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Beyond Linearity — When the Forward Model Is Nonlinear

So far we have treated the forward operator $\mathcal{A}$ as linear. Many RF imaging problems, however, involve nonlinear forward models:

Microwave tomography: The scattered field depends nonlinearly on the permittivity distribution (multiple scattering beyond the Born approximation).
Full-waveform inversion: The wave equation maps medium parameters to recorded waveforms nonlinearly via multiple reflections.
Phase retrieval: Only $|y|^2$ is measured (intensity-only), and the phase is lost — a fundamentally nonlinear problem.

The linearization-based approach — compute the Fréchet derivative, apply regularized linear inversion, update — is the workhorse of practical nonlinear imaging. Born iterative and distorted-Born iterative methods are the radar-specific instances.

This section is intentionally brief: most of our forward models in Parts II–III use the linearized (Born) approximation, and the full nonlinear theory is treated in Chapters 5–6 when we derive the scattering equations.

Definition:
Nonlinear Inverse Problems

A nonlinear inverse problem seeks $x$ from the equation

$\mathcal{F}(x) = y,$

where $\mathcal{F} \colon \mathcal{D}(\mathcal{F}) \subseteq \mathcal{X} \to \mathcal{Y}$ is a nonlinear operator. The Fréchet derivative $\mathcal{F}'(x)$ at a point $x$ is the bounded linear operator satisfying

$\mathcal{F}(x + h) = \mathcal{F}(x) + \mathcal{F}'(x)\,h + o(\|h\|).$

$\mathcal{F}'(x)$ is the Jacobian of the forward operator and plays the role of $\mathcal{A}$ in the linear theory.

Nonlinear ill-posedness adds to the linear difficulties:

Non-uniqueness from nonlinearity (multiple solutions, local minima).
Non-convexity of the residual functional.
Dependence of the linearization on the current iterate.

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Definition:
The Gauss–Newton Method and IRGNM

The Gauss–Newton method linearises $\mathcal{F}$ at the current iterate $x_n$ and solves the resulting linear inverse problem:

$x_{n+1} = x_n + \Delta x_n,$

where $\Delta x_n$ solves the linearised problem

$\mathcal{F}'(x_n)\,\Delta x = y^\delta - \mathcal{F}(x_n).$

Since $\mathcal{F}'(x_n)$ is typically ill-posed (a compact operator), the linearised problem must itself be regularised. Using Tikhonov regularisation on each linearised step gives the iteratively regularised Gauss–Newton method (IRGNM):

$\Delta x_n = \arg\min_{\Delta x} \left\{ \|\mathcal{F}'(x_n)\Delta x - (y^\delta - \mathcal{F}(x_n))\|^2 + \alpha_n \|\Delta x + x_n - x_0\|^2 \right\},$

where $\alpha_n$ is a decreasing sequence of regularisation parameters and $x_0$ is an initial guess (e.g., the matched-filter image).

The IRGNM is the standard method for nonlinear ill-posed problems. The decreasing $\alpha_n$ allows progressively finer resolution as the iterate approaches the solution. A typical choice is $\alpha_n = \alpha_0 q^n$ with $q \in (0,1)$ and $\alpha_0$ chosen via the discrepancy principle at the first iteration.

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Definition:
The Levenberg–Marquardt Method

The Levenberg–Marquardt method is closely related to Gauss–Newton but uses the discrepancy principle to select $\alpha_n$ adaptively at each step:

Choose $\alpha_n > 0$ such that

$\|\mathcal{F}'(x_n)(x_{n+1} - x_n) - (y^\delta - \mathcal{F}(x_n))\| = \tau_n \|y^\delta - \mathcal{F}(x_n)\|,$

where $\tau_n < 1$ . This ensures that each step reduces the residual by a fixed fraction. The update is:

$x_{n+1} = x_n + (\mathcal{F}'(x_n)^*\mathcal{F}'(x_n) + \alpha_n I)^{-1} \mathcal{F}'(x_n)^*(y^\delta - \mathcal{F}(x_n)).$

Levenberg–Marquardt interpolates between Gauss–Newton ( $\alpha_n \to 0$ ) and gradient descent ( $\alpha_n \to \infty$ ). Far from the solution (large residual), $\alpha_n$ is large and the step is short and robust. Near the solution (small residual), $\alpha_n$ is small and the method converges rapidly.

Theorem: Convergence of the IRGNM

Let $\mathcal{F}$ be Fréchet differentiable with a Lipschitz-continuous derivative in a ball $B_\rho(x^\dagger)$ . Assume a nonlinear source condition:

$x_0 - x^\dagger = (\mathcal{F}'(x^\dagger)^* \mathcal{F}'(x^\dagger))^{\mu/2} w, \qquad \|w\| \leq E,$

for some $\mu \in (0, 1]$ . If $\alpha_n = \alpha_0 q^n$ with appropriate $\alpha_0, q$ and the iteration is stopped by the discrepancy principle at $n^*(\delta)$ , then

$\|x_{n^*} - x^\dagger\| = O\bigl(\delta^{2\mu/(2\mu+1)}\bigr) \quad \text{as } \delta \to 0.$

The convergence rate matches the linear theory. The nonlinearity is handled by the fact that the linearisation error is of higher order: $\|\mathcal{F}(x_n) - \mathcal{F}(x^\dagger) - \mathcal{F}'(x^\dagger)(x_n - x^\dagger)\| = o(\|x_n - x^\dagger\|)$ , which becomes negligible as $x_n \to x^\dagger$ .

Proof

Error decomposition at each step

The error $e_n = x_n - x^\dagger$ satisfies a recurrence involving the linearisation residual. The linearisation error is bounded by the Lipschitz constant of $\mathcal{F}'$ times $\|e_n\|^2$ .

Induction and convergence

Under the source condition and with appropriate $\alpha_0$ , an induction argument shows $\|e_n\|$ decreases geometrically until the discrepancy stopping criterion is triggered, after which the error is $O(\delta^{2\mu/(2\mu+1)})$ . $\blacksquare$

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Example: Born Iterative Method for Microwave Tomography

In microwave tomography, the forward model maps the contrast function $\chi(\mathbf{r}) = \varepsilon_r(\mathbf{r}) - 1$ (relative permittivity minus 1) to the scattered field $E^s$ via the Lippmann–Schwinger integral equation:

$E^s(\mathbf{r}) = k_0^2 \int_\Omega G_0(\mathbf{r}, \mathbf{r}') \chi(\mathbf{r}') E(\mathbf{r}')\,d\mathbf{r}',$

where $G_0$ is the free-space Green's function and $E = E^i + E^s$ is the total field. The nonlinearity arises because $E$ depends on $\chi$ through the same equation.

Describe the Born iterative method (BIM) as a special case of IRGNM.

Solution

Linearisation (Born approximation)

At each iteration, approximate the total field by the field from the previous iterate: $E \approx E_{n}$ . This linearises the forward operator:

$\mathcal{F}'(x_n)\,\delta\chi = k_0^2 \int_\Omega G_0(\mathbf{r},\mathbf{r}')\,\delta\chi(\mathbf{r}')\, E_n(\mathbf{r}')\,d\mathbf{r}'.$

This is a linear integral operator (compact, hence ill-posed) — exactly the type of operator from Sections 2.1–2.4.

IRGNM update

Solve the Tikhonov-regularised linear problem:

$\chi_{n+1} = \chi_n + \arg\min_{\delta\chi}\left\{ \|\mathcal{F}'(x_n)\delta\chi - (E^{s,\delta} - E^s_n)\|^2 + \alpha_n\|\delta\chi + \chi_n\|^2\right\}.$

Then recompute the forward field $E_{n+1}$ using Maxwell's equations (or the Lippmann–Schwinger equation) with $\chi_{n+1}$ .

Convergence considerations

BIM converges when the contrast $\|\chi\|$ is small (weak scattering) so the Born approximation is accurate. For strong scatterers (metals, high-permittivity media), the distorted-Born iterative method (DBIM) uses the Green's function of the current background rather than the free-space $G_0$ , improving convergence significantly.

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⚠️Engineering Note

Computational Cost of Nonlinear Imaging

Each IRGNM iteration requires: (1) A forward solve: compute $\mathcal{F}(x_n)$ by solving Maxwell's equations (e.g., FDTD or FEM) — $O(N^3)$ for 3D problems. (2) A Jacobian solve: compute $\mathcal{F}'(x_n)$ and apply it — another $O(N^3)$ solve per measurement. (3) A regularised linear solve: the Tikhonov normal equation at the current linearisation point.

For a $100^3$ grid (modest 3D imaging domain), a single forward solve takes seconds to minutes on current hardware. The IRGNM typically needs $10$ – $50$ outer iterations, each with multiple inner CG iterations.

This cost is why most operational RF imaging systems use the Born approximation (single linearisation, no iteration) or limit iteration to 3–5 steps. Full nonlinear inversion is feasible only offline (post-processing) or with GPU-accelerated solvers and reduced-order models.

Common Mistake: Nonlinear Imaging Can Converge to a Local Minimum

Mistake:

Initialising the IRGNM from a zero starting point $x_0 = 0$ (no prior information) for a strongly nonlinear problem, expecting global convergence.

Correction:

The IRGNM convergence theorem (TConvergence of the IRGNM) assumes that the initial guess $x_0$ is sufficiently close to $x^\dagger$ (within the ball $B_\rho(x^\dagger)$ ). For strongly scattering media, $\rho$ can be very small and $x_0 = 0$ may not satisfy this condition.

Practical mitigation strategies:

Use the matched-filter (backprojection) image as $x_0$ — it is often close enough to the true solution for the linearisation to be valid.
Perform a frequency continuation sweep: start at low frequencies (large $\lambda$ , mild nonlinearity, large convergence basin) and progressively increase frequency.
Use a multi-scale approach: solve a coarsely discretised problem first, then refine the grid and warm-start from the coarse solution.

Methods for Nonlinear Inverse Problems

Method	Key Idea	Convergence Rate	Computational Cost	Best For
Born approximation (single)	Linearise once, no iteration	Not convergent (one step)	One forward solve + one linear solve	Weak scattering; real-time
Born iterative (BIM)	Iterate linearisation with free-space Green's function	Geometric (slow, weak scattering)	$O(N_{\text{iter}})$ forward solves	Moderate contrast
Distorted-Born iterative (DBIM)	Iterate with background Green's function	Faster than BIM	More expensive per step (Jacobian with background)	Higher contrast
IRGNM (Tikhonov inner)	Tikhonov at each linearisation	$O(\delta^{2\mu/(2\mu+1)})$ optimal	$O(N_{\text{iter}})$ forward + adjoint solves	General nonlinear ill-posed problems
Levenberg–Marquardt	Adaptive $\alpha_n$ via discrepancy	Same as IRGNM	Similar to IRGNM	Robust to poor initialisation

Key Takeaway

Nonlinear inverse problems arise whenever the forward model involves multiple scattering, nonlinear constitutive relations, or intensity-only measurements. The iteratively regularised Gauss–Newton method (IRGNM) is the standard approach: linearise at each step and apply Tikhonov regularisation to the linearised problem. Convergence rates match the linear theory under appropriate source conditions — the nonlinearity becomes negligible near the solution. In RF imaging, the Born iterative method is the specific instantiation of IRGNM for electromagnetic scattering, using the Lippmann–Schwinger equation as the forward model. Computational cost limits full nonlinear inversion to offline processing or GPU-accelerated implementations.

Nonlinear Inverse Problems