Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 335 / 2

Table of contents
- i 2
  - Solution i
- ii 2
  - Solution ii
- iii 2
  - Solution iii

i

Solution

With the scattering potential

V (r) = k_{0}^{2} (1 - n^{2} (r))

, the total field obeys

(\nabla^{2} + k_{0}^{2}) ψ = V ψ .

The outgoing Green function is

G_{0} (r) = e^{i k_{0} ∣ r ∣} / (4 π ∣ r ∣)

. The Lippmann-Schwinger equation and the first Born approximation give

ψ_{s} (r) s im e q - \int_{D} G_{0} (r - r^{'}) V (r^{'}) e^{i k_{0} x_{0} \cdot r^{'}}, d r^{'} .

For

r \to \infty

,

G_{0} (r - r^{'}) \sim \frac{e ^{i k_{0} r}}{4 π r} e^{- i k_{0} r \cdot r^{'}},

so

ψ_{s} (r) \sim \frac{e ^{i k_{0} r}}{r} f_{\infty} (x_{0}, r), f_{\infty} = - \frac{1}{4 π} \int_{D} V (r^{'}) e^{- i q \cdot r^{'}} d r^{'},

where

q = k_{0} (r - x_{0})

is the momentum transfer. Thus the far-field pattern is

- 1/ (4 π)

times the Fourier transform of

V

at the measured transfer vectors. If sufficiently many incident directions and frequencies supply all

q

, the formal reconstruction is

V (r) = - 4 π F^{- 1} {f_{\infty} (q)} (r),

with constants adjusted to the chosen Fourier convention.

ii

Solution

Use

(F g) (q) = \int_{R^{3}} g (r) e^{- i q \cdot r} d r, F^{- 1} h = \frac{1}{( 2 π ) ^{3}} \int h (q) e^{i q \cdot r} d q .

The adjoint operator is consequently

F^{*} = (2 π)^{3} F^{- 1} .

With a unitary Fourier normalization, this is simply

F^{*} = F^{- 1}

.

At fixed incident direction and wavenumber, define

(T V) (r) = - \frac{1}{4 π} \int_{D} e^{- i k_{0} (r - x_{0}) \cdot r} V (r) d r .

Taking the complex conjugate of the kernel gives

(T^{*} g) (r) = - \frac{1}{4 π} \int_{S^{2}} e^{i k_{0} (r - x_{0}) \cdot r} g (r) d S (r), r \in D .

The least-squares minimizer of

∥ T V - f_{\infty} ∥^{2}

satisfies the normal equation for a linear inverse problem

T^{*} T V = T^{*} f_{\infty}

. Fourier inversion on the measured transfer-vector set is exactly the corresponding Moore--Penrose reconstruction

T^{†} f_{\infty}

; hence the formal solution in part (i) is the minimum-norm least-squares solution when the data are incomplete or inconsistent.

iii

Solution

The inverse problem is ill posed for several related reasons. At one frequency and one incident direction, the data sample

V

only on the two-dimensional Ewald surface

q = k_{0} (r - x_{0})

, so three-dimensional reconstruction is nonunique without more illuminations or prior information. Finite aperture and a bounded frequency band omit additional Fourier components and limit resolution. The forward map is compact, so its singular values tend to zero and inversion strongly amplifies measurement noise. Finally, the Born approximation neglects multiple scattering; model error becomes significant when the contrast or support is too large, while uncertainty in the incident field, measurement geometry, and domain

D

creates further instability.

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