12 Section 2] . For every x ∈ R 2 x\in\mathbb{R}^2 x ∈ R 2 ω ∈ [ 0 , 2 π ) \omega\in[0,2\pi) ω ∈ [ 0 , 2 π ) (3) gives the backward-ray representation
u ( x , θ ( ω ) ) = ∫ 0 ∞ f ( x − s θ ( ω ) ) exp ( − ∫ 0 s a ( x − σ θ ( ω ) ) d σ ) d s . (4) u(x,\theta(\omega))=\int_0^\infty f(x-s\theta(\omega))\exp\left(-\int_0^s a(x-\sigma\theta(\omega))\,d\sigma\right)\,ds. \tag*{(4)} u ( x , θ ( ω )) = ∫ 0 ∞ f ( x − s θ ( ω )) exp ( − ∫ 0 s a ( x − σ θ ( ω )) d σ ) d s . (4) For y ∈ R 2 y\in\mathbb{R}^2 y ∈ R 2 θ ∈ S 1 \theta\in\mathbb{S}^1 θ ∈ S 1 s ≥ 0 s\geq0 s ≥ 0
D a ( y − s θ , θ ) = ∫ 0 s a ( y − σ θ ) d σ + D a ( y , θ ) . D_a(y-s\theta,\theta)=\int_0^s a(y-\sigma\theta)\,d\sigma+D_a(y,\theta). D a ( y − s θ , θ ) = ∫ 0 s a ( y − σ θ ) d σ + D a ( y , θ ) . Consequently, setting s = − t s=-t s = − t (4) give
∫ − ∞ 0 f ( y + t θ ) e − D a ( y + t θ , θ ) d t = e − D a ( y , θ ) ∫ 0 ∞ f ( y − s θ ) exp ( − ∫ 0 s a ( y − σ θ ) d σ ) d s = e − D a ( y , θ ) u ( y , θ ) . \begin{aligned}
\int_{-\infty}^{0}f(y+t\theta)e^{-D_a(y+t\theta,\theta)}\,dt=e^{-D_a(y,\theta)}\int_0^\infty f(y-s\theta)\exp\left(-\int_0^s a(y-\sigma\theta)\,d\sigma\right)\,ds \\
&=e^{-D_a(y,\theta)}u(y,\theta).
\end{aligned} ∫ − ∞ 0 f ( y + tθ ) e − D a ( y + tθ , θ ) d t = e − D a ( y , θ ) ∫ 0 ∞ f ( y − s θ ) exp ( − ∫ 0 s a ( y − σ θ ) d σ ) d s = e − D a ( y , θ ) u ( y , θ ) . Splitting the integral defining R a f R_af R a f t = 0 t=0 t = 0
R a f ( y , θ ) = e − D a ( y , θ ) u ( y , θ ) + ∫ 0 ∞ f ( y + t θ ) e − D a ( y + t θ , θ ) d t . R_af(y,\theta)=e^{-D_a(y,\theta)}u(y,\theta)+\int_0^\infty f(y+t\theta)e^{-D_a(y+t\theta,\theta)}\,dt. R a f ( y , θ ) = e − D a ( y , θ ) u ( y , θ ) + ∫ 0 ∞ f ( y + tθ ) e − D a ( y + tθ , θ ) d t . Let x ∈ ∂ Ω x\in\partial\Omega x ∈ ∂ Ω ν ( x ) \nu(x) ν ( x ) θ ⋅ ν ( x ) ≥ 0 \theta\cdot\nu(x)\geq0 θ ⋅ ν ( x ) ≥ 0 { x + t θ : t > 0 } \{x+t\theta:t>0\} { x + tθ : t > 0 } Ω \Omega Ω a a a D a ( x , θ ) = 0 D_a(x,\theta)=0 D a ( x , θ ) = 0 u u u
R a f ( x , θ ) = u ( x , θ ) . R_af(x,\theta)=u(x,\theta). R a f ( x , θ ) = u ( x , θ ) . Geometric setting of PGET
Fuel rods are enclosed in metal cladding and arranged in assemblies containing dozens or hundreds of rods. Assembly designs include rectangular and hexagonal arrangements, water channels of different sizes, and partial-length rods. For details, see [28 ] . For a specified assembly and axial slice, the rod positions and diameters are known. Gamma-ray measurements are acquired with collimated detectors located on a ring outside the assembly. Figure 1 shows a PGET device.
Figure 1. PGET device with its outer cover removed and a shortened dummy assembly. Measurements are collected around a horizontal slice of the assembly. Photograph by Dean Calma / IAEA.
We model the slice as disks of equal radius with constant coefficients, surrounded by a moderator with known attenuation and zero source. The model omits cladding, axial variation, scattering, and the finite collimator response. Known moderator-filled positions, including water channels and partial rods fully outside the field of view, belong to the moderator. Every other non-emitting rod remains a rod in the coefficient model. We also assume that the location of the assembly within the domain is known.
We now specify this geometry. Fix an outer radius r o u t > 0 r_{\mathrm{out}} > 0 r out > 0 Ω = B ( 0 , r o u t ) \Omega= B(0,r_{\mathrm{out}}) Ω = B ( 0 , r out )
∂ Ω = { x ( ψ ) : x ( ψ ) = r o u t ( cos ψ , sin ψ ) , ψ ∈ [ 0 , 2 π ) } . \partial\Omega= \{x(\psi) : x(\psi) = r_{\mathrm{out}}(\cos\psi,\sin\psi),\ \psi\in[0,2\pi)\}. ∂ Ω = { x ( ψ ) : x ( ψ ) = r out ( cos ψ , sin ψ ) , ψ ∈ [ 0 , 2 π )} . Fix the number of rods K ≥ 2 K \ge2 K ≥ 2 r > 0 r > 0 r > 0 c 1 , … , c K c_1,\ldots,c_K c 1 , … , c K
B ‾ q = B ( c q , r ) ⊂ Ω , q = 1 , … , K , \overline{B}_q = B(c_q,r) \subset\Omega,\qquad q=1,\ldots,K, B q = B ( c q , r ) ⊂ Ω , q = 1 , … , K , are pairwise disjoint. Equivalently, ∣ c q − c q ′ ∣ > 2 r |c_q-c_{q'}| > 2r ∣ c q − c q ′ ∣ > 2 r q ≠ q ′ q \ne q' q = q ′ ∣ c q ∣ + r < r o u t |c_q|+r < r_{\mathrm{out}} ∣ c q ∣ + r < r out q q q ∂ B q \partial B_q ∂ B q q = 1 , … , K q=1,\ldots,K q = 1 , … , K Ω 0 = Ω ∖ ⋃ q = 1 K B ‾ q \Omega_0 = \Omega\setminus\bigcup_{q=1}^{K}\overline{B}_q Ω 0 = Ω ∖ ⋃ q = 1 K B q B 1 , … , B K B_1,\ldots,B_K B 1 , … , B K Ω 0 \Omega_0 Ω 0 Figure 2 . These parameters define the geometry
Figure 2. Measurement geometry with disjoint rods B q B_q B q Ω 0 \Omega_0 Ω 0 ψ \psi ψ x ( ψ ) x(\psi) x ( ψ ) ω \omega ω L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω )
G = ( r o u t , r , { c q } q = 1 K ) , (5) \mathcal{G} = \left(r_{\mathrm{out}},r,\{c_q\}_{q=1}^{K}\right), \tag*{(5)} G = ( r out , r , { c q } q = 1 K ) , (5) which remains fixed throughout the article. For ( ψ , ω ) ∈ [ 0 , 2 π ) 2 (\psi,\omega) \in[0,2\pi)^2 ( ψ , ω ) ∈ [ 0 , 2 π ) 2 L ( ψ , ω ) : = L ( x ( ψ ) , θ ( ω ) ) L(\psi,\omega) := L(x(\psi),\theta(\omega)) L ( ψ , ω ) := L ( x ( ψ ) , θ ( ω ))
Tangencies of the rods
We regard the angular parameter space as the torus ( R / 2 π Z ) 2 (\mathbb{R}/2\pi\mathbb{Z})^2 ( R /2 π Z ) 2 [ 0 , 2 π ) 2 [0,2\pi)^2 [ 0 , 2 π ) 2
For a rod B p B_p B p c p − x ( ψ ) c_p-x(\psi) c p − x ( ψ ) { θ ( ω ) , θ ⊥ ( ω ) } \{\theta(\omega),\theta^\perp(\omega)\} { θ ( ω ) , θ ⊥ ( ω )}
t p ( ψ , ω ) : = θ ( ω ) ⋅ ( c p − x ( ψ ) ) , n p ( ψ , ω ) : = θ ⊥ ( ω ) ⋅ ( c p − x ( ψ ) ) . t_p(\psi,\omega):=\theta(\omega)\cdot(c_p-x(\psi)),\qquad n_p(\psi,\omega):=\theta^\perp(\omega)\cdot(c_p-x(\psi)). t p ( ψ , ω ) := θ ( ω ) ⋅ ( c p − x ( ψ )) , n p ( ψ , ω ) := θ ⊥ ( ω ) ⋅ ( c p − x ( ψ )) . The point x ( ψ ) + t p ( ψ , ω ) θ ( ω ) x(\psi)+t_p(\psi,\omega)\theta(\omega) x ( ψ ) + t p ( ψ , ω ) θ ( ω ) c p c_p c p L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) ∣ n p ( ψ , ω ) ∣ |n_p(\psi,\omega)| ∣ n p ( ψ , ω ) ∣ c p c_p c p L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) ∣ t p ( ψ , ω ) ∣ |t_p(\psi,\omega)| ∣ t p ( ψ , ω ) ∣ x ( ψ ) x(\psi) x ( ψ ) L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) B p B_p B p ∣ n p ( ψ , ω ) ∣ < r |n_p(\psi,\omega)|<r ∣ n p ( ψ , ω ) ∣ < r ∂ B p \partial B_p ∂ B p ∣ n p ( ψ , ω ) ∣ = r |n_p(\psi,\omega)|=r ∣ n p ( ψ , ω ) ∣ = r B p ‾ \overline{B_p} B p ∣ n p ( ψ , ω ) ∣ > r |n_p(\psi,\omega)|>r ∣ n p ( ψ , ω ) ∣ > r
We define the tangency set of B p B_p B p
T p : = { ( ψ , ω ) ∈ ( R / 2 π Z ) 2 : ∣ n p ( ψ , ω ) ∣ = r , t p ( ψ , ω ) > 0 } . \mathcal{T}_p:=\left\{(\psi,\omega)\in(\mathbb{R}/2\pi\mathbb{Z})^2:|n_p(\psi,\omega)|=r,\ t_p(\psi,\omega)>0\right\}. T p := { ( ψ , ω ) ∈ ( R /2 π Z ) 2 : ∣ n p ( ψ , ω ) ∣ = r , t p ( ψ , ω ) > 0 } . For τ = ( ψ , ω ) ∈ T p \tau=(\psi,\omega)\in\mathcal{T}_p τ = ( ψ , ω ) ∈ T p ∂ B p \partial B_p ∂ B p
y τ : = x ( ψ ) + t p ( ψ , ω ) θ ( ω ) ∈ ∂ B p . (6) y_\tau:=x(\psi)+t_p(\psi,\omega)\theta(\omega)\in\partial B_p. \tag*{(6)} y τ := x ( ψ ) + t p ( ψ , ω ) θ ( ω ) ∈ ∂ B p . (6) If a line is tangent to more than one rod, y τ y_\tau y τ
At τ = ( ψ , ω ) ∈ T p \tau=(\psi,\omega)\in\mathcal{T}_p τ = ( ψ , ω ) ∈ T p t p = t p ( ψ , ω ) > 0 t_p=t_p(\psi,\omega)>0 t p = t p ( ψ , ω ) > 0 y τ ∈ Ω y_\tau\in\Omega y τ ∈ Ω ∣ x ( ψ ) ∣ = r o u t |x(\psi)|=r_{\mathrm{out}} ∣ x ( ψ ) ∣ = r out
2 t p x ( ψ ) ⋅ θ ( ω ) + t p 2 = ∣ y τ ∣ 2 − r o u t 2 < 0. 2t_p x(\psi)\cdot\theta(\omega)+t_p^2=|y_\tau|^2-r_{\mathrm{out}}^2<0. 2 t p x ( ψ ) ⋅ θ ( ω ) + t p 2 = ∣ y τ ∣ 2 − r out 2 < 0. Hence x ( ψ ) ⋅ θ ( ω ) < 0 x(\psi)\cdot\theta(\omega)<0 x ( ψ ) ⋅ θ ( ω ) < 0 θ ( ω ) \theta(\omega) θ ( ω ) Ω \Omega Ω x ( ψ ) x(\psi) x ( ψ ) L ( ψ , ω ) ∩ Ω L(\psi,\omega)\cap\Omega L ( ψ , ω ) ∩ Ω x ( ψ ) x(\psi) x ( ψ ) θ ( ω ) \theta(\omega) θ ( ω ) L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) ∂ B q \partial B_q ∂ B q ∂ B q \partial B_q ∂ B q Ω \Omega Ω t q ( ψ , ω ) > 0 t_q(\psi,\omega)>0 t q ( ψ , ω ) > 0
Equivalently, the tangency set is given by
T p = { G p = 0 } ∩ { t p > 0 } , \mathcal{T}_p=\{G_p=0\}\cap\{t_p>0\}, T p = { G p = 0 } ∩ { t p > 0 } , where G p ( ψ , ω ) : = n p ( ψ , ω ) 2 − r 2 G_p(\psi,\omega):=n_p(\psi,\omega)^2-r^2 G p ( ψ , ω ) := n p ( ψ , ω ) 2 − r 2 G p G_p G p L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) B p B_p B p B p ‾ \overline{B_p} B p ∂ ω θ = θ ⊥ \partial_\omega\theta=\theta^\perp ∂ ω θ = θ ⊥ ∂ ω θ ⊥ = − θ \partial_\omega\theta^\perp=-\theta ∂ ω θ ⊥ = − θ
∂ ω G p ( ψ , ω ) = − 2 n p ( ψ , ω ) t p ( ψ , ω ) . (7) \partial_\omega G_p(\psi,\omega)=-2n_p(\psi,\omega)t_p(\psi,\omega). \tag*{(7)} ∂ ω G p ( ψ , ω ) = − 2 n p ( ψ , ω ) t p ( ψ , ω ) . (7) In particular, we have
∣ ∂ ω G p ( ψ , ω ) ∣ = 2 r t p ( ψ , ω ) > 0 \left|\partial_{\omega}G_p(\psi,\omega)\right|=2r\,t_p(\psi,\omega)>0 ∣ ∂ ω G p ( ψ , ω ) ∣ = 2 r t p ( ψ , ω ) > 0 for any ( ψ , ω ) ∈ T p (\psi,\omega)\in T_p ( ψ , ω ) ∈ T p
Definition 2.1 (Simple tangency). Let m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } τ = ( ψ , ω ) ∈ T m \tau=(\psi,\omega)\in T_m τ = ( ψ , ω ) ∈ T m τ \tau τ simple if L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) ∂ B q \partial B_q ∂ B q q ≠ m q\ne m q = m τ ∈ T m \tau\in T_m τ ∈ T m
τ ∉ ⋃ q ≠ m T q . \tau\notin\bigcup_{q\ne m}T_q. τ ∈ / q = m ⋃ T q . We denote the set of simple tangencies of B m B_m B m S m S_m S m
Lemma 2.2. Let p , q ∈ { 1 , … , K } p,q\in\{1,\ldots,K\} p , q ∈ { 1 , … , K } T p T_p T p ( R / 2 π Z ) 2 (\mathbb{R}/2\pi\mathbb{Z})^2 ( R /2 π Z ) 2 p ≠ q p\ne q p = q T p ∩ T q T_p\cap T_q T p ∩ T q S p ≠ ∅ S_p\ne\varnothing S p = ∅
Proof. Equation (7) gives ∂ ω G p ≠ 0 \partial_{\omega}G_p\ne0 ∂ ω G p = 0 T p T_p T p [13 Theorem 2.3.5] ) implies that T p T_p T p T p T_p T p ∂ Ω \partial\Omega ∂ Ω B p B_p B p ∂ B p \partial B_p ∂ B p
Now let p ≠ q p\ne q p = q T p ∩ T q T_p\cap T_q T p ∩ T q ∂ B p \partial B_p ∂ B p ∂ B q \partial B_q ∂ B q ∣ c p − c q ∣ > 2 r \lvert c_p-c_q\rvert>2r ∣ c p − c q ∣ > 2 r Ω \Omega Ω ∂ Ω \partial\Omega ∂ Ω Ω \Omega Ω ψ \psi ψ ω \omega ω ∣ T p ∩ T q ∣ = 4 ⋅ 2 = 8 \lvert T_p\cap T_q\rvert=4\cdot2=8 ∣ T p ∩ T q ∣ = 4 ⋅ 2 = 8 T p T_p T p
To define one-sided angular limits, we need a neighborhood of a tangency containing no other tangent direction when the boundary point x ( ψ ) x(\psi) x ( ψ ) Figure 3 .
Figure 3. An angular neighborhood of a tangency τ = ( ψ , ω 0 ) \tau=(\psi,\omega_0) τ = ( ψ , ω 0 ) x ( ψ ) x(\psi) x ( ψ ) ω 0 \omega_0 ω 0
Lemma 2.3 (Angular window). Let m ∈ { 1 , … , K } m \in\{1,\ldots,K\} m ∈ { 1 , … , K } τ = ( ψ , ω 0 ) ∈ T m \tau=(\psi,\omega_0)\in\mathcal{T}_m τ = ( ψ , ω 0 ) ∈ T m δ τ ∈ ( 0 , π ) \delta_\tau\in(0,\pi) δ τ ∈ ( 0 , π ) G \mathcal{G} G τ \tau τ
( ψ , ω ) ∉ ⋃ q = 1 K T q whenever 0 < ∣ ω − ω 0 ∣ < δ τ . (\psi,\omega)\notin\bigcup_{q=1}^{K}\mathcal{T}_q \qquad\text{whenever} \qquad0<|\omega-\omega_0|<\delta_\tau. ( ψ , ω ) ∈ / q = 1 ⋃ K T q whenever 0 < ∣ ω − ω 0 ∣ < δ τ . Moreover, for every ∣ ω − ω 0 ∣ < δ τ |\omega-\omega_0|<\delta_\tau ∣ ω − ω 0 ∣ < δ τ
t m ( ψ , ω ) > 0 , θ ( ω ) ⋅ x ( ψ ) < 0. t_m(\psi,\omega)>0,\qquad\theta(\omega)\cdot x(\psi)<0. t m ( ψ , ω ) > 0 , θ ( ω ) ⋅ x ( ψ ) < 0. Proof. Fix q ∈ { 1 , … , K } q\in\{1,\ldots,K\} q ∈ { 1 , … , K } B ‾ q ⊂ Ω \overline{B}_q\subset\Omega B q ⊂ Ω x ( ψ ) ∈ ∂ Ω x(\psi)\in\partial\Omega x ( ψ ) ∈ ∂ Ω b q = ∣ c q − x ( ψ ) ∣ > r b_q=|c_q-x(\psi)|>r b q = ∣ c q − x ( ψ ) ∣ > r α q \alpha_q α q c q − x ( ψ ) = b q θ ( α q ) c_q-x(\psi)=b_q\theta(\alpha_q) c q − x ( ψ ) = b q θ ( α q ) n q ( ψ , ω ) = b q sin ( α q − ω ) n_q(\psi,\omega)=b_q\sin(\alpha_q-\omega) n q ( ψ , ω ) = b q sin ( α q − ω )
G q ( ψ , ω ) = b q 2 sin 2 ( α q − ω ) − r 2 . G_q(\psi,\omega)=b_q^2\sin^2(\alpha_q-\omega)-r^2. G q ( ψ , ω ) = b q 2 sin 2 ( α q − ω ) − r 2 . Thus, the full line L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) ∂ B q \partial B_q ∂ B q
∣ sin ( α q − ω ) ∣ = r b q . |\sin(\alpha_q-\omega)|=\frac{r}{b_q}. ∣ sin ( α q − ω ) ∣ = b q r . Since 0 < r / b q < 1 0<r/b_q<1 0 < r / b q < 1 2 π 2\pi 2 π T q \mathcal{T}_q T q t q > 0 t_q>0 t q > 0 x ( ψ ) x(\psi) x ( ψ ) ω 0 \omega_0 ω 0 δ τ > 0 \delta_\tau>0 δ τ > 0 τ \tau τ δ τ < π \delta_\tau<\pi δ τ < π
t m ( ψ , ω ) > 0 , θ ( ω ) ⋅ x ( ψ ) < 0 , when ∣ ω − ω 0 ∣ < δ τ . t_m(\psi,\omega)>0,\qquad\theta(\omega)\cdot x(\psi)<0,\quad\text{when }|\omega-\omega_0|<\delta_\tau. t m ( ψ , ω ) > 0 , θ ( ω ) ⋅ x ( ψ ) < 0 , when ∣ ω − ω 0 ∣ < δ τ . The restriction δ τ < π \delta_\tau<\pi δ τ < π 2 π 2\pi 2 π
We next identify the side of a tangency on which a small angular perturbation causes the ray to enter the rod. This is the side from which the singular limits in § 2.7.1 are taken.
Definition 2.4 (Singular side). Let m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } τ = ( ψ , ω 0 ) ∈ T m \tau=(\psi,\omega_0)\in\mathcal{T}_m τ = ( ψ , ω 0 ) ∈ T m
ϵ τ : = sgn n m ( ψ , ω 0 ) ∈ { + 1 , − 1 } . (8) \epsilon_\tau:=\operatorname{sgn}n_m(\psi,\omega_0)\in\{+1,-1\}. \tag*{(8)} ϵ τ := sgn n m ( ψ , ω 0 ) ∈ { + 1 , − 1 } . (8) This is well defined because ∣ n m ( ψ , ω 0 ) ∣ = r > 0 |n_m(\psi,\omega_0)|=r>0 ∣ n m ( ψ , ω 0 ) ∣ = r > 0 singular side of τ \tau τ ω \omega ω Lemma 2.3 with ϵ τ ( ω − ω 0 ) > 0 \epsilon_\tau(\omega-\omega_0)>0 ϵ τ ( ω − ω 0 ) > 0 Lemma 2.5 (ii), this is the side on which L ( ψ , ω ) L(\psi,\omega) L ( ψ , ω ) B m B_m B m F m ( a , f ) ( τ ) ≠ 0 F_m^{(a,f)}(\tau)\ne0 F m ( a , f ) ( τ ) = 0 Proposition 2.19 identifies the nonzero square-root singular coefficient of the data on this side. The term singular side refers to this geometric choice, including when the leading coefficient vanishes. We write ω → ω 0 ϵ \omega\to\omega_0^{\epsilon} ω → ω 0 ϵ
The following lemma identifies the inward normal at the tangency point and verifies that the singular side is the side on which the perturbed line enters the rod.
Lemma 2.5. Let τ = ( ψ , ω 0 ) ∈ T m \tau=(\psi,\omega_0)\in\mathcal{T}_m τ = ( ψ , ω 0 ) ∈ T m y τ y_\tau y τ δ τ \delta_\tau δ τ Lemma 2.3 . Then the following statements hold.
(i) We have c m − y τ = ϵ τ r θ ⊥ ( ω 0 ) c_m-y_\tau=\epsilon_\tau r\theta^\perp(\omega_0) c m − y τ = ϵ τ r θ ⊥ ( ω 0 ) ϵ τ θ ⊥ ( ω 0 ) \epsilon_\tau\theta^\perp(\omega_0) ϵ τ θ ⊥ ( ω 0 ) ∂ B m \partial B_m ∂ B m y τ y_\tau y τ c m c_m c m
(ii) For 0 < ∣ ω − ω 0 ∣ < δ τ 0<|\omega-\omega_0|<\delta_\tau 0 < ∣ ω − ω 0 ∣ < δ τ L ( x ( ψ ) , θ ( ω ) ) L(x(\psi),\theta(\omega)) L ( x ( ψ ) , θ ( ω )) B m B_m B m ϵ τ ( ω − ω 0 ) > 0 \epsilon_\tau(\omega-\omega_0)>0 ϵ τ ( ω − ω 0 ) > 0
Proof. (i) Since the line L ( x ( ψ ) , θ ( ω 0 ) ) L(x(\psi),\theta(\omega_0)) L ( x ( ψ ) , θ ( ω 0 )) ∂ B m \partial B_m ∂ B m y τ y_\tau y τ θ ( ω 0 ) ⋅ ( c m − y τ ) = 0 \theta(\omega_0)\cdot(c_m-y_\tau)=0 θ ( ω 0 ) ⋅ ( c m − y τ ) = 0 y τ ∈ ∂ B m y_\tau\in\partial B_m y τ ∈ ∂ B m ∣ c m − y τ ∣ = r |c_m-y_\tau|=r ∣ c m − y τ ∣ = r c m − y τ = ± r θ ⊥ ( ω 0 ) c_m-y_\tau=\pm r\theta^\perp(\omega_0) c m − y τ = ± r θ ⊥ ( ω 0 ) y τ = x ( ψ ) + t m ( ψ , ω 0 ) θ ( ω 0 ) y_\tau=x(\psi)+t_m(\psi,\omega_0)\theta(\omega_0) y τ = x ( ψ ) + t m ( ψ , ω 0 ) θ ( ω 0 ) θ ⊥ ( ω 0 ) \theta^\perp(\omega_0) θ ⊥ ( ω 0 )
θ ⊥ ( ω 0 ) ⋅ ( c m − y τ ) = θ ⊥ ( ω 0 ) ⋅ ( c m − x ( ψ ) ) = n m ( ψ , ω 0 ) . \theta^\perp(\omega_0)\cdot(c_m-y_\tau)=\theta^\perp(\omega_0)\cdot(c_m-x(\psi))=n_m(\psi,\omega_0). θ ⊥ ( ω 0 ) ⋅ ( c m − y τ ) = θ ⊥ ( ω 0 ) ⋅ ( c m − x ( ψ )) = n m ( ψ , ω 0 ) . By the definition of ϵ τ \epsilon_\tau ϵ τ n m ( ψ , ω 0 ) = ϵ τ r n_m(\psi,\omega_0)=\epsilon_\tau r n m ( ψ , ω 0 ) = ϵ τ r
(ii) Recall that G m ( ψ , ω ) = n m ( ψ , ω ) 2 − r 2 G_m(\psi,\omega)=n_m(\psi,\omega)^2-r^2 G m ( ψ , ω ) = n m ( ψ , ω ) 2 − r 2 L ( x ( ψ ) , θ ( ω ) ) L(x(\psi),\theta(\omega)) L ( x ( ψ ) , θ ( ω )) B m B_m B m G m ( ψ , ω ) < 0 G_m(\psi,\omega)<0 G m ( ψ , ω ) < 0 G m ( ψ , ω 0 ) = 0 G_m(\psi,\omega_0)=0 G m ( ψ , ω 0 ) = 0
∂ ω G m ( ψ , ω 0 ) = − 2 n m ( ψ , ω 0 ) t m ( ψ , ω 0 ) = − 2 ϵ τ r t m ( ψ , ω 0 ) ≠ 0. \partial_\omega G_m(\psi,\omega_0)=-2n_m(\psi,\omega_0)t_m(\psi,\omega_0)=-2\epsilon_\tau r t_m(\psi,\omega_0)\ne0. ∂ ω G m ( ψ , ω 0 ) = − 2 n m ( ψ , ω 0 ) t m ( ψ , ω 0 ) = − 2 ϵ τ r t m ( ψ , ω 0 ) = 0. because t m ( ψ , ω 0 ) > 0 t_m(\psi,\omega_0)>0 t m ( ψ , ω 0 ) > 0 G m G_m G m ω 0 \omega_0 ω 0 G m G_m G m t m > 0 t_m>0 t m > 0 G m G_m G m ( ψ , ω ) ∈ T m (\psi,\omega)\in T_m ( ψ , ω ) ∈ T m G m G_m G m − ϵ τ ( ω − ω 0 ) -\epsilon_\tau(\omega-\omega_0) − ϵ τ ( ω − ω 0 )
G m ( ψ , ω ) < 0 ⟺ ϵ τ ( ω − ω 0 ) > 0 , G_m(\psi,\omega)<0\quad\Longleftrightarrow\quad\epsilon_\tau(\omega-\omega_0)>0, G m ( ψ , ω ) < 0 ⟺ ϵ τ ( ω − ω 0 ) > 0 , which proves (ii).
We collect the geometric quantities attached to a tangency.
Notation 2.6. Let m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } τ = ( ψ , ω ) ∈ T m \tau=(\psi,\omega)\in T_m τ = ( ψ , ω ) ∈ T m
x τ : = x ( ψ ) , θ τ : = θ ( ω ) , L τ : = t m ( ψ , ω ) . x_\tau:=x(\psi),\qquad\theta_\tau:=\theta(\omega),\qquad L_\tau:=t_m(\psi,\omega). x τ := x ( ψ ) , θ τ := θ ( ω ) , L τ := t m ( ψ , ω ) . At a configuration tangent to more than one rod, the notation is understood relative to the specified rod. The tangency point y τ y_\tau y τ δ τ \delta_\tau δ τ ϵ τ \epsilon_\tau ϵ τ
y τ = x τ + L τ θ τ , L τ = ∣ y τ − x τ ∣ > 0. y_\tau=x_\tau+L_\tau\theta_\tau,\qquad L_\tau=|y_\tau-x_\tau|>0. y τ = x τ + L τ θ τ , L τ = ∣ y τ − x τ ∣ > 0. Lemma 2.7. Let m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } τ ∈ T m \tau\in T_m τ ∈ T m
0 < L min ≤ L τ ≤ L max < ∞ , 0<L_{\min}\le L_\tau\le L_{\max}<\infty, 0 < L m i n ≤ L τ ≤ L m a x < ∞ , where
L min : = min 1 ≤ q ≤ K ( r o u t − ∣ c q ∣ − r ) , L max : = 2 r o u t . L_{\min}:=\min_{1\le q\le K}(r_{\mathrm{out}}-|c_q|-r),\qquad L_{\max}:=2r_{\mathrm{out}}. L m i n := 1 ≤ q ≤ K min ( r out − ∣ c q ∣ − r ) , L m a x := 2 r out . The bounds L min L_{\min} L m i n L max L_{\max} L m a x G \mathcal{G} G
Proof. Since x τ ∈ ∂ Ω x_\tau\in\partial\Omega x τ ∈ ∂ Ω y τ ∈ ∂ B m y_\tau\in\partial B_m y τ ∈ ∂ B m
L τ = ∣ y τ − x τ ∣ ≥ dist ( ∂ Ω , B m ‾ ) = r o u t − ∣ c m ∣ − r ≥ L min > 0. L_\tau=|y_\tau-x_\tau|\ge\operatorname{dist}(\partial\Omega,\overline{B_m})=r_{\mathrm{out}}-|c_m|-r\ge L_{\min}>0. L τ = ∣ y τ − x τ ∣ ≥ dist ( ∂ Ω , B m ) = r out − ∣ c m ∣ − r ≥ L m i n > 0. Since x τ , y τ ∈ Ω ‾ x_\tau,y_\tau\in\overline{\Omega} x τ , y τ ∈ Ω L τ ≤ diam ( Ω ) = 2 r o u t = L max L_\tau\le\operatorname{diam}(\Omega)=2r_{\mathrm{out}}=L_{\max} L τ ≤ diam ( Ω ) = 2 r out = L m a x
Chords along a backward ray
The representation (4) expresses the solution u u u (2) at y y y y y y − θ ( ω ) -\theta(\omega) − θ ( ω )
For y ∈ R 2 y \in\mathbb{R}^2 y ∈ R 2 ω ∈ [ 0 , 2 π ) \omega\in[0,2\pi) ω ∈ [ 0 , 2 π ) backward ray
R ( y , ω ) : = { y − s θ ( ω ) : s ≥ 0 } . \mathcal{R}(y,\omega):=\{y-s\theta(\omega):s\ge0\}. R ( y , ω ) := { y − s θ ( ω ) : s ≥ 0 } . For q ∈ { 1 , … , K } q \in\{1,\ldots,K\} q ∈ { 1 , … , K }
σ q ( y , ω ) : = − ( c q − y ) ⋅ θ ( ω ) , ρ q ( y , ω ) : = ( c q − y ) ⋅ θ ⊥ ( ω ) . \sigma_q(y,\omega):=-(c_q-y)\cdot\theta(\omega),\qquad\rho_q(y,\omega):=(c_q-y)\cdot\theta^\perp(\omega). σ q ( y , ω ) := − ( c q − y ) ⋅ θ ( ω ) , ρ q ( y , ω ) := ( c q − y ) ⋅ θ ⊥ ( ω ) . The quantity ∣ ρ q ( y , ω ) ∣ |\rho_q(y,\omega)| ∣ ρ q ( y , ω ) ∣ c q c_q c q L ( y , θ ( ω ) ) L(y,\theta(\omega)) L ( y , θ ( ω )) σ q ( y , ω ) \sigma_q(y,\omega) σ q ( y , ω ) c q − y c_q-y c q − y − θ ( ω ) -\theta(\omega) − θ ( ω ) ρ q \rho_q ρ q σ q \sigma_q σ q y y y cos ω \cos\omega cos ω sin ω \sin\omega sin ω y ∈ L ( ψ , ω ) y \in L(\psi,\omega) y ∈ L ( ψ , ω ) ρ q ( y , ω ) = n q ( ψ , ω ) \rho_q(y,\omega)=n_q(\psi,\omega) ρ q ( y , ω ) = n q ( ψ , ω ) ∣ c q − y ∣ 2 = σ q ( y , ω ) 2 + ρ q ( y , ω ) 2 |c_q-y|^2=\sigma_q(y,\omega)^2+\rho_q(y,\omega)^2 ∣ c q − y ∣ 2 = σ q ( y , ω ) 2 + ρ q ( y , ω ) 2
Lemma 2.8. Let ω ∈ [ 0 , 2 π ) \omega\in[0,2\pi) ω ∈ [ 0 , 2 π ) y ∈ R 2 y \in\mathbb{R}^2 y ∈ R 2 q ∈ { 1 , … , K } q \in\{1,\ldots,K\} q ∈ { 1 , … , K } ∣ c q − y ∣ > r |c_q-y|>r ∣ c q − y ∣ > r
(i) The line L ( y , θ ( ω ) ) L(y,\theta(\omega)) L ( y , θ ( ω )) ∂ B q \partial B_q ∂ B q ∣ ρ q ( y , ω ) ∣ ≤ r |\rho_q(y,\omega)|\le r ∣ ρ q ( y , ω ) ∣ ≤ r
(ii) The line L ( y , θ ( ω ) ) L(y,\theta(\omega)) L ( y , θ ( ω )) ∂ B q \partial B_q ∂ B q ∣ ρ q ( y , ω ) ∣ = r |\rho_q(y,\omega)|=r ∣ ρ q ( y , ω ) ∣ = r
(iii) The backward ray R ( y , ω ) \mathcal{R}(y,\omega) R ( y , ω ) B q B_q B q
∣ ρ q ( y , ω ) ∣ < r and σ q ( y , ω ) > 0. (9) |\rho_q(y,\omega)|<r\qquad\text{and}\qquad\sigma_q(y,\omega)>0. \tag*{(9)} ∣ ρ q ( y , ω ) ∣ < r and σ q ( y , ω ) > 0. (9) Under (9) , let d q ( y , ω ) d_q(y,\omega) d q ( y , ω ) s ≥ 0 s\ge0 s ≥ 0 y − s θ ( ω ) ∈ ∂ B q y-s\theta(\omega)\in\partial B_q y − s θ ( ω ) ∈ ∂ B q ℓ q ( y , ω ) \ell_q(y,\omega) ℓ q ( y , ω ) R ( y , ω ) ∩ B q \mathcal{R}(y,\omega)\cap B_q R ( y , ω ) ∩ B q
d q ( y , ω ) = σ q ( y , ω ) − r 2 − ρ q ( y , ω ) 2 , ℓ q ( y , ω ) = 2 r 2 − ρ q ( y , ω ) 2 . (10) d_q(y,\omega)=\sigma_q(y,\omega)-\sqrt{r^2-\rho_q(y,\omega)^2},\qquad\ell_q(y,\omega)=2\sqrt{r^2-\rho_q(y,\omega)^2}. \tag*{(10)} d q ( y , ω ) = σ q ( y , ω ) − r 2 − ρ q ( y , ω ) 2 , ℓ q ( y , ω ) = 2 r 2 − ρ q ( y , ω ) 2 . (10) Proof. Write ρ q = ρ q ( y , ω ) \rho_q=\rho_q(y,\omega) ρ q = ρ q ( y , ω ) σ q = σ q ( y , ω ) \sigma_q=\sigma_q(y,\omega) σ q = σ q ( y , ω ) s ∈ R s\in\mathbb{R} s ∈ R
∣ y − s θ ( ω ) − c q ∣ 2 = ( s − σ q ) 2 + ρ q 2 . \lvert y-s\theta(\omega)-c_q\rvert^2=(s-\sigma_q)^2+\rho_q^2. ∣ y − s θ ( ω ) − c q ∣ 2 = ( s − σ q ) 2 + ρ q 2 . (i) The line L ( y , θ ( ω ) ) L(y,\theta(\omega)) L ( y , θ ( ω )) ∂ B q \partial B_q ∂ B q ( s − σ q ) 2 = r 2 − ρ q 2 (s-\sigma_q)^2=r^2-\rho_q^2 ( s − σ q ) 2 = r 2 − ρ q 2 ∣ ρ q ∣ ≤ r \lvert\rho_q\rvert\le r ∣ ρ q ∣ ≤ r
(ii) When ∣ ρ q ∣ ≤ r \lvert\rho_q\rvert\le r ∣ ρ q ∣ ≤ r
s ± = σ q ± r 2 − ρ q 2 . s_{\pm}=\sigma_q\pm\sqrt{r^2-\rho_q^2}. s ± = σ q ± r 2 − ρ q 2 . The parameters s − s_- s − s + s_+ s + ∣ ρ q ∣ = r \lvert\rho_q\rvert=r ∣ ρ q ∣ = r L ( y , θ ( ω ) ) L(y,\theta(\omega)) L ( y , θ ( ω )) ∂ B q \partial B_q ∂ B q ∣ ρ q ∣ = r \lvert\rho_q\rvert=r ∣ ρ q ∣ = r
(iii) Suppose that ∣ ρ q ∣ < r \lvert\rho_q\rvert<r ∣ ρ q ∣ < r ∣ c q − y ∣ > r \lvert c_q-y\rvert>r ∣ c q − y ∣ > r
σ q 2 = ∣ c q − y ∣ 2 − ρ q 2 > r 2 − ρ q 2 > 0. \sigma_q^2=\lvert c_q-y\rvert^2-\rho_q^2>r^2-\rho_q^2>0. σ q 2 = ∣ c q − y ∣ 2 − ρ q 2 > r 2 − ρ q 2 > 0. Thus, s − s_- s − s + s_+ s + σ q \sigma_q σ q y − s θ ( ω ) y-s\theta(\omega) y − s θ ( ω ) B q B_q B q s − < s < s + s_-<s<s_+ s − < s < s + R ( y , ω ) \mathcal{R}(y,\omega) R ( y , ω ) s ≥ 0 s\ge0 s ≥ 0 (9) . Under this condition, the entry parameter is d q = s − d_q=s_- d q = s − s + s_+ s + ∣ θ ( ω ) ∣ = 1 \lvert\theta(\omega)\rvert=1 ∣ θ ( ω )∣ = 1 ℓ q = s + − s − \ell_q=s_+-s_- ℓ q = s + − s − s ± s_{\pm} s ± (10) . □
The chord geometry is illustrated in Figure 4 .
Figure 4. Geometry of the intersection of the backward ray R ( y , ω ) \mathcal{R}(y,\omega) R ( y , ω ) B q B_q B q B q B_q B q s − s_- s − s + s_+ s + y y y ∂ B q \partial B_q ∂ B q
Remark 2.9. Fix q ∈ { 1 , … , K } q\in\{1,\ldots,K\} q ∈ { 1 , … , K } 2 π 2\pi 2 π ρ q \rho_q ρ q σ q \sigma_q σ q (10) define real-analytic functions d q d_q d q ℓ q \ell_q ℓ q
{ ( y , ω ) ∈ R 2 × R | ∣ c q − y ∣ > r , ∣ ρ q ( y , ω ) ∣ < r } . (11) \left\{(y,\omega)\in\mathbb{R}^2\times\mathbb{R}\ \middle|\ \lvert c_q-y\rvert>r,\ \lvert\rho_q(y,\omega)\rvert<r\right\}. \tag*{(11)} { ( y , ω ) ∈ R 2 × R ∣ c q − y ∣ > r , ∣ ρ q ( y , ω )∣ < r } . (11) Indeed, ρ q \rho_q ρ q σ q \sigma_q σ q ( y , ω ) (y,\omega) ( y , ω ) (11) , the orthogonal decomposition of c q − y c_q-y c q − y
σ q ( y , ω ) 2 = ∣ c q − y ∣ 2 − ρ q ( y , ω ) 2 > r 2 − ρ q ( y , ω ) 2 > 0. \sigma_q(y,\omega)^2=\lvert c_q-y\rvert^2-\rho_q(y,\omega)^2>r^2-\rho_q(y,\omega)^2>0. σ q ( y , ω ) 2 = ∣ c q − y ∣ 2 − ρ q ( y , ω ) 2 > r 2 − ρ q ( y , ω ) 2 > 0. Hence, σ q ( y , ω ) ≠ 0 \sigma_q(y,\omega)\ne0 σ q ( y , ω ) = 0 (11) .
For y ∈ R 2 y\in\mathbb{R}^2 y ∈ R 2 ω ∈ [ 0 , 2 π ) \omega\in[0,2\pi) ω ∈ [ 0 , 2 π ) y y y R ( y , ω ) \mathcal{R}(y,\omega) R ( y , ω )
C ( y , ω ) : = { q ∈ { 1 , … , K } : ∣ c q − y ∣ > r , ∣ ρ q ( y , ω ) ∣ < r , σ q ( y , ω ) > 0 } . C(y,\omega):=\left\{q\in\{1,\ldots,K\}:\lvert c_q-y\rvert>r,\ \lvert\rho_q(y,\omega)\rvert<r,\ \sigma_q(y,\omega)>0\right\}. C ( y , ω ) := { q ∈ { 1 , … , K } : ∣ c q − y ∣ > r , ∣ ρ q ( y , ω )∣ < r , σ q ( y , ω ) > 0 } . If C ( y , ω ) C(y,\omega) C ( y , ω ) C ( y , ω ) = { q 1 , … , q N } C(y,\omega)=\{q_1,\ldots,q_N\} C ( y , ω ) = { q 1 , … , q N } R ( y , ω ) \mathcal{R}(y,\omega) R ( y , ω )
0 < d q 1 ( y , ω ) < ⋯ < d q N ( y , ω ) . (12) 0<d_{q_1}(y,\omega)<\cdots<d_{q_N}(y,\omega). \tag*{(12)} 0 < d q 1 ( y , ω ) < ⋯ < d q N ( y , ω ) . (12) The inequalities are strict because the rods are pairwise disjoint.
If y y y B m B_m B m q ≠ m q\ne m q = m
∣ c q − y ∣ ≥ ∣ c q − c m ∣ − r > r . \lvert c_q-y\rvert\ge\lvert c_q-c_m\rvert-r>r. ∣ c q − y ∣ ≥ ∣ c q − c m ∣ − r > r . Hence, Lemma 2.8 applies to every q ≠ m q\ne m q = m B m B_m B m
Parameter space
Definition 2.10. Fix a known moderator attenuation a 0 > 0 a_0 > 0 a 0 > 0 G \mathcal{G} G (5) , and let 0 < a min ≤ a max 0 < a_{\min} \le a_{\max} 0 < a m i n ≤ a m a x 0 ≤ f min ≤ f max 0 \le f_{\min} \le f_{\max} 0 ≤ f m i n ≤ f m a x parameter space X = X ( a min , a max , f min , f max , a 0 , G ) \mathcal{X} = \mathcal{X}(a_{\min}, a_{\max}, f_{\min}, f_{\max}, a_0, \mathcal{G}) X = X ( a m i n , a m a x , f m i n , f m a x , a 0 , G ) ( a , f ) ∈ L ∞ ( Ω ) × L ∞ ( Ω ) (a, f) \in L^\infty(\Omega) \times L^\infty(\Omega) ( a , f ) ∈ L ∞ ( Ω ) × L ∞ ( Ω )
a = a 0 χ Ω 0 + ∑ j = 1 K a j χ B j , f = ∑ j = 1 K f j χ B j , (13) a = a_0\chi_{\Omega_0} + \sum_{j=1}^{K} a_j\chi_{B_j}, \qquad f = \sum_{j=1}^{K} f_j\chi_{B_j}, \tag*{(13)} a = a 0 χ Ω 0 + j = 1 ∑ K a j χ B j , f = j = 1 ∑ K f j χ B j , (13) with
a min ≤ a j ≤ a max , f min ≤ f j ≤ f max , j = 1 , … , K . a_{\min} \le a_j \le a_{\max}, \qquad f_{\min} \le f_j \le f_{\max}, \qquad j = 1,\ldots,K. a m i n ≤ a j ≤ a m a x , f m i n ≤ f j ≤ f m a x , j = 1 , … , K . In the whole-space formulas (1) –(4) , we extend both a a a f f f Ω \Omega Ω a 0 a_0 a 0
Remark 2.11. For fixed G \mathcal{G} G (13) is unique up to sets of measure zero. The coefficient map ( a , f ) ⟼ ( a 1 , … , a K , f 1 , … , f K ) (a,f) \longmapsto(a_1,\ldots,a_K,f_1,\ldots,f_K) ( a , f ) ⟼ ( a 1 , … , a K , f 1 , … , f K ) X \mathcal{X} X [ a min , a max ] K × [ f min , f max ] K ⊂ R 2 K [a_{\min},a_{\max}]^K \times[f_{\min},f_{\max}]^K \subset\mathbb{R}^{2K} [ a m i n , a m a x ] K × [ f m i n , f m a x ] K ⊂ R 2 K X \mathcal{X} X a , f a,f a , f X \mathcal{X} X a 0 a_0 a 0
∥ a − a ~ ∥ ℓ ∞ = ∥ a − a ~ ∥ L ∞ ( Ω ) , ∥ f − f ~ ∥ ℓ ∞ = ∥ f − f ~ ∥ L ∞ ( Ω ) . \|a-\widetilde{a}\|_{\ell^\infty} = \|a-\widetilde{a}\|_{L^\infty(\Omega)}, \qquad\|f-\widetilde{f}\|_{\ell^\infty} = \|f-\widetilde{f}\|_{L^\infty(\Omega)}. ∥ a − a ∥ ℓ ∞ = ∥ a − a ∥ L ∞ ( Ω ) , ∥ f − f ∥ ℓ ∞ = ∥ f − f ∥ L ∞ ( Ω ) . We equip X \mathcal{X} X d X ( ( a , f ) , ( a ~ , f ~ ) ) : = ∥ a − a ~ ∥ ℓ ∞ + ∥ f − f ~ ∥ ℓ ∞ d_{\mathcal{X}}\big((a,f),(\widetilde{a},\widetilde{f})\big) := \|a-\widetilde{a}\|_{\ell^\infty} + \|f-\widetilde{f}\|_{\ell^\infty} d X ( ( a , f ) , ( a , f ) ) := ∥ a − a ∥ ℓ ∞ + ∥ f − f ∥ ℓ ∞ L ∞ ( Ω ) × L ∞ ( Ω ) L^\infty(\Omega) \times L^\infty(\Omega) L ∞ ( Ω ) × L ∞ ( Ω ) X \mathcal{X} X
The solution of the transport equation and its tangential trace
Since a a a f f f (4) reduces to a finite sum over the chords described in Subsection 2.4. Fix a rod B m B_m B m τ = ( ψ , ω 0 ) ∈ T m \tau= (\psi,\omega_0) \in T_m τ = ( ψ , ω 0 ) ∈ T m y τ y_\tau y τ θ τ \theta_\tau θ τ y τ y_\tau y τ ω \omega ω ( a , f ) ∈ X (a,f) \in\mathcal{X} ( a , f ) ∈ X q ∈ C ( y τ , ω ) q \in C(y_\tau,\omega) q ∈ C ( y τ , ω )
β q , m ( ω ) : = ∫ 0 d q ( y τ , ω ) a ( y τ − ξ θ ( ω ) ) d ξ , W q , m ( ω ) : = f q 1 − e − a q ℓ q ( y τ , ω ) a q . \beta_{q,m}(\omega) := \int_{0}^{d_q(y_\tau,\omega)} a\big(y_\tau-\xi\theta(\omega)\big)\,\mathrm{d}\xi, \qquad W_{q,m}(\omega) := f_q\frac{1-e^{-a_q\ell_q(y_\tau,\omega)}}{a_q}. β q , m ( ω ) := ∫ 0 d q ( y τ , ω ) a ( y τ − ξ θ ( ω ) ) d ξ , W q , m ( ω ) := f q a q 1 − e − a q ℓ q ( y τ , ω ) . The quantity β q , m ( ω ) \beta_{q,m}(\omega) β q , m ( ω ) y τ y_\tau y τ y τ − d q ( y τ , ω ) θ ( ω ) y_\tau-d_q(y_\tau,\omega)\theta(\omega) y τ − d q ( y τ , ω ) θ ( ω ) W q , m ( ω ) W_{q,m}(\omega) W q , m ( ω ) B q B_q B q B q B_q B q β q , m \beta_{q,m} β q , m W q , m W_{q,m} W q , m ( a , f ) (a,f) ( a , f ) τ \tau τ β q , m \beta_{q,m} β q , m a a a d q ( y τ , ω ) d_q(y_\tau,\omega) d q ( y τ , ω ) W q , m W_{q,m} W q , m a q ≥ a min > 0 a_q \ge a_{\min} > 0 a q ≥ a m i n > 0
Since y τ ∈ ∂ B m y_\tau\in\partial B_m y τ ∈ ∂ B m R ( y τ , ω ) \mathcal{R}(y_\tau,\omega) R ( y τ , ω ) B m B_m B m σ m ( y τ , ω ) > 0 \sigma_m(y_\tau,\omega) > 0 σ m ( y τ , ω ) > 0 R ( y τ , ω ) ∩ B m \mathcal{R}(y_\tau,\omega) \cap B_m R ( y τ , ω ) ∩ B m 2 σ m ( y τ , ω ) 2\sigma_m(y_\tau,\omega) 2 σ m ( y τ , ω )
ℓ m + ( ω ) : = 2 max { σ m ( y τ , ω ) , 0 } . \ell_m^+(\omega) := 2\max\{\sigma_m(y_\tau,\omega),0\}. ℓ m + ( ω ) := 2 max { σ m ( y τ , ω ) , 0 } . Lemma 2.12. For every ( a , f ) ∈ X (a,f) \in\mathcal{X} ( a , f ) ∈ X ω ∈ [ 0 , 2 π ) \omega\in[0,2\pi) ω ∈ [ 0 , 2 π ) u u u (2) , given by (4) , satisfies
u ( y τ , θ ( ω ) ) = f m 1 − e − a m ℓ m + ( ω ) a m + ∑ q ∈ C ( y τ , ω ) W q , m ( ω ) e − β q , m ( ω ) . (14) u(y_\tau,\theta(\omega)) = f_m\frac{1-e^{-a_m\ell_m^+(\omega)}}{a_m} + \sum_{q\in C(y_\tau,\omega)} W_{q,m}(\omega)e^{-\beta_{q,m}(\omega)}. \tag*{(14)} u ( y τ , θ ( ω )) = f m a m 1 − e − a m ℓ m + ( ω ) + q ∈ C ( y τ , ω ) ∑ W q , m ( ω ) e − β q , m ( ω ) . (14) :::`
Proof. By (4) , we have
u ( y τ , θ ( ω ) ) = ∫ 0 ∞ f ( y τ − s θ ( ω ) ) exp ( − ∫ 0 s a ( y τ − ξ θ ( ω ) ) d ξ ) d s . u(y_\tau,\theta(\omega))=\int_{0}^{\infty}f(y_\tau-s\theta(\omega))\exp\left(-\int_{0}^{s}a(y_\tau-\xi\theta(\omega))\,\mathrm{d}\xi\right)\,\mathrm{d}s. u ( y τ , θ ( ω )) = ∫ 0 ∞ f ( y τ − s θ ( ω )) exp ( − ∫ 0 s a ( y τ − ξ θ ( ω )) d ξ ) d s . The source vanishes outside the rods. Hence only the portions of R ( y τ , ω ) \mathcal{R}(y_\tau,\omega) R ( y τ , ω ) B m B_m B m
∫ 0 ℓ m + ( ω ) f m e − a m s d s = f m 1 − e − a m ℓ m + ( ω ) a m . \int_{0}^{\ell_m^+(\omega)}f_m e^{-a_m s}\,\mathrm{d}s=f_m\frac{1-e^{-a_m\ell_m^+(\omega)}}{a_m}. ∫ 0 ℓ m + ( ω ) f m e − a m s d s = f m a m 1 − e − a m ℓ m + ( ω ) . This expression is zero when the backward ray does not enter B m B_m B m
For q ∈ C ( y τ , ω ) q\in\mathcal{C}(y_\tau,\omega) q ∈ C ( y τ , ω ) B q B_q B q
[ d q ( y τ , ω ) , d q ( y τ , ω ) + ℓ q ( y τ , ω ) ] . [d_q(y_\tau,\omega),d_q(y_\tau,\omega)+\ell_q(y_\tau,\omega)]. [ d q ( y τ , ω ) , d q ( y τ , ω ) + ℓ q ( y τ , ω )] . Set s = d q ( y τ , ω ) + κ s=d_q(y_\tau,\omega)+\kappa s = d q ( y τ , ω ) + κ a = a q a=a_q a = a q B q B_q B q
∫ d q ( y τ , ω ) d q ( y τ , ω ) + ℓ q ( y τ , ω ) f q exp ( − ∫ 0 s a ( y τ − ξ θ ( ω ) ) d ξ ) d s = f q e − β q , m ( ω ) ∫ 0 ℓ q ( y τ , ω ) e − a q κ d κ = W q , m ( ω ) e − β q , m ( ω ) . \begin{aligned}
&\int_{d_q(y_\tau,\omega)}^{d_q(y_\tau,\omega)+\ell_q(y_\tau,\omega)}
f_q\exp\left(-\int_{0}^{s}a(y_\tau-\xi\theta(\omega))\,\mathrm{d}\xi\right)\,\mathrm{d}s\\
&=f_qe^{-\beta_{q,m}(\omega)}\int_{0}^{\ell_q(y_\tau,\omega)}e^{-a_q\kappa}\,\mathrm{d}\kappa
=W_{q,m}(\omega)e^{-\beta_{q,m}(\omega)}.
\end{aligned} ∫ d q ( y τ , ω ) d q ( y τ , ω ) + ℓ q ( y τ , ω ) f q exp ( − ∫ 0 s a ( y τ − ξ θ ( ω )) d ξ ) d s = f q e − β q , m ( ω ) ∫ 0 ℓ q ( y τ , ω ) e − a q κ d κ = W q , m ( ω ) e − β q , m ( ω ) . The rod intervals are pairwise disjoint. Summing their contributions proves (14) .
Definition 2.13 (Tangential trace). Let ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X τ = ( ψ , ω 0 ) ∈ T m \tau=(\psi,\omega_0)\in\mathcal{T}_m τ = ( ψ , ω 0 ) ∈ T m u u u (2) given by (4) . We define the tangential trace of u u u τ \tau τ
u tan [ a , f ] ( y τ , θ τ ) : = u ( y τ , θ τ ) . u_{\tan}[a,f](y_\tau,\theta_\tau):=u(y_\tau,\theta_\tau). u t a n [ a , f ] ( y τ , θ τ ) := u ( y τ , θ τ ) . Corollary 2.14. Let τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in\mathcal{S}_m τ = ( ψ , ω 0 ) ∈ S m ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X U ∋ ω 0 U\ni\omega_0 U ∋ ω 0
For every ω ∈ U \omega\in U ω ∈ U C ( y τ , ω ) = C ( y τ , ω 0 ) \mathcal{C}(y_\tau,\omega)=\mathcal{C}(y_\tau,\omega_0) C ( y τ , ω ) = C ( y τ , ω 0 ) (12) of the indices in C ( y τ , ω 0 ) \mathcal{C}(y_\tau,\omega_0) C ( y τ , ω 0 ) U U U
For every q ∈ C ( y τ , ω 0 ) q\in\mathcal{C}(y_\tau,\omega_0) q ∈ C ( y τ , ω 0 ) d q ( y τ , ⋅ ) d_q(y_\tau,\cdot) d q ( y τ , ⋅ ) ℓ q ( y τ , ⋅ ) \ell_q(y_\tau,\cdot) ℓ q ( y τ , ⋅ ) β q , m \beta_{q,m} β q , m W q , m W_{q,m} W q , m U U U ℓ m + \ell_m^+ ℓ m + U U U ℓ m + ( ω 0 ) = 0 \ell_m^+(\omega_0)=0 ℓ m + ( ω 0 ) = 0
The function ω ↦ u ( y τ , θ ( ω ) ) \omega\mapsto u(y_\tau,\theta(\omega)) ω ↦ u ( y τ , θ ( ω )) U U U
u tan [ a , f ] ( y τ , θ τ ) = ∑ q ∈ C ( y τ , ω 0 ) W q , m ( ω 0 ) e − β q , m ( ω 0 ) . (15) u_{\tan}[a,f](y_\tau,\theta_\tau)=\sum_{q\in\mathcal{C}(y_\tau,\omega_0)}W_{q,m}(\omega_0)e^{-\beta_{q,m}(\omega_0)}. \tag*{(15)} u t a n [ a , f ] ( y τ , θ τ ) = q ∈ C ( y τ , ω 0 ) ∑ W q , m ( ω 0 ) e − β q , m ( ω 0 ) . (15) Proof. (i) Fix q ≠ m q\ne m q = m τ \tau τ ∣ ρ q ( y τ , ω 0 ) ∣ = ∣ n q ( ψ , ω 0 ) ∣ ≠ r |\rho_q(y_\tau,\omega_0)|=|n_q(\psi,\omega_0)|\ne r ∣ ρ q ( y τ , ω 0 ) ∣ = ∣ n q ( ψ , ω 0 ) ∣ = r ρ q ( y τ , ω ) 2 − r 2 \rho_q(y_\tau,\omega)^2-r^2 ρ q ( y τ , ω ) 2 − r 2 ω 0 \omega_0 ω 0 ∣ ρ q ( y τ , ω 0 ) ∣ < r |\rho_q(y_\tau,\omega_0)|<r ∣ ρ q ( y τ , ω 0 ) ∣ < r σ q ( y τ , ω 0 ) ≠ 0 \sigma_q(y_\tau,\omega_0)\ne0 σ q ( y τ , ω 0 ) = 0 σ q ( y τ , ω ) \sigma_q(y_\tau,\omega) σ q ( y τ , ω ) ω 0 \omega_0 ω 0 (9) therefore shows that membership of q q q C ( y τ , ω ) \mathcal{C}(y_\tau,\omega) C ( y τ , ω ) m m m ∣ c m − y τ ∣ = r |c_m-y_\tau|=r ∣ c m − y τ ∣ = r
Since there are finitely many rods, we can choose a common open interval U ∋ ω 0 U\ni\omega_0 U ∋ ω 0 C ( y τ , ω ) = C ( y τ , ω 0 ) \mathcal{C}(y_\tau,\omega)=\mathcal{C}(y_\tau,\omega_0) C ( y τ , ω ) = C ( y τ , ω 0 ) d q ( y τ , ⋅ ) d_q(y_\tau,\cdot) d q ( y τ , ⋅ ) q ∈ C ( y τ , ω 0 ) q\in\mathcal{C}(y_\tau,\omega_0) q ∈ C ( y τ , ω 0 ) U U U U U U (12) remain valid.
(ii) Remark 2.9 also gives continuity of ℓ q ( y τ , ⋅ ) \ell_q(y_\tau,\cdot) ℓ q ( y τ , ⋅ ) q ∈ C ( y τ , ω 0 ) q\in\mathcal{C}(y_\tau,\omega_0) q ∈ C ( y τ , ω 0 ) W q , m W_{q,m} W q , m W q , m W_{q,m} W q , m U U U ℓ m + ( ω ) = \ell_m^+(\omega)= ℓ m + ( ω ) = 2 max { σ m ( y τ , ω ) , 0 } 2\max\{\sigma_m(y_\tau,\omega),0\} 2 max { σ m ( y τ , ω ) , 0 } ℓ m + \ell_m^+ ℓ m + σ m ( y τ , ω 0 ) = 0 \sigma_m(y_\tau,\omega_0)=0 σ m ( y τ , ω 0 ) = 0 ℓ m + ( ω 0 ) = 0 \ell_m^+(\omega_0)=0 ℓ m + ( ω 0 ) = 0
If C ( y τ , ω 0 ) C(y_\tau,\omega_0) C ( y τ , ω 0 ) q 1 , … , q N q_1,\ldots,q_N q 1 , … , q N (12) . For ω ∈ U \omega\in U ω ∈ U 1 ≤ k ≤ N 1\le k\le N 1 ≤ k ≤ N y τ y_\tau y τ y τ − d q k ( y τ , ω ) θ ( ω ) y_\tau-d_{q_k}(y_\tau,\omega)\theta(\omega) y τ − d q k ( y τ , ω ) θ ( ω ) Ω \Omega Ω B m B_m B m ℓ m + ( ω ) \ell_m^+(\omega) ℓ m + ( ω ) j < k j<k j < k B q j B_{q_j} B q j ℓ q j ( y τ , ω ) \ell_{q_j}(y_\tau,\omega) ℓ q j ( y τ , ω )
Splitting the attenuation integral into moderator and rod contributions, we obtain
β q k , m ( ω ) = a 0 d q k ( y τ , ω ) + ( a m − a 0 ) ℓ m + ( ω ) + ∑ j < k ( a q j − a 0 ) ℓ q j ( y τ , ω ) , ω ∈ U . \beta_{q_k,m}(\omega)=a_0d_{q_k}(y_\tau,\omega)+(a_m-a_0)\ell_m^+(\omega)+\sum_{j<k}(a_{q_j}-a_0)\ell_{q_j}(y_\tau,\omega),\qquad\omega\in U. β q k , m ( ω ) = a 0 d q k ( y τ , ω ) + ( a m − a 0 ) ℓ m + ( ω ) + j < k ∑ ( a q j − a 0 ) ℓ q j ( y τ , ω ) , ω ∈ U . The continuity of the terms on the right-hand side gives continuity of β q k , m \beta_{q_k,m} β q k , m U U U
(iii) By parts (i) and (ii), every term in (14) is continuous on U U U ω ↦ u ( y τ , θ ( ω ) ) \omega\mapsto u(y_\tau,\theta(\omega)) ω ↦ u ( y τ , θ ( ω )) U U U (14) at ω = ω 0 \omega=\omega_0 ω = ω 0 ℓ m + ( ω 0 ) = 0 \ell_m^+(\omega_0)=0 ℓ m + ( ω 0 ) = 0 (15) . □
Remark 2.15. For every ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X y ∈ R 2 y\in\mathbb{R}^2 y ∈ R 2 θ ∈ S 1 \theta\in\mathbb{S}^1 θ ∈ S 1 (4) gives
0 ≤ u ( y , θ ) ≤ f max a min . 0\le u(y,\theta)\le\frac{f_{\max}}{a_{\min}}. 0 ≤ u ( y , θ ) ≤ a m i n f m a x . To see this, put A ( s ) = ∫ 0 s a ( y − σ θ ) d σ A(s)=\int_0^s a(y-\sigma\theta)\,\mathrm{d}\sigma A ( s ) = ∫ 0 s a ( y − σ θ ) d σ A ( 0 ) = 0 A(0)=0 A ( 0 ) = 0 A ′ ( s ) = a ( y − s θ ) A'(s)=a(y-s\theta) A ′ ( s ) = a ( y − s θ ) T > 0 T>0 T > 0
∫ 0 T A ′ ( s ) e − A ( s ) d s = 1 − e − A ( T ) ≤ 1. \int_0^T A'(s)e^{-A(s)}\,\mathrm{d}s=1-e^{-A(T)}\le1. ∫ 0 T A ′ ( s ) e − A ( s ) d s = 1 − e − A ( T ) ≤ 1. Since the source is supported in the rods, f ( y − s θ ) ≤ ( f max / a min ) a ( y − s θ ) f(y-s\theta)\le(f_{\max}/a_{\min})a(y-s\theta) f ( y − s θ ) ≤ ( f m a x / a m i n ) a ( y − s θ )
u ( y , θ ) ≤ f max a min ∫ 0 ∞ A ′ ( s ) e − A ( s ) d s ≤ f max a min . u(y,\theta)\le\frac{f_{\max}}{a_{\min}}\int_0^\infty A'(s)e^{-A(s)}\,\mathrm{d}s\le\frac{f_{\max}}{a_{\min}}. u ( y , θ ) ≤ a m i n f m a x ∫ 0 ∞ A ′ ( s ) e − A ( s ) d s ≤ a m i n f m a x . The same bound holds for the tangential trace defined in Definition 2.13 .
Data space
Definition 2.16. The data associated with ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X
R a f ( ψ , ω ) : = R a f ( x ( ψ ) , θ ( ω ) ) , ( ψ , ω ) ∈ M , R_{af}(\psi,\omega):=R_{af}(x(\psi),\theta(\omega)),\qquad(\psi,\omega)\in\mathcal{M}, R a f ( ψ , ω ) := R a f ( x ( ψ ) , θ ( ω )) , ( ψ , ω ) ∈ M , where M ⊂ [ 0 , 2 π ) 2 \mathcal{M}\subset[0,2\pi)^2 M ⊂ [ 0 , 2 π ) 2
For every selected simple tangency τ = ( ψ , ω 0 ) \tau=(\psi,\omega_0) τ = ( ψ , ω 0 )
{ ψ } × ( ω 0 − δ τ , ω 0 + δ τ ) ⊂ M . (16) \{\psi\}\times(\omega_0-\delta_\tau,\omega_0+\delta_\tau)\subset\mathcal{M}. \tag*{(16)} { ψ } × ( ω 0 − δ τ , ω 0 + δ τ ) ⊂ M . (16) Angular intervals in (16) are understood modulo 2 π 2\pi 2 π Assumption 4.2 .
The singular functionals
Fix a simple tangency τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in S_m τ = ( ψ , ω 0 ) ∈ S m
I τ ∘ : = ( ω 0 − δ τ , ω 0 + δ τ ) ∖ { ω 0 } . I_\tau^\circ:=(\omega_0-\delta_\tau,\omega_0+\delta_\tau)\setminus\{\omega_0\}. I τ ∘ := ( ω 0 − δ τ , ω 0 + δ τ ) ∖ { ω 0 } . The boundary point x τ x_\tau x τ ω \omega ω ω → ω 0 ϵ τ \omega\to\omega_0^{\epsilon_\tau} ω → ω 0 ϵ τ ϵ τ ( ω − ω 0 ) > 0 \epsilon_\tau(\omega-\omega_0)>0 ϵ τ ( ω − ω 0 ) > 0 2 ω 0 − ω 2\omega_0-\omega 2 ω 0 − ω L ( ψ , 2 ω 0 − ω ) L(\psi,2\omega_0-\omega) L ( ψ , 2 ω 0 − ω ) B m B_m B m L ( ψ , 2 ω 0 − ω ) L(\psi,2\omega_0-\omega) L ( ψ , 2 ω 0 − ω )
Definition 2.17. Let V τ V_\tau V τ H ∈ C 2 ( I τ ) H \in C^2(I_\tau) H ∈ C 2 ( I τ )
Φ D [ H ] : = ϵ τ lim ω → ω 0 ϵ τ ∣ ω − ω 0 ∣ 1 / 2 H ′ ( ω ) (17) \Phi_D[H] := \epsilon_\tau\lim_{\omega\to\omega_0^{\epsilon_\tau}} |\omega-\omega_0|^{1/2}H'(\omega) \tag*{(17)} Φ D [ H ] := ϵ τ ω → ω 0 ϵ τ lim ∣ ω − ω 0 ∣ 1/2 H ′ ( ω ) (17) Φ N [ H ] : = − lim ω → ω 0 ϵ τ [ ∣ ω − ω 0 ∣ H ′ ′ ( ω ) + ϵ τ 2 ( H ′ ( ω ) − H ′ ( 2 ω 0 − ω ) ) ] (18) \Phi_N[H] := -\lim_{\omega\to\omega_0^{\epsilon_\tau}}\left[|\omega-\omega_0|H''(\omega)+\frac{\epsilon_\tau}{2}\left(H'(\omega)-H'(2\omega_0-\omega)\right)\right] \tag*{(18)} Φ N [ H ] := − ω → ω 0 ϵ τ lim [ ∣ ω − ω 0 ∣ H ′′ ( ω ) + 2 ϵ τ ( H ′ ( ω ) − H ′ ( 2 ω 0 − ω ) ) ] (18) exist and are finite. We define
∥ H ∥ τ : = ∣ Φ D [ H ] ∣ + ∣ Φ N [ H ] ∣ (19) \|H\|_\tau:= |\Phi_D[H]|+|\Phi_N[H]| \tag*{(19)} ∥ H ∥ τ := ∣ Φ D [ H ] ∣ + ∣ Φ N [ H ] ∣ (19) for any H ∈ V τ H \in V_\tau H ∈ V τ
Restricting H H H Φ D [ H ] \Phi_D[H] Φ D [ H ] Φ N [ H ] \Phi_N[H] Φ N [ H ] ∥ H ∥ τ \|H\|_\tau ∥ H ∥ τ H ′ ( 2 ω 0 − ω ) H'(2\omega_0-\omega) H ′ ( 2 ω 0 − ω ) H ′ H' H ′ 2 ω 0 − ω 2\omega_0-\omega 2 ω 0 − ω Φ D \Phi_D Φ D Φ N \Phi_N Φ N τ = ( ψ , ω 0 ) \tau=(\psi,\omega_0) τ = ( ψ , ω 0 ) Φ D [ H ] ≠ 0 \Phi_D[H]\ne0 Φ D [ H ] = 0 a m − a 0 a_m-a_0 a m − a 0 Φ D [ H ] \Phi_D[H] Φ D [ H ] Φ N [ H ] \Phi_N[H] Φ N [ H ]
For ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in S_m τ = ( ψ , ω 0 ) ∈ S m
F m ( a , f ) ( τ ) : = f m − ( a m − a 0 ) u tan [ a , f ] ( y τ , θ τ ) . F_m^{(a,f)}(\tau):=f_m-(a_m-a_0)u_{\tan}[a,f](y_\tau,\theta_\tau). F m ( a , f ) ( τ ) := f m − ( a m − a 0 ) u t a n [ a , f ] ( y τ , θ τ ) . We call F m ( a , f ) ( τ ) F_m^{(a,f)}(\tau) F m ( a , f ) ( τ ) non-cancellation coefficient of ( a , f ) (a,f) ( a , f ) τ \tau τ
The following proposition specializes the local singularity calculations of Holman and Richardson [12 ] to the circular interfaces of our model. Fix a simple tangency τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in S_m τ = ( ψ , ω 0 ) ∈ S m ∂ B m \partial B_m ∂ B m 1 / r 1/r 1/ r ω \omega ω L ( x τ , θ ( ω ) ) L(x_\tau,\theta(\omega)) L ( x τ , θ ( ω )) x τ x_\tau x τ ∣ x τ − y τ ∣ = L τ |x_\tau-y_\tau|=L_\tau ∣ x τ − y τ ∣ = L τ y τ y_\tau y τ f m f_m f m y τ y_\tau y τ a m − a 0 a_m-a_0 a m − a 0 u ( y τ , θ τ ) u(y_\tau,\theta_\tau) u ( y τ , θ τ )
The change of angular coordinate η = ϵ τ ( ω − ω 0 ) \eta=\epsilon_\tau(\omega-\omega_0) η = ϵ τ ( ω − ω 0 ) η > 0 \eta>0 η > 0
Proposition 2.19. Let ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in S_m τ = ( ψ , ω 0 ) ∈ S m H ( ω ) : = R a f ( x τ , θ ( ω ) ) H(\omega):=R_af(x_\tau,\theta(\omega)) H ( ω ) := R a f ( x τ , θ ( ω )) 0 < ∣ ω − ω 0 ∣ < δ τ 0<|\omega-\omega_0|<\delta_\tau 0 < ∣ ω − ω 0 ∣ < δ τ H ∈ V τ H\in V_\tau H ∈ V τ Φ D \Phi_D Φ D Φ N \Phi_N Φ N
Φ D [ H ] = 2 r L τ F m ( a , f ) ( τ ) e − D a ( y τ , θ τ ) (20) \Phi_D[H]=\sqrt{2rL_\tau}\,F_m^{(a,f)}(\tau)e^{-Da(y_\tau,\theta_\tau)} \tag*{(20)} Φ D [ H ] = 2 r L τ F m ( a , f ) ( τ ) e − D a ( y τ , θ τ ) (20) Φ N [ H ] = ( a m − a 0 ) 2 r L τ Φ D [ H ] . (21) \Phi_N[H]=(a_m-a_0)\sqrt{2rL_\tau}\,\Phi_D[H]. \tag*{(21)} Φ N [ H ] = ( a m − a 0 ) 2 r L τ Φ D [ H ] . (21) On the nonintersecting side, we have
lim ω → ω 0 − ϵ τ ∣ ω − ω 0 ∣ 1 / 2 H ′ ( ω ) = 0. \lim_{\omega\to\omega_0^{-\epsilon_\tau}}|\omega-\omega_0|^{1/2}H'(\omega)=0. ω → ω 0 − ϵ τ lim ∣ ω − ω 0 ∣ 1/2 H ′ ( ω ) = 0. In particular, if Φ D [ H ] ≠ 0 \Phi_D[H] \ne0 Φ D [ H ] = 0
a m − a 0 = Φ N [ H ] 2 r L τ Φ D [ H ] . (22) a_m-a_0=\frac{\Phi_N[H]}{\sqrt{2rL_\tau}\,\Phi_D[H]}. \tag*{(22)} a m − a 0 = 2 r L τ Φ D [ H ] Φ N [ H ] . (22) Uniqueness and reconstruction procedure
By Proposition 2.19, attenuation recovery reduces to finding, for each rod B p B_p B p τ ∈ S p \tau\in S_p τ ∈ S p F p ( a , f ) ( τ ) F_p^{(a,f)}(\tau) F p ( a , f ) ( τ ) H ( ω ) = R a f ( x τ , θ ( ω ) ) H(\omega)=R_af(x_\tau,\theta(\omega)) H ( ω ) = R a f ( x τ , θ ( ω )) Φ D [ H ] ≠ 0 \Phi_D[H]\ne0 Φ D [ H ] = 0 (22) gives a p a_p a p F p ( a , f ) ( τ ) = 0 F_p^{(a,f)}(\tau)=0 F p ( a , f ) ( τ ) = 0 τ \tau τ cancellation point . In this case, Φ D [ H ] = Φ N [ H ] = 0 \Phi_D[H]=\Phi_N[H]=0 Φ D [ H ] = Φ N [ H ] = 0 (22) is undefined.
We give a sufficient condition on the geometry and coefficients for non-cancellation on a nonempty relatively open family of simple tangencies of a rod. If this condition holds for every rod, the quotient (22) determines all attenuation coefficients from the full data.
Non-cancellation along a rod boundary
Definition 3.1 (Non-cancellation condition). Let ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X τ = ( ψ , ω 0 ) ∈ S p \tau=(\psi,\omega_0)\in S_p τ = ( ψ , ω 0 ) ∈ S p ( a , f ) (a,f) ( a , f ) non-cancellation condition at τ \tau τ
F p ( a , f ) ( τ ) = f p − ( a p − a 0 ) u tan [ a , f ] ( y τ , θ τ ) ≠ 0. (23) F_p^{(a,f)}(\tau)=f_p-(a_p-a_0)u_{\operatorname{tan}}[a,f](y_\tau,\theta_\tau)\ne0. \tag*{(23)} F p ( a , f ) ( τ ) = f p − ( a p − a 0 ) u tan [ a , f ] ( y τ , θ τ ) = 0. (23) In [12 (11)] , the non-cancellation condition is stated in terms of u ( x ∗ , θ ∗ ) u(x^*,\theta^*) u ( x ∗ , θ ∗ ) (23) expresses that condition in our jump convention.
Fix a rod B p B_p B p ϕ ∈ R / 2 π Z \phi\in\mathbb{R}/2\pi\mathbb{Z} ϕ ∈ R /2 π Z ∂ B p \partial B_p ∂ B p c p + r θ ( ϕ ) c_p+r\theta(\phi) c p + r θ ( ϕ ) θ ⊥ ( ϕ ) \theta^\perp(\phi) θ ⊥ ( ϕ ) x ( ψ ( ϕ ) ) x(\psi(\phi)) x ( ψ ( ϕ )) R ( c p + r θ ( ϕ ) , ϕ + π / 2 ) \mathcal{R}(c_p+r\theta(\phi),\phi+\pi/2) R ( c p + r θ ( ϕ ) , ϕ + π /2 ) Ω \Omega Ω
τ ( ϕ ) : = ( ψ ( ϕ ) , ϕ + π 2 ) ∈ T p , y τ ( ϕ ) = c p + r θ ( ϕ ) , θ τ ( ϕ ) = θ ⊥ ( ϕ ) , (24) \tau(\phi):=(\psi(\phi),\phi+\frac{\pi}{2})\in T_p,\qquad y_{\tau(\phi)}=c_p+r\theta(\phi),\qquad\theta_{\tau(\phi)}=\theta^\perp(\phi), \tag*{(24)} τ ( ϕ ) := ( ψ ( ϕ ) , ϕ + 2 π ) ∈ T p , y τ ( ϕ ) = c p + r θ ( ϕ ) , θ τ ( ϕ ) = θ ⊥ ( ϕ ) , (24) and all quantities attached to τ ( ϕ ) \tau(\phi) τ ( ϕ ) ϕ \phi ϕ c p − y τ ( ϕ ) = − r θ ( ϕ ) = r θ ⊥ ( ϕ + π / 2 ) c_p-y_{\tau(\phi)}=-r\theta(\phi)=r\theta^\perp(\phi+\pi/2) c p − y τ ( ϕ ) = − r θ ( ϕ ) = r θ ⊥ ( ϕ + π /2 ) ϵ τ ( ϕ ) = + 1 \epsilon_{\tau(\phi)}=+1 ϵ τ ( ϕ ) = + 1 L ( τ ( ϕ ) ) L(\tau(\phi)) L ( τ ( ϕ )) y τ ( ϕ ) y_{\tau(\phi)} y τ ( ϕ ) ∂ B p \partial B_p ∂ B p
The map ϕ ↦ τ ( ϕ ) \phi\mapsto\tau(\phi) ϕ ↦ τ ( ϕ ) R / 2 π Z \mathbb{R}/2\pi\mathbb{Z} R /2 π Z { τ ∈ T p : ϵ τ = + 1 } \{\tau\in T_p:\epsilon_\tau=+1\} { τ ∈ T p : ϵ τ = + 1 } ( ψ , ω 0 ) ↦ ω 0 − π / 2 (\psi,\omega_0)\mapsto\omega_0-\pi/2 ( ψ , ω 0 ) ↦ ω 0 − π /2 ϵ τ = + 1 \epsilon_\tau=+1 ϵ τ = + 1 y τ = c p + r θ ( ϕ ) y_\tau=c_p+r\theta(\phi) y τ = c p + r θ ( ϕ ) ϕ \phi ϕ ω 0 = ϕ + π / 2 \omega_0=\phi+\pi/2 ω 0 = ϕ + π /2 x τ x_\tau x τ y τ y_\tau y τ T p T_p T p ϵ τ \epsilon_\tau ϵ τ ϕ ↦ τ ( ϕ ) \phi\mapsto\tau(\phi) ϕ ↦ τ ( ϕ ) T p T_p T p
The next assumption uses a line tangent to both B p B_p B p B p B_p B p
Assumption 3.2. For the rod B p B_p B p
The attenuation jump is nonzero, so that
a p − a 0 ≠ 0. a_p-a_0\ne0. a p − a 0 = 0. (ii) There exist an index m ∈ { 1 , … , K } ∖ { p } m \in\{1,\ldots,K\} \setminus\{p\} m ∈ { 1 , … , K } ∖ { p } ϕ ∗ ∈ R / 2 π Z \phi_* \in\mathbb{R}/2\pi\mathbb{Z} ϕ ∗ ∈ R /2 π Z τ ∗ : = τ ( ϕ ∗ ) = ( ψ ∗ , ω ∗ ) \tau_* := \tau(\phi_*) = (\psi_*,\omega_*) τ ∗ := τ ( ϕ ∗ ) = ( ψ ∗ , ω ∗ )
∣ ρ m ( y τ ∗ , ω ∗ ) ∣ = r , σ m ( y τ ∗ , ω ∗ ) > 0 , f m ≠ 0. |\rho_m(y_{\tau_*},\omega_*)|=r,\qquad\sigma_m(y_{\tau_*},\omega_*)>0,\qquad f_m\ne0. ∣ ρ m ( y τ ∗ , ω ∗ ) ∣ = r , σ m ( y τ ∗ , ω ∗ ) > 0 , f m = 0. For every q ∉ { p , m } q\notin\{p,m\} q ∈ / { p , m }
∣ ρ q ( y τ ∗ , ω ∗ ) ∣ ≠ r . |\rho_q(y_{\tau_*},\omega_*)|\ne r. ∣ ρ q ( y τ ∗ , ω ∗ ) ∣ = r . Thus, L ( ψ ∗ , ω ∗ ) L(\psi_*,\omega_*) L ( ψ ∗ , ω ∗ ) ∂ B p \partial B_p ∂ B p ∂ B m \partial B_m ∂ B m ∂ B m \partial B_m ∂ B m R ( y τ ∗ , ω ∗ ) \mathcal{R}(y_{\tau_*},\omega_*) R ( y τ ∗ , ω ∗ ) y τ ∗ y_{\tau_*} y τ ∗
(iii) Every rod crossed by R ( y τ ∗ , ω ∗ ) \mathcal{R}(y_{\tau_*},\omega_*) R ( y τ ∗ , ω ∗ ) ∂ B m \partial B_m ∂ B m q ∈ C ( y τ ∗ , ω ∗ ) q\in\mathcal{C}(y_{\tau_*},\omega_*) q ∈ C ( y τ ∗ , ω ∗ )
d q ( y τ ∗ , ω ∗ ) + ℓ q ( y τ ∗ , ω ∗ ) < σ m ( y τ ∗ , ω ∗ ) . d_q(y_{\tau_*},\omega_*)+\ell_q(y_{\tau_*},\omega_*)<\sigma_m(y_{\tau_*},\omega_*). d q ( y τ ∗ , ω ∗ ) + ℓ q ( y τ ∗ , ω ∗ ) < σ m ( y τ ∗ , ω ∗ ) . Lemma 3.3. Let ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X B p B_p B p T p T_p T p
Proof. We show that the short chord created near the common tangent makes the tangential trace nonconstant. Since the attenuation jump is nonzero, the non-cancellation coefficient then cannot vanish identically.
Let m m m ϕ ∗ \phi_* ϕ ∗ τ ( ϕ ) \tau(\phi) τ ( ϕ ) ρ q ( ϕ ) \rho_q(\phi) ρ q ( ϕ ) σ q ( ϕ ) \sigma_q(\phi) σ q ( ϕ ) d q ( ϕ ) d_q(\phi) d q ( ϕ ) ℓ q ( ϕ ) \ell_q(\phi) ℓ q ( ϕ ) ( y τ ( ϕ ) , ϕ + π / 2 ) (y_{\tau(\phi)},\phi+\pi/2) ( y τ ( ϕ ) , ϕ + π /2 ) ϕ \phi ϕ θ ⊥ ( ϕ + π / 2 ) = − θ ( ϕ ) \theta^\perp(\phi+\pi/2)=-\theta(\phi) θ ⊥ ( ϕ + π /2 ) = − θ ( ϕ )
ρ q ( ϕ ) = r − ( c q − c p ) ⋅ θ ( ϕ ) , σ q ( ϕ ) = − ( c q − c p ) ⋅ θ ⊥ ( ϕ ) , ρ q ′ ( ϕ ) = σ q ( ϕ ) . \rho_q(\phi)=r-(c_q-c_p)\cdot\theta(\phi),\qquad\sigma_q(\phi)=-(c_q-c_p)\cdot\theta^\perp(\phi),\qquad\rho_q'(\phi)=\sigma_q(\phi). ρ q ( ϕ ) = r − ( c q − c p ) ⋅ θ ( ϕ ) , σ q ( ϕ ) = − ( c q − c p ) ⋅ θ ⊥ ( ϕ ) , ρ q ′ ( ϕ ) = σ q ( ϕ ) . Assumption (ii) gives ρ m ( ϕ ∗ ) = ± r \rho_m(\phi_*)=\pm r ρ m ( ϕ ∗ ) = ± r ρ m ′ ( ϕ ∗ ) = σ m ( ϕ ∗ ) > 0 \rho_m'(\phi_*)=\sigma_m(\phi_*)>0 ρ m ′ ( ϕ ∗ ) = σ m ( ϕ ∗ ) > 0
ρ m ( ϕ ) 2 − r 2 = 2 ρ m ( ϕ ∗ ) σ m ( ϕ ∗ ) ( ϕ − ϕ ∗ ) + O ( ( ϕ − ϕ ∗ ) 2 ) . (25) \rho_m(\phi)^2-r^2=2\rho_m(\phi_*)\sigma_m(\phi_*)(\phi-\phi_*)+O((\phi-\phi_*)^2). \tag*{(25)} ρ m ( ϕ ) 2 − r 2 = 2 ρ m ( ϕ ∗ ) σ m ( ϕ ∗ ) ( ϕ − ϕ ∗ ) + O (( ϕ − ϕ ∗ ) 2 ) . (25) Hence, the tangent lines L ( τ ( ϕ ) ) L(\tau(\phi)) L ( τ ( ϕ )) B m B_m B m ϕ ∗ \phi_* ϕ ∗ ρ m ( ϕ ∗ ) = r \rho_m(\phi_*)=r ρ m ( ϕ ∗ ) = r ϕ < ϕ ∗ \phi<\phi_* ϕ < ϕ ∗ ρ m ( ϕ ∗ ) = − r \rho_m(\phi_*)=-r ρ m ( ϕ ∗ ) = − r ϕ > ϕ ∗ \phi>\phi_* ϕ > ϕ ∗
Set S 0 : = C ( y τ ∗ , ω ∗ ) S_0:=\mathcal{C}(y_{\tau_*},\omega_*) S 0 := C ( y τ ∗ , ω ∗ ) q ∉ { p , m } q\notin\{p,m\} q ∈ / { p , m } ∣ ρ q ( ϕ ∗ ) ∣ > r |\rho_q(\phi_*)|>r ∣ ρ q ( ϕ ∗ ) ∣ > r L ( τ ∗ ) L(\tau_*) L ( τ ∗ ) B q ‾ \overline{B_q} B q ∣ ρ q ( ϕ ∗ ) ∣ < r |\rho_q(\phi_*)|<r ∣ ρ q ( ϕ ∗ ) ∣ < r L ( τ ∗ ) L(\tau_*) L ( τ ∗ ) B q B_q B q σ q ( ϕ ∗ ) ≠ 0 \sigma_q(\phi_*)\ne0 σ q ( ϕ ∗ ) = 0 σ q ( ϕ ∗ ) > 0 \sigma_q(\phi_*)>0 σ q ( ϕ ∗ ) > 0 y τ ∗ y_{\tau_*} y τ ∗ σ q ( ϕ ∗ ) < 0 \sigma_q(\phi_*)<0 σ q ( ϕ ∗ ) < 0 y τ ∗ y_{\tau_*} y τ ∗ ϕ \phi ϕ ϕ ∗ \phi_* ϕ ∗ C ( y τ ( ϕ ) , ϕ + π / 2 ) ∖ { m } = S 0 \mathcal{C}(y_{\tau(\phi)},\phi+\pi/2)\setminus\{m\}=S_0 C ( y τ ( ϕ ) , ϕ + π /2 ) ∖ { m } = S 0 ϕ ∗ \phi_* ϕ ∗ I I I ϕ ∗ \phi_* ϕ ∗
∣ ρ m ( ϕ ) ∣ < r , σ m ( ϕ ) > 0 , C ( y τ ( ϕ ) , ϕ + π / 2 ) = S 0 ∪ { m } , ϕ ∈ I . |\rho_m(\phi)|<r,\qquad\sigma_m(\phi)>0,\qquad\mathcal{C}(y_{\tau(\phi)},\phi+\pi/2)=S_0\cup\{m\},\qquad\phi\in I. ∣ ρ m ( ϕ ) ∣ < r , σ m ( ϕ ) > 0 , C ( y τ ( ϕ ) , ϕ + π /2 ) = S 0 ∪ { m } , ϕ ∈ I . All tangencies τ ( ϕ ) \tau(\phi) τ ( ϕ ) ϕ ∈ I \phi\in I ϕ ∈ I B p B_p B p I I I B m B_m B m q ∈ S 0 q\in S_0 q ∈ S 0 R ( y τ ( ϕ ) , ϕ + π / 2 ) \mathcal{R}(y_{\tau(\phi)},\phi+\pi/2) R ( y τ ( ϕ ) , ϕ + π /2 ) B q B_q B q ϕ ∗ \phi_* ϕ ∗ ϕ ∗ \phi_* ϕ ∗ S 0 S_0 S 0
In this proof, g ∼ h g \sim h g ∼ h g / h → 1 g/h \to1 g / h → 1 ϕ → ϕ ∗ \phi\to\phi_\ast ϕ → ϕ ∗ I I I B m B_m B m (25) gives, for some constant C > 0 C > 0 C > 0
ℓ m ( ϕ ) = 2 r 2 − ρ m ( ϕ ) 2 ∼ C ∣ ϕ − ϕ ∗ ∣ 1 / 2 . (26) \ell_m(\phi) = 2\sqrt{r^2-\rho_m(\phi)^2} \sim C|\phi-\phi_\ast|^{1/2}. \tag*{(26)} ℓ m ( ϕ ) = 2 r 2 − ρ m ( ϕ ) 2 ∼ C ∣ ϕ − ϕ ∗ ∣ 1/2 . (26) Using the exact derivative
ℓ m ′ ( ϕ ) = − 2 ρ m ( ϕ ) ρ m ′ ( ϕ ) r 2 − ρ m ( ϕ ) 2 , \ell_m'(\phi)=-\frac{2\rho_m(\phi)\rho_m'(\phi)}{\sqrt{r^2-\rho_m(\phi)^2}}, ℓ m ′ ( ϕ ) = − r 2 − ρ m ( ϕ ) 2 2 ρ m ( ϕ ) ρ m ′ ( ϕ ) , we also obtain
∣ ℓ m ′ ( ϕ ) ∣ ∼ C 2 ∣ ϕ − ϕ ∗ ∣ − 1 / 2 ⟶ ∞ (27) |\ell_m'(\phi)| \sim\frac{C}{2}|\phi-\phi_\ast|^{-1/2}\longrightarrow\infty \tag*{(27)} ∣ ℓ m ′ ( ϕ ) ∣ ∼ 2 C ∣ ϕ − ϕ ∗ ∣ − 1/2 ⟶ ∞ (27) as ϕ → ϕ ∗ \phi\to\phi_\ast ϕ → ϕ ∗ I I I
Let β ( ϕ ) \beta(\phi) β ( ϕ ) y τ ( ϕ ) y_{\tau(\phi)} y τ ( ϕ ) y τ ( ϕ ) − d m ( ϕ ) θ τ ( ϕ ) y_{\tau(\phi)}-d_m(\phi)\theta_{\tau(\phi)} y τ ( ϕ ) − d m ( ϕ ) θ τ ( ϕ ) β m , p \beta_{m,p} β m , p τ ( ϕ ) \tau(\phi) τ ( ϕ ) Ω \Omega Ω B m B_m B m S 0 S_0 S 0
β ( ϕ ) = a 0 d m ( ϕ ) + ∑ q ∈ S 0 ( a q − a 0 ) ℓ q ( ϕ ) . (28) \beta(\phi)=a_0d_m(\phi)+\sum_{q\in S_0}(a_q-a_0)\ell_q(\phi). \tag*{(28)} β ( ϕ ) = a 0 d m ( ϕ ) + q ∈ S 0 ∑ ( a q − a 0 ) ℓ q ( ϕ ) . (28) The ray remains tangent to B p B_p B p B p B_p B p
d m ( ϕ ) = σ m ( ϕ ) − ℓ m ( ϕ ) 2 . d_m(\phi)=\sigma_m(\phi)-\frac{\ell_m(\phi)}{2}. d m ( ϕ ) = σ m ( ϕ ) − 2 ℓ m ( ϕ ) . The derivatives of σ m \sigma_m σ m q ∈ S 0 q\in S_0 q ∈ S 0
β ′ ( ϕ ) = O ( ∣ ϕ − ϕ ∗ ∣ − 1 / 2 ) . (29) \beta'(\phi)=O\bigl(|\phi-\phi_\ast|^{-1/2}\bigr). \tag*{(29)} β ′ ( ϕ ) = O ( ∣ ϕ − ϕ ∗ ∣ − 1/2 ) . (29) Equation (28) also shows that β \beta β ϕ → ϕ ∗ \phi\to\phi_\ast ϕ → ϕ ∗
Let U ( ϕ ) U(\phi) U ( ϕ ) S 0 S_0 S 0 W q , p e − β q , p W_{q,p}e^{-\beta_{q,p}} W q , p e − β q , p q ∈ S 0 q\in S_0 q ∈ S 0 τ ( ϕ ) \tau(\phi) τ ( ϕ )
u tan [ a , f ] ( y τ ( ϕ ) , θ τ ( ϕ ) ) = U ( ϕ ) + f m a m ( 1 − e − a m ℓ m ( ϕ ) ) e − β ( ϕ ) , ϕ ∈ I . (30) u_{\tan}[a,f](y_{\tau(\phi)},\theta_{\tau(\phi)})=U(\phi)+\frac{f_m}{a_m}\left(1-e^{-a_m\ell_m(\phi)}\right)e^{-\beta(\phi)},\qquad\phi\in I. \tag*{(30)} u t a n [ a , f ] ( y τ ( ϕ ) , θ τ ( ϕ ) ) = U ( ϕ ) + a m f m ( 1 − e − a m ℓ m ( ϕ ) ) e − β ( ϕ ) , ϕ ∈ I . (30) The contributions to U U U B m B_m B m U U U
U ′ ( ϕ ) = O ( 1 ) . (31) U'(\phi)=O(1). \tag*{(31)} U ′ ( ϕ ) = O ( 1 ) . (31) Differentiating the contribution from B m B_m B m
d d ϕ [ f m a m ( 1 − e − a m ℓ m ( ϕ ) ) e − β ( ϕ ) ] = e − β ( ϕ ) [ f m e − a m ℓ m ( ϕ ) ℓ m ′ ( ϕ ) − f m a m ( 1 − e − a m ℓ m ( ϕ ) ) β ′ ( ϕ ) ] . \frac{d}{d\phi}\left[\frac{f_m}{a_m}\left(1-e^{-a_m\ell_m(\phi)}\right)e^{-\beta(\phi)}\right]
=e^{-\beta(\phi)}\left[f_me^{-a_m\ell_m(\phi)}\ell_m'(\phi)-\frac{f_m}{a_m}\left(1-e^{-a_m\ell_m(\phi)}\right)\beta'(\phi)\right]. d ϕ d [ a m f m ( 1 − e − a m ℓ m ( ϕ ) ) e − β ( ϕ ) ] = e − β ( ϕ ) [ f m e − a m ℓ m ( ϕ ) ℓ m ′ ( ϕ ) − a m f m ( 1 − e − a m ℓ m ( ϕ ) ) β ′ ( ϕ ) ] . By (26) , we have
1 − e − a m ℓ m ( ϕ ) = O ( ∣ ϕ − ϕ ∗ ∣ 1 / 2 ) . 1-e^{-a_m\ell_m(\phi)}=O\bigl(|\phi-\phi_\ast|^{1/2}\bigr). 1 − e − a m ℓ m ( ϕ ) = O ( ∣ ϕ − ϕ ∗ ∣ 1/2 ) . Together with (3.7), this shows that the second term in brackets is O ( 1 ) O(1) O ( 1 ) f m ≠ 0 f_m \ne0 f m = 0 e − β ( ϕ ) e^{-\beta(\phi)} e − β ( ϕ )
∣ d d ϕ u tan [ a , f ] ( y τ ( ϕ ) , θ τ ( ϕ ) ) ∣ ⟶ ∞ . \left|\frac{d}{d\phi}u_{\tan}[a,f](y_{\tau(\phi)},\theta_{\tau(\phi)})\right| \longrightarrow\infty. d ϕ d u t a n [ a , f ] ( y τ ( ϕ ) , θ τ ( ϕ ) ) ⟶ ∞. Thus, the tangential trace is nonconstant on I I I a p − a 0 ≠ 0 a_p-a_0\ne0 a p − a 0 = 0 ϕ ↦ F p ( a , f ) ( τ ( ϕ ) ) \phi\mapsto F_p^{(a,f)}(\tau(\phi)) ϕ ↦ F p ( a , f ) ( τ ( ϕ )) I I I J ⊂ I J\subset I J ⊂ I F p ( a , f ) ( τ ( ϕ ) ) ≠ 0 F_p^{(a,f)}(\tau(\phi))\ne0 F p ( a , f ) ( τ ( ϕ )) = 0 ϕ ↦ τ ( ϕ ) \phi\mapsto\tau(\phi) ϕ ↦ τ ( ϕ ) J J J T p T_p T p □ \square □
Theorem 3.5 (Unique recovery). Let ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X B p B_p B p p = 1 , … , K p=1,\ldots,K p = 1 , … , K ( a , f ) (a,f) ( a , f )
{ R a f ( ψ , ω ) : ( ψ , ω ) ∈ [ 0 , 2 π ) 2 } . \{R_af(\psi,\omega):(\psi,\omega)\in[0,2\pi)^2\}. { R a f ( ψ , ω ) : ( ψ , ω ) ∈ [ 0 , 2 π ) 2 } . Proof. Fix p ∈ { 1 , … , K } p\in\{1,\ldots,K\} p ∈ { 1 , … , K } τ = ( ψ , ω 0 ) ∈ S p \tau=(\psi,\omega_0)\in S_p τ = ( ψ , ω 0 ) ∈ S p F p ( a , f ) ( τ ) ≠ 0 F_p^{(a,f)}(\tau)\ne0 F p ( a , f ) ( τ ) = 0 H ( ω ) : = R a f ( ψ , ω ) H(\omega):=R_af(\psi,\omega) H ( ω ) := R a f ( ψ , ω ) Φ D [ H ] ≠ 0 \Phi_D[H]\ne0 Φ D [ H ] = 0
a p = a 0 + Φ N [ H ] 2 r L τ Φ D [ H ] . a_p=a_0+\frac{\Phi_N[H]}{\sqrt{2rL_\tau}\,\Phi_D[H]}. a p = a 0 + 2 r L τ Φ D [ H ] Φ N [ H ] . The geometry is known, and the data determine both functionals, so this formula reconstructs a p a_p a p Φ D [ H ] ≠ 0 \Phi_D[H]\ne0 Φ D [ H ] = 0 a a a
Once a a a f f f a a a f f f L ∞ , 1 + ε ( R 2 , R ) L^{\infty,1+\varepsilon}(\mathbb{R}^2,\mathbb{R}) L ∞ , 1 + ε ( R 2 , R ) [19 (1.9)–(1.10)] , for every ε > 0 \varepsilon>0 ε > 0 Ω \Omega Ω Ω \Omega Ω
coincide with P a f P_af P a f [19 (1.1)–(1.2)] . The inversion formula, with the distributional interpretation in [19 Theorem 2.1 and Remark 2.6] , therefore recovers f f f
Under the assumptions of Section 4, Remark 4.6 gives explicit formulas for both coefficients and a self-contained proof of uniqueness in that setting.
Singular stability estimates
For stability, we use tangencies at which the backward ray meets no other rod. The tangential trace then vanishes, and a positive lower bound on the rod sources gives a uniform non-cancellation bound.
Definition 4.1 (Admissible tangency). A simple tangency τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in S_m τ = ( ψ , ω 0 ) ∈ S m admissible if the backward ray R ( y τ , ω 0 ) \mathcal{R}(y_\tau,\omega_0) R ( y τ , ω 0 ) B m B_m B m
R ( y τ , ω 0 ) ∩ B q ‾ = ∅ , q ≠ m . (32) \mathcal{R}(y_\tau,\omega_0)\cap\overline{B_q}=\varnothing,\qquad q\ne m. \tag*{(32)} R ( y τ , ω 0 ) ∩ B q = ∅ , q = m . (32) Assumption 4.2. (A1) The geometry G = ( r o u t , r , { c q } q = 1 K ) \mathcal{G}=(r_{\mathrm{out}},r,\{c_q\}_{q=1}^{K}) G = ( r out , r , { c q } q = 1 K ) X = X ( a min , a max , f min , f max , a 0 , G ) X=X(a_{\min},a_{\max},f_{\min},f_{\max},a_0,\mathcal{G}) X = X ( a m i n , a m a x , f m i n , f m a x , a 0 , G )
(A2) Emitting rods. We assume f min > 0 f_{\min}>0 f m i n > 0 X X X
(A3) Row separation. The rod centers lie on parallel lines, called rows . Any two distinct rows are separated by more than one rod diameter. Equivalently, there is a unit vector n ∈ S 1 n\in S^1 n ∈ S 1 p , q ∈ { 1 , … , K } p,q\in\{1,\ldots,K\} p , q ∈ { 1 , … , K }
( c q − c p ) ⋅ n = 0 or ∣ ( c q − c p ) ⋅ n ∣ > 2 r . (33) (c_q-c_p)\cdot n=0\qquad\text{or}\qquad|(c_q-c_p)\cdot n|>2r. \tag*{(33)} ( c q − c p ) ⋅ n = 0 or ∣ ( c q − c p ) ⋅ n ∣ > 2 r . (33) The vector n n n
(A4) Selected tangencies. For every m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } τ m = ( ψ m , ω m ) ∈ S m \tau_m=(\psi_m,\omega_m)\in S_m τ m = ( ψ m , ω m ) ∈ S m
(A5) Data. The measurement set M \mathcal{M} M
{ ψ m } × ( ω m − δ τ m , ω m + δ τ m ) ⊂ M , m = 1 , … , K , \{\psi_m\}\times(\omega_m-\delta_{\tau_m},\omega_m+\delta_{\tau_m})\subset\mathcal{M},\qquad m=1,\ldots,K, { ψ m } × ( ω m − δ τ m , ω m + δ τ m ) ⊂ M , m = 1 , … , K , that is, (2.16) holds for τ = τ m \tau=\tau_m τ = τ m
Proposition 4.4 below shows that (A3) guarantees an admissible tangency for every rod, as required by (A4). The tangencies τ 1 , … , τ K \tau_1,\ldots,\tau_K τ 1 , … , τ K K K K
The lower bound (A2) makes the estimates uniform over X X X f m ≥ f min f_m\ge f_{\min} f m ≥ f m i n
Proposition 4.4 (Existence of admissible tangencies). Suppose that assumptions (A1) and (A3) hold. Then every rod B m B_m B m τ ∈ S m \tau\in\mathcal{S}_m τ ∈ S m
F m ( a , f ) ( τ ) = f m for all ( a , f ) ∈ X . F_m^{(a,f)}(\tau)=f_m \qquad\text{for all }(a,f)\in\mathcal{X}. F m ( a , f ) ( τ ) = f m for all ( a , f ) ∈ X . In particular, under (A2), F m ( a , f ) ( τ ) ≥ f min > 0 F_m^{(a,f)}(\tau)\ge f_{\min}>0 F m ( a , f ) ( τ ) ≥ f m i n > 0 ( a , f ) ∈ X (a,f)\in\mathcal{X} ( a , f ) ∈ X
Proof. We construct the tangent direction by tilting it slightly from the row direction, as illustrated in Figure 5 . Let n n n e e e π / 2 \pi/2 π /2 n n n d > 2 r d>2r d > 2 r d d d
Figure 5. Admissible tangencies obtained by tilting the tangent lines relative to the row direction. Each line is tangent to exactly one rod. The backward ray from each tangency point meets no other rod.
w e : = max p , q ∣ ( c q − c p ) ⋅ e ∣ . w_e:=\max_{p,q}\left|(c_q-c_p)\cdot e\right|. w e := p , q max ∣ ( c q − c p ) ⋅ e ∣ . Choose α ∈ ( 0 , π / 2 ) \alpha\in(0,\pi/2) α ∈ ( 0 , π /2 )
d cos α − w e sin α > 2 r , w e sin α < 2 r . d\cos\alpha-w_e\sin\alpha>2r,\qquad w_e\sin\alpha<2r. d cos α − w e sin α > 2 r , w e sin α < 2 r . This is possible by continuity at α = 0 \alpha=0 α = 0
ϑ : = cos α e + sin α n , ϖ : = − sin α e + cos α n . \vartheta:=\cos\alpha\,e+\sin\alpha\,n,\qquad\varpi:=-\sin\alpha\,e+\cos\alpha\,n. ϑ := cos α e + sin α n , ϖ := − sin α e + cos α n . Fix m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } y : = c m − r ϖ y:=c_m-r\varpi y := c m − r ϖ L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) ∂ B m \partial B_m ∂ B m q ≠ m q\ne m q = m c q − c m = ξ e + η n c_q-c_m=\xi e+\eta n c q − c m = ξ e + η n c q c_q c q L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ )
∣ ( c q − y ) ⋅ ϖ ∣ = ∣ r − ξ sin α + η cos α ∣ . \left|(c_q-y)\cdot\varpi\right|=\left|r-\xi\sin\alpha+\eta\cos\alpha\right|. ∣ ( c q − y ) ⋅ ϖ ∣ = ∣ r − ξ sin α + η cos α ∣ . If this distance is greater than r r r L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) B q ‾ \overline{B_q} B q r r r L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) B q B_q B q
If B q B_q B q ∣ η ∣ ≥ d |\eta|\ge d ∣ η ∣ ≥ d
∣ r − ξ sin α + η cos α ∣ ≥ ∣ η ∣ cos α − ∣ r − ξ sin α ∣ ≥ d cos α − r − ∣ ξ ∣ sin α ≥ d cos α − r − w e sin α > r . \left|r-\xi\sin\alpha+\eta\cos\alpha\right|
\ge|\eta|\cos\alpha-\left|r-\xi\sin\alpha\right|
\ge d\cos\alpha-r-|\xi|\sin\alpha
\ge d\cos\alpha-r-w_e\sin\alpha>r. ∣ r − ξ sin α + η cos α ∣ ≥ ∣ η ∣ cos α − ∣ r − ξ sin α ∣ ≥ d cos α − r − ∣ ξ ∣ sin α ≥ d cos α − r − w e sin α > r . Thus, L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) B q ‾ \overline{B_q} B q
If B q B_q B q η = 0 \eta=0 η = 0 ξ ≠ 0 \xi\ne0 ξ = 0 ξ < 0 \xi<0 ξ < 0
∣ r − ξ sin α ∣ = r + ∣ ξ ∣ sin α > r . \left|r-\xi\sin\alpha\right|=r+|\xi|\sin\alpha>r. ∣ r − ξ sin α ∣ = r + ∣ ξ ∣ sin α > r . Thus, L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) B q ‾ \overline{B_q} B q ξ > 0 \xi>0 ξ > 0
0 < ξ sin α ≤ w e sin α < 2 r , ∣ r − ξ sin α ∣ < r . 0<\xi\sin\alpha\le w_e\sin\alpha<2r,\qquad\left|r-\xi\sin\alpha\right|<r. 0 < ξ sin α ≤ w e sin α < 2 r , ∣ r − ξ sin α ∣ < r . The line L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) B q B_q B q
( y − c q ) ⋅ ϑ = − ξ cos α < 0. (y-c_q)\cdot\vartheta=-\xi\cos\alpha<0. ( y − c q ) ⋅ ϑ = − ξ cos α < 0. Since the closed rods are pairwise disjoint, we have ∣ y − c q ∣ > r |y-c_q|>r ∣ y − c q ∣ > r s ≥ 0 s\ge0 s ≥ 0
∣ y − s ϑ − c q ∣ 2 = ∣ y − c q ∣ 2 − 2 s ( y − c q ) ⋅ ϑ + s 2 > r 2 . |y-s\vartheta-c_q|^2=|y-c_q|^2-2s(y-c_q)\cdot\vartheta+s^2>r^2. ∣ y − s ϑ − c q ∣ 2 = ∣ y − c q ∣ 2 − 2 s ( y − c q ) ⋅ ϑ + s 2 > r 2 . Hence, the backward ray { y − s ϑ : s ≥ 0 } \{y-s\vartheta:s\ge0\} { y − s ϑ : s ≥ 0 } B q ‾ \overline{B_q} B q L ( y , ϑ ) ∩ B q L(y,\vartheta)\cap B_q L ( y , ϑ ) ∩ B q y y y
These cases cover every q ≠ m q\ne m q = m L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) ∂ B m \partial B_m ∂ B m { y − s ϑ : s ≥ 0 } \{y-s\vartheta:s\ge0\} { y − s ϑ : s ≥ 0 } L ( y , ϑ ) L(y,\vartheta) L ( y , ϑ ) Ω \Omega Ω τ \tau τ y τ = y y_\tau=y y τ = y θ τ = ϑ \theta_\tau=\vartheta θ τ = ϑ
For any admissible tangency τ = ( ψ , ω 0 ) ∈ S m \tau=(\psi,\omega_0)\in\mathcal{S}_m τ = ( ψ , ω 0 ) ∈ S m (32) gives C ( y τ , ω 0 ) = ∅ C(y_\tau,\omega_0)=\varnothing C ( y τ , ω 0 ) = ∅ ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X
u tan [ a , f ] ( y τ , θ τ ) = 0. u_{\tan}[a,f](y_\tau,\theta_\tau)=0. u t a n [ a , f ] ( y τ , θ τ ) = 0. Consequently, F m ( a , f ) ( τ ) = f m F_m^{(a,f)}(\tau)=f_m F m ( a , f ) ( τ ) = f m F m ( a , f ) ( τ ) ≥ f min > 0 F_m^{(a,f)}(\tau)\ge f_{\min}>0 F m ( a , f ) ( τ ) ≥ f m i n > 0 ( a , f ) ∈ X (a,f)\in X ( a , f ) ∈ X
The stability seminorm and the main estimate
Let Assumption 4.2 hold. Since the selection τ 1 , … , τ K \tau_1,\ldots,\tau_K τ 1 , … , τ K
min 1 ≤ m ≤ K δ τ m > 0 , \min_{1\le m\le K}\delta_{\tau_m}>0, 1 ≤ m ≤ K min δ τ m > 0 , and Lemma 2.7 gives
0 < L min ≤ L τ m ≤ L max < ∞ , m = 1 , … , K . (34) 0<L_{\min}\le L_{\tau_m}\le L_{\max}<\infty,\qquad m=1,\ldots,K. \tag*{(34)} 0 < L m i n ≤ L τ m ≤ L m a x < ∞ , m = 1 , … , K . (34) Definition 4.5 (Stability seminorm). Let Assumption 4.2 hold. For a real-valued function H H H M \mathcal{M} M H ( ψ m , ⋅ ) ∣ I τ m ∈ V τ m H(\psi_m,\cdot)|_{I_{\tau_m}}\in\mathcal{V}_{\tau_m} H ( ψ m , ⋅ ) ∣ I τ m ∈ V τ m m m m
∥ H ∥ ⋆ : = max 1 ≤ m ≤ K ∥ H ( ψ m , ⋅ ) ∥ τ m . (35) \|H\|_{\star}:=\max_{1\le m\le K}\|H(\psi_m,\cdot)\|_{\tau_m}. \tag*{(35)} ∥ H ∥ ⋆ := 1 ≤ m ≤ K max ∥ H ( ψ m , ⋅ ) ∥ τ m . (35) These functions form a vector space on which the finite maximum of the local seminorms is a seminorm. For two pairs in X X X V τ m \mathcal{V}_{\tau_m} V τ m
Theorem 4.7. Let Assumption 4.2 hold. Then there exist constants C a , C f > 0 C_a,C_f>0 C a , C f > 0 X X X G \mathcal{G} G τ 1 , … , τ K \tau_1,\ldots,\tau_K τ 1 , … , τ K
∥ a ( 1 ) − a ( 2 ) ∥ ∞ ≤ C a ∥ R a ( 1 ) f ( 1 ) − R a ( 2 ) f ( 2 ) ∥ ⋆ (36) \lVert a^{(1)}-a^{(2)}\rVert_{\infty}\leq C_a\lVert R_{a^{(1)}}f^{(1)}-R_{a^{(2)}}f^{(2)}\rVert_{\star} \tag*{(36)} ∥ a ( 1 ) − a ( 2 ) ∥ ∞ ≤ C a ∥ R a ( 1 ) f ( 1 ) − R a ( 2 ) f ( 2 ) ∥ ⋆ (36) and
∥ f ( 1 ) − f ( 2 ) ∥ ∞ ≤ C f ∥ R a ( 1 ) f ( 1 ) − R a ( 2 ) f ( 2 ) ∥ ⋆ \lVert f^{(1)}-f^{(2)}\rVert_{\infty}\leq C_f\lVert R_{a^{(1)}}f^{(1)}-R_{a^{(2)}}f^{(2)}\rVert_{\star} ∥ f ( 1 ) − f ( 2 ) ∥ ∞ ≤ C f ∥ R a ( 1 ) f ( 1 ) − R a ( 2 ) f ( 2 ) ∥ ⋆ for every ( a ( 1 ) , f ( 1 ) ) , ( a ( 2 ) , f ( 2 ) ) ∈ X (a^{(1)},f^{(1)}),(a^{(2)},f^{(2)})\in X ( a ( 1 ) , f ( 1 ) ) , ( a ( 2 ) , f ( 2 ) ) ∈ X
Remark 4.8. The proof gives explicit formulas for C a C_a C a C f C_f C f
Proof. We first estimate the attenuation using (21) –(22) , and then the source using (20) . Fix m ∈ { 1 , … , K } m\in\{1,\ldots,K\} m ∈ { 1 , … , K } H : = H ( 1 ) − H ( 2 ) H:=H^{(1)}-H^{(2)} H := H ( 1 ) − H ( 2 )
H ( i ) : = R a ( i ) f ( i ) ( ψ m , ⋅ ) , i = 1 , 2. H^{(i)}:=R_{a^{(i)}}f^{(i)}(\psi_m,\cdot),\qquad i=1,2. H ( i ) := R a ( i ) f ( i ) ( ψ m , ⋅ ) , i = 1 , 2. The bounds (34) hold for L τ m L_{\tau_m} L τ m
A Δ : = max { ∣ a min − a 0 ∣ , ∣ a max − a 0 ∣ } , A : = max { a 0 , a max } , b i : = D a ( i ) ( y τ m , θ τ m ) . A_{\Delta}:=\max\{|a_{\min}-a_0|,|a_{\max}-a_0|\},\qquad A:=\max\{a_0,a_{\max}\},\qquad b_i:=D^{a^{(i)}}(y_{\tau_m},\theta_{\tau_m}). A Δ := max { ∣ a m i n − a 0 ∣ , ∣ a m a x − a 0 ∣ } , A := max { a 0 , a m a x } , b i := D a ( i ) ( y τ m , θ τ m ) . Then ∣ a m ( i ) − a 0 ∣ ≤ A Δ |a_m^{(i)}-a_0|\leq A_{\Delta} ∣ a m ( i ) − a 0 ∣ ≤ A Δ 2 r o u t 2r_{\mathrm{out}} 2 r out Ω \Omega Ω
0 ≤ b i ≤ 2 r o u t A , i = 1 , 2. (37) 0\leq b_i\leq2r_{\mathrm{out}}A,\qquad i=1,2. \tag*{(37)} 0 ≤ b i ≤ 2 r out A , i = 1 , 2. (37) Proposition 4.4 gives F m ( a ( i ) , f ( i ) ) ( τ m ) = f m ( i ) ≥ f min F_m^{(a^{(i)},f^{(i)})}(\tau_m)=f_m^{(i)}\geq f_{\min} F m ( a ( i ) , f ( i ) ) ( τ m ) = f m ( i ) ≥ f m i n (20) , (4.6) and (34) , we obtain
Φ D [ H ( i ) ] = 2 r L τ m f m ( i ) e − b i , ∣ Φ D [ H ( i ) ] ∣ ≥ 2 r L min f min e − 2 r o u t A = : B D > 0. \Phi_D[H^{(i)}]=\sqrt{2rL_{\tau_m}}f_m^{(i)}e^{-b_i},\qquad|\Phi_D[H^{(i)}]|\geq\sqrt{2rL_{\min}}f_{\min}e^{-2r_{\mathrm{out}}A}=:B_D>0. Φ D [ H ( i ) ] = 2 r L τ m f m ( i ) e − b i , ∣ Φ D [ H ( i ) ] ∣ ≥ 2 r L m i n f m i n e − 2 r out A =: B D > 0. By (21) , we have
Φ N [ H ( i ) ] = ( a m ( i ) − a 0 ) 2 r L τ m Φ D [ H ( i ) ] . \Phi_N[H^{(i)}]=(a_m^{(i)}-a_0)\sqrt{2rL_{\tau_m}}\Phi_D[H^{(i)}]. Φ N [ H ( i ) ] = ( a m ( i ) − a 0 ) 2 r L τ m Φ D [ H ( i ) ] . Subtracting the two identities, we obtain
( a m ( 1 ) − a m ( 2 ) ) 2 r L τ m Φ D [ H ( 1 ) ] = ( Φ N [ H ( 1 ) ] − Φ N [ H ( 2 ) ] ) − ( a m ( 2 ) − a 0 ) 2 r L τ m ( Φ D [ H ( 1 ) ] − Φ D [ H ( 2 ) ] ) . (a_m^{(1)}-a_m^{(2)})\sqrt{2rL_{\tau_m}}\Phi_D[H^{(1)}]
=
\bigl(\Phi_N[H^{(1)}]-\Phi_N[H^{(2)}]\bigr)
-(a_m^{(2)}-a_0)\sqrt{2rL_{\tau_m}}\bigl(\Phi_D[H^{(1)}]-\Phi_D[H^{(2)}]\bigr). ( a m ( 1 ) − a m ( 2 ) ) 2 r L τ m Φ D [ H ( 1 ) ] = ( Φ N [ H ( 1 ) ] − Φ N [ H ( 2 ) ] ) − ( a m ( 2 ) − a 0 ) 2 r L τ m ( Φ D [ H ( 1 ) ] − Φ D [ H ( 2 ) ] ) . By linearity of the functionals at τ m \tau_m τ m
Φ N [ H ( 1 ) ] − Φ N [ H ( 2 ) ] = Φ N [ H ] , \Phi_N[H^{(1)}]-\Phi_N[H^{(2)}]=\Phi_N[H], Φ N [ H ( 1 ) ] − Φ N [ H ( 2 ) ] = Φ N [ H ] , Φ D [ H ( 1 ) ] − Φ D [ H ( 2 ) ] = Φ D [ H ] . \Phi_D[H^{(1)}]-\Phi_D[H^{(2)}]=\Phi_D[H]. Φ D [ H ( 1 ) ] − Φ D [ H ( 2 ) ] = Φ D [ H ] . Consequently, we obtain
∣ a m ( 1 ) − a m ( 2 ) ∣ ≤ ∣ Φ N [ H ] ∣ + A Δ 2 r L max ∣ Φ D [ H ] ∣ 2 r L min B D ≤ max { 1 , A Δ 2 r L max } 2 r L min f min e − 2 r o u t A ∥ H ∥ τ m . |a_m^{(1)}-a_m^{(2)}|
\leq
\frac{|\Phi_N[H]|+A_{\Delta}\sqrt{2rL_{\max}}|\Phi_D[H]|}{\sqrt{2rL_{\min}}B_D}
\leq
\frac{\max\{1,A_{\Delta}\sqrt{2rL_{\max}}\}}{2rL_{\min}f_{\min}e^{-2r_{\mathrm{out}}A}}\lVert H\rVert_{\tau_m}. ∣ a m ( 1 ) − a m ( 2 ) ∣ ≤ 2 r L m i n B D ∣ Φ N [ H ] ∣ + A Δ 2 r L m a x ∣ Φ D [ H ] ∣ ≤ 2 r L m i n f m i n e − 2 r out A max { 1 , A Δ 2 r L m a x } ∥ H ∥ τ m . Taking the maximum over m m m
C a : = e 2 r o u t A 2 r L min f min max { 1 , A Δ 2 r L max } . (38) C_a:=\frac{e^{2r_{\mathrm{out}}A}}{2rL_{\min}f_{\min}}\max\{1,A_{\Delta}\sqrt{2rL_{\max}}\}. \tag*{(38)} C a := 2 r L m i n f m i n e 2 r out A max { 1 , A Δ 2 r L m a x } . (38) To estimate the source, recall that
Φ D [ H ( i ) ] = 2 r L τ m f m ( i ) e − b i . \Phi_D[H^{(i)}]=\sqrt{2rL_{\tau_m}}f_m^{(i)}e^{-b_i}. Φ D [ H ( i ) ] = 2 r L τ m f m ( i ) e − b i . Since the estimate is symmetric in the two pairs, we may assume b 1 ≤ b 2 b_1\leq b_2 b 1 ≤ b 2
f m ( 1 ) − f m ( 2 ) = e b 1 2 r L τ m Φ D [ H ] + f m ( 2 ) ( e b 1 − b 2 − 1 ) . (39) f_m^{(1)}-f_m^{(2)}
=
\frac{e^{b_1}}{\sqrt{2rL_{\tau_m}}}\Phi_D[H]+f_m^{(2)}(e^{b_1-b_2}-1). \tag*{(39)} f m ( 1 ) − f m ( 2 ) = 2 r L τ m e b 1 Φ D [ H ] + f m ( 2 ) ( e b 1 − b 2 − 1 ) . (39) Since 1 − e − t ≤ t 1-e^{-t}\le t 1 − e − t ≤ t t ≥ 0 t\ge0 t ≥ 0
∣ f m ( 1 ) − f m ( 2 ) ∣ ≤ e 2 r o u t A 2 r L min ∣ Φ D [ H ] ∣ + f max ∣ b 1 − b 2 ∣ . \left|f_m^{(1)}-f_m^{(2)}\right|\le\frac{e^{2r_{\mathrm{out}}A}}{\sqrt{2rL_{\min}}}\left|\Phi_D[H]\right|+f_{\max}|b_1-b_2|. f m ( 1 ) − f m ( 2 ) ≤ 2 r L m i n e 2 r out A ∣ Φ D [ H ] ∣ + f m a x ∣ b 1 − b 2 ∣. The moderator attenuation is the same for both pairs. Therefore, we obtain
∣ b 1 − b 2 ∣ ≤ ∫ 0 ∞ ∣ a ( 1 ) ( y τ m + s θ τ m ) − a ( 2 ) ( y τ m + s θ τ m ) ∣ d s ≤ 2 r o u t ∥ a ( 1 ) − a ( 2 ) ∥ ℓ ∞ . |b_1-b_2|\le\int_0^\infty\left|a^{(1)}(y_{\tau_m}+s\theta_{\tau_m})-a^{(2)}(y_{\tau_m}+s\theta_{\tau_m})\right|\,\mathrm{d}s
\le2r_{\mathrm{out}}\left\|a^{(1)}-a^{(2)}\right\|_{\ell^\infty}. ∣ b 1 − b 2 ∣ ≤ ∫ 0 ∞ a ( 1 ) ( y τ m + s θ τ m ) − a ( 2 ) ( y τ m + s θ τ m ) d s ≤ 2 r out a ( 1 ) − a ( 2 ) ℓ ∞ . Combining this bound with Remark 2.18 and (36) , and taking the maximum over m m m
C f : = e 2 r o u t A 2 r L min + 2 r o u t f max C a . (40) C_f:=\frac{e^{2r_{\mathrm{out}}A}}{\sqrt{2rL_{\min}}}+2r_{\mathrm{out}}f_{\max}C_a. \tag*{(40)} C f := 2 r L m i n e 2 r out A + 2 r out f m a x C a . (40) Both constants depend only on the global geometric and coefficient bounds and are therefore independent of the selected tangencies. □
Numerical experiment
To complement the stability estimates, we conduct numerical experiments to examine the asymptotic behavior of the singular functionals.
By Definition 2.17 and (35) , we have
∥ R a f − R a ~ f ~ ∥ ⋆ = max 1 ≤ m ≤ K ∥ R a f ( ψ m , ⋅ ) − R a ~ f ~ ( ψ m , ⋅ ) ∥ τ m . \left\|R_a f-R_{\widetilde a}\widetilde f\right\|_\star=\max_{1\le m\le K}\left\|R_a f(\psi_m,\cdot)-R_{\widetilde a}\widetilde f(\psi_m,\cdot)\right\|_{\tau_m}. R a f − R a f ⋆ = 1 ≤ m ≤ K max R a f ( ψ m , ⋅ ) − R a f ( ψ m , ⋅ ) τ m . for simple tangencies τ m ∈ S m \tau_m\in\mathcal{S}_m τ m ∈ S m (32) . Here we consider the case where ( a , f ) (a,f) ( a , f ) ( a ~ , f ~ ) (\widetilde a,\widetilde f) ( a , f ) B p B_p B p
Given the chosen parameters, we can estimate the differences Φ D [ H ] \Phi_D[H] Φ D [ H ] Φ N [ H ] \Phi_N[H] Φ N [ H ] (17) and (18) by analytically computing the ray and chord lengths for the given ray angle and fixed boundary point x ( ψ ) x(\psi) x ( ψ ) ∂ ω R a f ( ψ p , ω ) \partial_\omega R_a f(\psi_p,\omega) ∂ ω R a f ( ψ p , ω ) ω → ω 0 ε τ \omega\to\omega_0^{\varepsilon_\tau} ω → ω 0 ε τ γ 2 = ∣ ω − ω 0 ∣ \gamma^2=|\omega-\omega_0| γ 2 = ∣ ω − ω 0 ∣ ∣ ω − ω 0 ∣ = ε τ ( ω − ω 0 ) |\omega-\omega_0|=\varepsilon_\tau(\omega-\omega_0) ∣ ω − ω 0 ∣ = ε τ ( ω − ω 0 ) ω → ω 0 ε τ \omega\to\omega_0^{\varepsilon_\tau} ω → ω 0 ε τ
Now for γ > 0 \gamma>0 γ > 0 φ + ( γ ) : = H ( ω 0 + ε τ γ 2 ) \varphi_+(\gamma):=H(\omega_0+\varepsilon_\tau\gamma^2) φ + ( γ ) := H ( ω 0 + ε τ γ 2 )
φ + ′ ( γ ) = 2 ε τ γ H ′ ( ω 0 + ε τ γ 2 ) ⇒ lim γ → 0 + 1 2 φ + ′ ( γ ) = lim ω → ω 0 ε τ ∣ ω − ω 0 ∣ 1 2 H ′ ( ω 0 + ( ω − ω 0 ) ) = Φ D [ H ] , \begin{aligned}
\varphi_+'(\gamma)=2\varepsilon_\tau\gamma H'(\omega_0+\varepsilon_\tau\gamma^2) \\
&\Rightarrow\lim_{\gamma\to0^+}\frac{1}{2}\varphi_+'(\gamma)=\lim_{\omega\to\omega_0^{\varepsilon_\tau}}|\omega-\omega_0|^{\frac{1}{2}}H'(\omega_0+(\omega-\omega_0))=\Phi_D[H],
\end{aligned} φ + ′ ( γ ) = 2 ε τ γ H ′ ( ω 0 + ε τ γ 2 ) ⇒ γ → 0 + lim 2 1 φ + ′ ( γ ) = ω → ω 0 ε τ lim ∣ ω − ω 0 ∣ 2 1 H ′ ( ω 0 + ( ω − ω 0 )) = Φ D [ H ] , where we estimate φ + ′ \varphi_+' φ + ′ h h h
1 2 φ + ′ ( γ ) ≈ 1 2 φ + ( γ + h ) − φ + ( γ ) h = 1 2 H ( ω 0 + ε τ ( γ + h ) 2 ) − H ( ω 0 + ε τ γ 2 ) h . (41) \frac{1}{2}\varphi_+'(\gamma)\approx\frac{1}{2}\frac{\varphi_+(\gamma+h)-\varphi_+(\gamma)}{h}=\frac{1}{2}\frac{H(\omega_0+\varepsilon_\tau(\gamma+h)^2)-H(\omega_0+\varepsilon_\tau\gamma^2)}{h}. \tag*{(41)} 2 1 φ + ′ ( γ ) ≈ 2 1 h φ + ( γ + h ) − φ + ( γ ) = 2 1 h H ( ω 0 + ε τ ( γ + h ) 2 ) − H ( ω 0 + ε τ γ 2 ) . (41) Similarly, we can define a second auxiliary function φ − ( γ ) : = H ( ω 0 − ϵ τ γ 2 ) \varphi_{-}(\gamma) := H(\omega_{0} - \epsilon_{\tau}\gamma^{2}) φ − ( γ ) := H ( ω 0 − ϵ τ γ 2 )
φ + ′ ′ ( γ ) = 4 γ 2 H ′ ′ ( ω 0 + ϵ τ γ 2 ) + 2 ϵ τ H ′ ( ω 0 + ϵ τ γ 2 ) , \varphi_{+}^{\prime\prime}(\gamma) = 4\gamma^{2}H^{\prime\prime}(\omega_{0} + \epsilon_{\tau}\gamma^{2}) + 2\epsilon_{\tau}H^{\prime}(\omega_{0} + \epsilon_{\tau}\gamma^{2}), φ + ′′ ( γ ) = 4 γ 2 H ′′ ( ω 0 + ϵ τ γ 2 ) + 2 ϵ τ H ′ ( ω 0 + ϵ τ γ 2 ) , φ − ′ ( γ ) = − 2 ϵ τ γ H ′ ( ω 0 − ϵ τ γ 2 ) . \varphi_{-}^{\prime}(\gamma) = -2\epsilon_{\tau}\gamma H^{\prime}(\omega_{0} - \epsilon_{\tau}\gamma^{2}). φ − ′ ( γ ) = − 2 ϵ τ γ H ′ ( ω 0 − ϵ τ γ 2 ) . Then
lim γ → 0 + 1 4 φ + ′ ′ ( γ ) + 1 4 γ φ − ′ ( γ ) = lim ω → ω 0 + γ 2 H ′ ′ ( ω 0 + ( ω − ω 0 ) ) + ϵ τ 2 H ′ ( ω 0 + ( ω − ω 0 ) ) − ϵ τ 2 H ′ ( ω 0 − ( ω − ω 0 ) ) = Φ N [ H ] . \begin{aligned}
\lim_{\gamma\to0+} \frac{1}{4}\varphi_{+}^{\prime\prime}(\gamma) + \frac{1}{4\gamma}\varphi_{-}^{\prime}(\gamma) \\
&= \lim_{\omega\to\omega_{0}^{+}} \gamma^{2}H^{\prime\prime}(\omega_{0} + (\omega-\omega_{0})) + \frac{\epsilon_{\tau}}{2}H^{\prime}(\omega_{0} + (\omega-\omega_{0})) - \frac{\epsilon_{\tau}}{2}H^{\prime}(\omega_{0} - (\omega-\omega_{0})) = \Phi_{N}[H].
\end{aligned} γ → 0 + lim 4 1 φ + ′′ ( γ ) + 4 γ 1 φ − ′ ( γ ) = ω → ω 0 + lim γ 2 H ′′ ( ω 0 + ( ω − ω 0 )) + 2 ϵ τ H ′ ( ω 0 + ( ω − ω 0 )) − 2 ϵ τ H ′ ( ω 0 − ( ω − ω 0 )) = Φ N [ H ] . These we can estimate using first and second order finite differences
1 4 φ + ′ ′ ( γ ) + 1 4 γ φ − ′ ( γ ) ≈ 1 4 φ + ( γ + h ) − 2 φ + ( γ ) + φ + ( γ − h ) h 2 + 1 4 γ φ − ( γ + h ) − φ − ( γ ) h = 1 4 H ( ω 0 + ϵ τ ( γ + h ) 2 ) − 2 H ( ω 0 + ϵ τ γ 2 ) + H ( ω 0 + ϵ τ ( γ − h ) 2 ) h 2 \begin{aligned}
\frac{1}{4}\varphi_{+}^{\prime\prime}(\gamma) + \frac{1}{4\gamma}\varphi_{-}^{\prime}(\gamma) \approx\frac{1}{4}\frac{\varphi_{+}(\gamma+h)-2\varphi_{+}(\gamma)+\varphi_{+}(\gamma-h)}{h^{2}} + \frac{1}{4\gamma}\frac{\varphi_{-}(\gamma+h)-\varphi_{-}(\gamma)}{h} \\
&= \frac{1}{4}\frac{H(\omega_{0}+\epsilon_{\tau}(\gamma+h)^{2})-2H(\omega_{0}+\epsilon_{\tau}\gamma^{2})+H(\omega_{0}+\epsilon_{\tau}(\gamma-h)^{2})}{h^{2}}
\end{aligned} 4 1 φ + ′′ ( γ ) + 4 γ 1 φ − ′ ( γ ) ≈ 4 1 h 2 φ + ( γ + h ) − 2 φ + ( γ ) + φ + ( γ − h ) + 4 γ 1 h φ − ( γ + h ) − φ − ( γ ) = 4 1 h 2 H ( ω 0 + ϵ τ ( γ + h ) 2 ) − 2 H ( ω 0 + ϵ τ γ 2 ) + H ( ω 0 + ϵ τ ( γ − h ) 2 ) + 1 4 γ H ( ω 0 − ϵ τ ( γ + h ) 2 ) − H ( ω 0 − ϵ τ γ 2 ) h . (42) \quad+ \frac{1}{4\gamma}\frac{H(\omega_{0}-\epsilon_{\tau}(\gamma+h)^{2})-H(\omega_{0}-\epsilon_{\tau}\gamma^{2})}{h}. \tag*{(42)} + 4 γ 1 h H ( ω 0 − ϵ τ ( γ + h ) 2 ) − H ( ω 0 − ϵ τ γ 2 ) . (42) We can also use φ − \varphi_{-} φ − φ + \varphi_{+} φ + Φ D \Phi_{D} Φ D Φ N \Phi_{N} Φ N
With the approximations (5.1) and (5.2), we fix the finite difference step length h = 10 − 4 h = 10^{-4} h = 1 0 − 4 φ + ′ ( γ ) \varphi_{+}^{\prime}(\gamma) φ + ′ ( γ ) φ + ′ ′ ( γ ) + γ − 1 φ − ′ ( γ ) \varphi_{+}^{\prime\prime}(\gamma) + \gamma^{-1}\varphi_{-}^{\prime}(\gamma) φ + ′′ ( γ ) + γ − 1 φ − ′ ( γ ) γ 0 \gamma_{0} γ 0 B p B_{p} B p r o u t = 165 r_{\mathrm{out}} = 165 r out = 165 a 0 = 0.0085 a_{0} = 0.0085 a 0 = 0.0085 − 1 ^{-1} − 1 a p = 0.14 a_{p} = 0.14 a p = 0.14 − 1 ^{-1} − 1 r = 8.25 r = 8.25 r = 8.25
We also consider two different types of simple tangencies illustrated in Figure 6 : (i) a ray which is only tangent to the perturbed rod and never meets any other rods; and (ii) a ray which is tangent to the perturbed rod but first passes through another rod earlier along the ray. Note that in both cases the forward ray from the tangency to the detector meets no other rods as required. The numerical derivative estimates of the different perturbations for the two rays are shown in Figure 7 and Figure 8 , respectively. Proposition 2.19 also gives an alternative way of obtaining the limit values Φ D \Phi_{D} Φ D Φ N \Phi_{N} Φ N ω = ω 0 \omega= \omega_{0} ω = ω 0
Figure 6. Geometric setup illustrating two different simple tangencies. In (i), the tangent ray meets no other rods. In (ii), the tangent ray passes through a second rod.
Figure 7. Numerical estimates of Φ D \Phi_D Φ D Φ N \Phi_N Φ N
Figure 8. Numerical estimates of Φ D \Phi_D Φ D Φ N \Phi_N Φ N
As can be seen from Figure 7 and Figure 8 , the singularity at the tangential ray and vanishing data are captured well, and the angular derivative estimates tend to the values predicted by Proposition 2.19 . However, at that scale it is still hard to judge how accurately the limit value is obtained, in particular if the jump in attenuation values should be computed.
Figure 9 shows the relative difference between the estimated values of Φ D [ H ] \Phi_{D}[H] Φ D [ H ] Φ N [ H ] \Phi_{N}[H] Φ N [ H ] γ = ∣ ω − ω 0 ∣ \gamma= \sqrt{|\omega-\omega_{0}|} γ = ∣ ω − ω 0 ∣ γ < 10 − 2 \gamma< 10^{-2} γ < 1 0 − 2 ∣ ω − ω 0 ∣ < 10 − 4 |\omega-\omega_0| < 10^{-4} ∣ ω − ω 0 ∣ < 1 0 − 4 γ ≈ 10 − 4 \gamma\approx10^{-4} γ ≈ 1 0 − 4 h h h Φ N \Phi_N Φ N h h h γ < h \gamma< h γ < h
Figure 9. Plot of the relative difference between the estimated and known values of Φ D \Phi_D Φ D Φ N \Phi_N Φ N ω \omega ω O ( γ ) = O ( ∣ ω − ω 0 ∣ ) O(\gamma)=O(\sqrt{|\omega-\omega_0|}) O ( γ ) = O ( ∣ ω − ω 0 ∣ )
Rather than a difference in perturbed inputs, from now on we only consider single input ( a , f ) (a,f) ( a , f ) H ( ω ) = R a f ( ψ , ω ) H(\omega)=R_a f(\psi,\omega) H ( ω ) = R a f ( ψ , ω ) γ ∈ [ 5 ⋅ 10 − 4 , 10 − 2 ] \gamma\in[5\cdot10^{-4},10^{-2}] γ ∈ [ 5 ⋅ 1 0 − 4 , 1 0 − 2 ] γ → 0 \gamma\to0 γ → 0 a p a_p a p f p f_p f p B p B_p B p
a p = a 0 + Φ N [ H ] 2 r L τ Φ D [ H ] , f p = e D a ( y τ , θ τ ) 2 r L τ Φ D [ H ] + ( a p − a 0 ) u ( y τ , θ τ ) . a_p=a_0+\frac{\Phi_N[H]}{\sqrt{2rL_\tau\Phi_D[H]}},\qquad f_p=\frac{e^{Da(y_\tau,\theta_\tau)}}{\sqrt{2rL_\tau}}\Phi_D[H]+(a_p-a_0)u(y_\tau,\theta_\tau). a p = a 0 + 2 r L τ Φ D [ H ] Φ N [ H ] , f p = 2 r L τ e D a ( y τ , θ τ ) Φ D [ H ] + ( a p − a 0 ) u ( y τ , θ τ ) . These values with the relative error are shown in Table 1 for the three different variants of ( a ~ , f ~ ) (\tilde{a},\tilde{f}) ( a ~ , f ~ )
Table 1. Attenuation and emission values of one rod reconstructed using the asymptotic limits of Φ D \Phi_D Φ D Φ N \Phi_N Φ N
Acknowledgments
The work of T. Heikkilä, T. Helin, S.R.J., and J.R. was supported by the Research Council of Finland through the Flagship of Advanced Mathematics for Sensing, Imaging and Modelling (decision number 359183). In addition the work of T. Heikkilä and T. Helin was supported by the Business Finland TomoHQ project. The work of S.R.J. and J.R. was also supported by the Jenny and Antti Wihuri Foundation.
The work of S.H. was supported by the Engineering and Physical Sciences Research Council (EPSRC) through grant EP/V007742/1. S.R.J. acknowledges the support of the DKO Trust and the European Mathematical Society for his visit to the University of Manchester, where part of this work was completed, and thanks the University of Manchester for its hospitality. In addition, J.R. was supported by the Emil Aaltonen Foundation.
Data availability statement
Data availability statement. The Python code for our numerical examples with instructions is available at github.com/tommheik/PGET_stability.
Conflict of interest
Conflict of interest. The authors declare no potential conflicts of interest with respect to the research, authorship, or publication of this article.
Declaration of generative AI use
Declaration of generative AI use. During the preparation of this work, the authors used Claude Opus 5/5.5 and Fable 5.1 to help simplify certain calculations, to improve the presentation and to help implement the numerical experiments. The authors reviewed and verified the AI-assisted content.
Computational Engineering, School of Engineering Sciences, Lappeenranta-Lahti University of Technology LUT, Lappeenranta, Finland
Email address: tommi.heikkila@lut.fi
Computational Engineering, School of Engineering Sciences, Lappeenranta-Lahti University of Technology LUT, Lappeenranta, Finland
Email address: tapio.helin@lut.fi
Department of Mathematics, University of Manchester, Manchester, United Kingdom
Email address: sean.holman@manchester.ac.uk
Computational Engineering, School of Engineering Sciences, Lappeenranta-Lahti University of Technology LUT, Lappeenranta, Finland
Email address: shubham.jathar7@gmail.com, shubham.jathar@lut.fi
Computational Engineering, School of Engineering Sciences, Lappeenranta-Lahti University of Technology LUT, Lappeenranta, Finland
Email address: jesse.railo@lut.fi
