BFN: Back-and-Forth Nudging for PDE Estimation

• Back-and-Forth Nudging (BFN) is an iterative observer-based methodology for recovering unknown initial states or parameters in time-dependent dynamical systems modeled by PDEs.
• It alternates forward and backward observer evolutions driven by measured outputs, ensuring the estimates converge robustly under proper observability or detectability conditions.
• BFN has been effectively applied in areas like crystallization monitoring, thermoacoustic tomography, and quantum state estimation, demonstrating its versatility and practical impact.

Back-and-Forth Nudging (BFN) is an iterative observer-based methodology for initial state or joint state-parameter estimation in time-dependent dynamical systems, particularly those modeled by partial differential equations (PDEs) or infinite-dimensional linear systems. BFN alternates forward and backward (time-reversed) observer evolutions, using measured outputs to drive the estimate towards consistency with observed data, and provides robust, provable convergence under appropriate observability or detectability assumptions. This framework is notable for theoretical guarantees in both finite- and infinite-dimensional settings, and for successful application to inverse problems where adjoint-based methods or full observability may be unavailable.

1. Mathematical Structure and Algorithmic Procedure

The BFN method addresses the recovery of an unknown initial state (or, more generally, initial condition and parameters) from noisy or incomplete output measurements over a finite time interval. For a broad class of linear (and mildly nonlinear) evolution equations, the system can be formulated as: $$\dot{z}(t) = A(t)z(t),\quad y(t) = C z(t),\quad t \in [0,T],\quad z(0) = z_0$$ where $A(t)$ is a (possibly time-varying) generator of a strongly continuous semigroup on Hilbert space $X$ and $C$ is a bounded linear output operator.

The BFN procedure iteratively alternates the following two steps, with iterates labeled by $n$:

• Forward Observer (Even Iterates $2n$):

$$\dot{z}^{2n}(t) = A(t)z^{2n}(t) - r C^* (C z^{2n}(t) - y(t)), \quad z^{2n}(0) = z^{2n-1}(0)$$

or, for initial iteration, $z^0(0)$ is an arbitrary guess.

• Backward Observer (Odd Iterates $2n+1$):

$$\dot{z}^{2n+1}(t) = A(t) z^{2n+1}(t) + r C^* (C z^{2n+1}(t) - y(t)), \quad z^{2n+1}(T) = z^{2n}(T)$$

This step is numerically equivalent to integrating a time-reversed observer, resetting the initial guess for the next iteration.

The gain parameter $r$ (or, in some formulations, a sequence of vanishing gains $r_j$) determines the contraction properties of the observer dynamics and affects both stability and convergence speed (Brivadis et al., 2020, Aalto, 2015).

Specialized variants exist for state/parameter estimation, such as when incorporating Gauss–Newton updates for unknown parameters between BFN steps (Aalto, 2016). The method extends to PDEs representing spatial dynamics with distributed measurements, such as the transport and wave equations (Brivadis et al., 2021, Bonnefond et al., 2011).

2. Observability, Detectability, and Theoretical Guarantees

The convergence properties of BFN derive from the interplay between system structure (skew-adjointness, dissipativity) and various notions of observability:

• Exact observability: Over $[0,T]$, the observability Gramian $W(0,T)$ is coercive, guaranteeing that the only initial state consistent with zero output is the trivial state. In this setting, BFN converges strongly to the unique minimizer of the quadratic output-error functional (Aalto, 2015).
• Approximate/finite-time observability: Weaker conditions suffice for weak convergence; strong convergence on the observable subspace is possible under periodicity or normality assumptions (Brivadis et al., 2020).
• Weak detectability: If $A$ is dissipative or satisfies a weaker coercivity bound, contraction is preserved for sufficiently large observer gain (Brivadis et al., 2020).

In PDE settings, approximate observability often replaces exact observability due to bounded measurement kernels (e.g., in crystallization models with chord-length measurements) (Brivadis et al., 2021).

A representative convergence theorem (for skew-adjoint generators and vanishing gains) states: If $(A,C)$ is exactly observable on $[0,T]$ and $r_j \to 0$ with $\sum_j r_j = \infty$, then the BFN iterates $z_j^-(T)$ converge to the unique minimum-norm solution of the quadratic output error, i.e.,

$$\min_{x_0} \frac{1}{2}\int_0^T \| y(t) - C x(t; x_0) \|^2 dt$$

with $x(t; x_0)$ the system trajectory from $x_0$ (Aalto, 2015).

3. Forward/Backward Observer Constructions

Across applications, the BFN algorithm is articulated in the Luenberger observer framework, where forward and backward observer equations mirror each other but exchange the sign in the correction term:

• Forward: $-\mu (\text{innovation})$
• Backward: $+\mu (\text{innovation})$

For example, in the transport PDE model for crystallization: $$\partial_t \hat\psi_i^{2n} + G_i(t,r) \partial_r \hat\psi_i^{2n} = -\mu \mathcal{K}^*(\mathcal{K}(\hat\psi^{2n}) - Q)$$ with the backward pass differing by the sign of $\mu$ (Brivadis et al., 2021).

In the context of quantum filtering, the BFN protocol is similarly constructed for the system's density matrix estimation, alternating forward and backward Lindblad-Kalman filters, with analytic convergence analysis for the two-level system (Leghtas et al., 2010).

In dissipative or essentially skew-adjoint systems, feedback correction by a time-dependent operator (e.g., $P(t) = e^{A t} e^{A^* t}$) is required to achieve unbiased convergence (Aalto, 2015).

4. Application Domains and Model-Specific Adaptations

BFN has been applied in diverse settings:

• Crystallization process monitoring: Estimation of particle-size distributions from chord length measurements using population-balance PDEs, with explicit construction of the forward measurement operator and its adjoint (Brivadis et al., 2021, Brivadis et al., 2020, Brivadis et al., 2020).
• Thermoacoustic and seismic tomography: Reconstruction of initial acoustic or displacement fields from surface measurements over time, relying on wave propagation PDEs and geometric observability (Bonnefond et al., 2011).
• Observer-based quantum state estimation: Reconstruction of quantum states from continuous weak measurement data in controlled Hamiltonian systems (Leghtas et al., 2010).
• Joint state and parameter estimation: Hybrid algorithms combining BFN for state recovery and Gauss–Newton updates for unknown parameters, with convergence results for linear and locally for nonlinear/bilinear systems (Aalto, 2016).

These implementations require domain-specific discretization schemes (e.g., characteristic-Euler, finite-difference time-domain, upwind finite-volume), tailored gain selection, and efficient computation of measurement operators and their adjoints.

5. Convergence Analysis and Practical Behavior

Convergence proofs for BFN leverage contraction estimates in suitable weighted norms. For strongly observable or exactly observable systems, each forward/backward sweep yields an exponentially contractive map for the estimation error. In practice, convergence is rapid for observably driven modes, with geometric decay of the $L^2$-error observed in numerical studies (Brivadis et al., 2021, Brivadis et al., 2020, Bonnefond et al., 2011). For systems with only weak detectability, convergence is guaranteed on the observable subspace, with rate depending on the minimal eigenvalue of the Gramian and the chosen gain.

Practical guidelines for implementation include:

• Choice of gain parameter $r$ or $\mu$ balances convergence speed against numerical stability; too large gains induce stiffness, while too small gains slow contraction.
• Number of iterations required is typically modest (10–100) for recovery of dominant state features in high-dimensional settings (Brivadis et al., 2021, Bonnefond et al., 2011).
• The method is robust to partial/incomplete measurements (e.g., sensor sparsity, partial data coverage) and demonstrates stability against moderate levels of measurement noise (Bonnefond et al., 2011, Brivadis et al., 2020).

6. Comparative Features, Limitations, and Extensions

Compared to adjoint-based or direct inversion approaches (e.g., conjugate gradient, Neumann series), BFN offers:

• No need for explicit inversion or adjoint of the measurement operator outside the observer correction.
• Flexibility in accommodating model/data mismatch via direct innovation feedback.
• Applicability to infinite-dimensional and time-varying systems where standard filtering or output-error minimization is ill-posed (Brivadis et al., 2020).

However, BFN may converge more slowly in full-data, noiseless scenarios than Neumann-series methods and requires careful attention to observer gain tuning, especially for discretized stiff PDEs (Bonnefond et al., 2011). In the absence of exact observability, only convergence in the observable subspace is assured. Extensions include hybrid protocols for joint state/parameter estimation, observers for nonlinear and bilinear systems, and adaptation to positivity constraints in quantum systems.

7. Representative Applications and Empirical Performance

Tabulated summary of BFN performance in key domains:

Application	Model Type	Observability/Convergence
Crystallization PSD from CLD	Transport PDE	Geometric decay, $O(50)$ iter.
Thermoacoustic tomography	Wave equation	Geometric decay, robust to noise/sparsity
Quantum state estimation	Lindblad equation	Monotonic Lyapunov decrease, high-fidelity state recovery

In all listed cases, BFN has demonstrated robust empirical performance, rapid error decay, and stability to incomplete/noisy observations (Leghtas et al., 2010, Brivadis et al., 2021, Bonnefond et al., 2011, Brivadis et al., 2020).

---

In summary, the Back-and-Forth Nudging framework universalizes Luenberger-type observer theory to the offline, initial-state inversion context in both finite and infinite-dimensional systems, with solid theoretical and empirical backing across varied disciplines (Brivadis et al., 2021, Brivadis et al., 2020, Aalto, 2015, Aalto, 2016, Leghtas et al., 2010, Bonnefond et al., 2011, Brivadis et al., 2020).