Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 70 / 4 / b

In first Born order the unknown potential enters linearly. Let $M_i$ multiply by the known incident field, $\mathcal{G}$ be the outgoing Green integral over the known support $D$, and $\mathcal{R}$ restrict the resulting field to the measurement set. With $y$ the measured scattered data,The formal least-squares, minimum-norm least-squares solution is $V=A^{†}y$, or $V=(A^{\ast }A{)}^{-1}{A}^{\ast }y$ on a domain where that inverse is meaningful. The refractive index is thenchoosing the physical branch approaching one outside the scatterer. At weak contrast $n-1\simeq V/(2k^2)$.

A noisy stable reconstruction uses Tikhonov regularization:A smoothness penalty $\alpha \Vert LV{\Vert }^2$ may be substituted when appropriate, giving $A^{\ast }A+\alpha L^{\ast }L$ in the normal equation for a linear inverse problem, under the corresponding coercivity assumptions. The parameter balances noise amplification and smoothing bias.

The data geometry matters: knowing a scattered field on a measurement surface does not automatically mean knowing it throughout the support. A single incident wave and fixed-frequency far field sample only a shifted sphere of the potential's Fourier transform, and in general do not determine an arbitrary three-dimensional potential uniquely. Tikhonov regularization stabilizes a selected minimum-norm least-squares solution; it cannot create missing information or guarantee uniqueness from inadequate data. If the total field and its derivatives were known throughout $D$, the first-Born equation could instead be inverted formally as $V\simeq -(\Delta +k^2){\psi }_s/{\psi }_i$ where ${\psi }_i\ne 0$, but differentiation still amplifies measurement noise.