Lower Confidence Bound Adaptive Search

• The Lower Confidence Bound-based adaptive search algorithm integrates Bayesian estimation with stochastic elliptical-fire models to forecast wildfire perimeters under uncertainty.
• It employs nonlinear elliptical dynamics, vertex-wise stochastic perturbations, and parallel computation for high-fidelity fire front tracking and adaptive monitoring.
• By optimizing sensor deployment via recursive filtering and policy evaluation, the algorithm significantly reduces estimation error and improves risk-informed fire planning.

Stochastic nonlinear elliptical-growth fire front models provide a mathematically rigorous framework for representing and forecasting wildfire perimeter evolution under environmental and fuel-driven variability. These models depart from classical isotropic or deterministically-parameterized fire spread equations by encoding ellipsoidal geometry, nonlinear empirical fire behavior relations, and environmental stochasticity. Their applications span risk-informed wildfire forecasting, strategic fuel management, adaptive sensor deployment, and uncertainty quantification in operational fire modeling (Pais et al., 2019, Papaioannou et al., 16 Jan 2026).

1. Mathematical Formulation of Stochastic Nonlinear Elliptical-Growth

The central object of stochastic nonlinear elliptical-growth models is a discretized fire perimeter or scarring region, whose propagation is governed by empirical, nonlinear elliptical dynamics subject to random fluctuations in wind, fuel, and spread parameters.

Let the fire front at time $t$ be represented by $N$ perimeter vertices $X_t = \left[x_t^1, \ldots, x_t^N\right]^\top$, $x^i_t \in \mathbb{R}^2$. Each vertex's advance is dictated by nonlinear, coupled systems such as those derived from Richards's elliptical-growth model. With a discrete-time increment $\Delta t$,

$x^i_{t} = x^i_{t-1} + \Delta t\,\dot x^i_{t-1},$

where $\dot x^i$ is a local nonlinear velocity dependent on ellipsoid parameters. These parameters—axis lengths, orientation, and eccentricity—are themselves functions of random environmental variables: wind direction $\theta(i)$, wind speed $w_s(i)$, and fuel-driven spread rate $r_f(i)$, often modeled as von Mises or rectified Gaussian random variables, $$\theta(i)\sim\mathrm{vonMises}(\mu_{\theta(i)},\kappa)$$ and $w_s(i), r_f(i)\sim \mathcal N_R(\cdot)$ (Papaioannou et al., 16 Jan 2026).

Within grid-based models such as Cell2Fire, each burning cell “deploys” an elliptical firelet whose axes evolve by

$$a = \tfrac{1}{2}(R_h + R_b)\Delta t, \qquad b = R_f\Delta t, \qquad R_f = \tfrac{R_h + R_b}{2\,LB},$$

where $R_h$, $R_b$ are head/back rates of spread, and $LB$ is the empirical length-to-breadth ratio from the Canadian FBP System (Pais et al., 2019).

The stochastic process kernel for one-step evolution is given by

$$p(X_t \mid X_{t-1}) = \int \delta(X_t - \xi(X_{t-1}, E))\, P_E(dE),$$

where $E$ collects the random environmental parameters and $\xi$ denotes the deterministic mapping from process state and environment (Papaioannou et al., 16 Jan 2026).

2. Nonlinearity and Environmental Coupling

Nonlinearity arises at several levels:

• Empirical Spread Functions: The rates $R_h$, $R_b$, and $R_f$ depend on complex, nonlinear empirical relations as a function of wind speed, direction, fuel, moisture, and slope. For example, $R_h = \mathrm{FBP}_h(U,\theta_w,s,\text{fuel},M)$ encodes regression-based dependencies from the FBP system (Pais et al., 2019).
• Geometric Coupling: The propagation in direction $\phi$ from each cell is typically modeled by

$$\mathrm{ROS}(\phi) = \frac{a\,b}{\sqrt{(b \cos \phi)^2 + (a \sin \phi)^2}},$$

with $a, b, \phi$ dynamically coupled to local environmental states and the geometry of the existing fire perimeter (Pais et al., 2019).

• Interacting Fronts: The aggregate fire perimeter is the result of simultaneous, interacting elliptical firelets with potentially overlapping, stochastically-perturbed envelopes.

Temporal nonlinearity is introduced as weather variables are allowed to change at fixed intervals, leading to time-varying stochasticity in axis growth and orientation at all perimeter locations (Pais et al., 2019).

3. Stochastic Components and Uncertainty Quantification

Stochasticity, essential for probabilistic risk analysis in fire modeling, is introduced at multiple layers:

• Ignition Sampling: Initial ignition sites are drawn according to user-specified spatial probability distributions over the landscape grid (Pais et al., 2019).
• Spread Rate Perturbation: Each time step, spread rates (e.g., $R_h$) are randomly perturbed, often as $R'_h \sim \mathcal{N}(R_h, \sigma^2)$, with coefficient of variation controlling stochastic intensity (Pais et al., 2019).
• Scenario-Based Weather Realizations: Fire progression is simulated under random draws from predefined weather scenario files, yielding burn-probability maps as empirical frequencies of cell fires across Monte Carlo runs (Pais et al., 2019).
• Vertexwise Stochasticity in Front Models: In vertex-based front-tracking formulations, environmental variables are modeled as independent random processes across vertices; independence and stationarity are adopted as simplifying assumptions (Papaioannou et al., 16 Jan 2026).

Uncertainty representation is further codified by posterior beliefs estimated via particle filtering or approximate Gaussian recursion, providing sample-based estimates of burn probabilities or perimeter location (Papaioannou et al., 16 Jan 2026).

4. Numerical Methods and Computational Implementation

The discrete-time, cell-based approach exemplified by Cell2Fire utilizes parallel, object-oriented architectures (Python and C++), where at each time step burning cells process neighborhood ignition and advance messages via an embarrassingly parallel message-passing algorithm. Profiling indicates 80% of runtime in message passing, supporting strong and weak scaling efficiencies of 75–80% up to 32 threads—achieving 15–20x speedups over serial implementations (Pais et al., 2019).

Front-tracking implementations for adaptive monitoring employ particle filters for fully nonlinear, non-Gaussian uncertainty propagation, using Sequential Importance Resampling (SIR) to approximate the evolving belief over fire perimeter states. Gaussian (e.g., EKF) filters are deployed only under linearized, compact-parameter regimes (Papaioannou et al., 16 Jan 2026).

5. Estimation, Control, and Adaptive Monitoring

A major extension of stochastic nonlinear elliptical-growth models is their integration into closed-loop estimation and adaptive control frameworks. The recursive Bayesian estimator is formulated as

$$\mathcal{B}_t^-(X_t) = \int p(X_t|X_{t-1}) \mathcal{B}_{t-1}(X_{t-1})\,dX_{t-1}, \quad \mathcal{B}_t(X_t) \propto p(Z_t|X_t, y_t) \mathcal{B}_t^-(X_t),$$

where $Z_t$ are sensor measurements and $y_t$ is the sensing agent's pose. The likelihood $p(Z_t|X_t, y_t)$ encodes detection rates and spatial point-process measurement models (Papaioannou et al., 16 Jan 2026).

Adaptive monitoring is posed as a stochastic optimal control problem: the finite-horizon Markov decision process (MDP) state combines the Bayesian belief and the agent's pose. The cost function is the posterior uncertainty-weighted risk,

$$\mathcal{C}(s_{t+1}) = \frac{1}{\omega} \sum_{\text{cell } \varepsilon} \mathcal{R}(\varepsilon) \det(\Sigma_{t+1}^{\varepsilon}),$$

with $\Sigma^{\varepsilon}$ the sample covariance over perimeter particles in cell $\varepsilon$ (Papaioannou et al., 16 Jan 2026).

Policy optimization leverages a lower confidence bound (LCB) algorithm over open-loop action sequences, guaranteeing asymptotic policy selection optimality with convergence rate $O(\ln n/n)$ (Papaioannou et al., 16 Jan 2026). Empirically, non-myopic ($T>1$) control policies sharply decrease estimation RMSE by focusing sensing on high-uncertainty, high-risk regions.

6. Validation, Benchmarking, and Performance Metrics

Stochastic nonlinear elliptical-growth models are validated against established fire simulators (e.g., Prometheus) and empirical fire scars. Cell2Fire achieves high agreement with Prometheus, with $1-\mathrm{MSE} \approx 0.90$–$0.98$ and SSIM 0.70–0.95 across multiple fire cases (400–550k ha domains) (Pais et al., 2019). Real scar overlays on Landsat-derived burn maps indicate vector perimeter agreement within 10–20%.

Simulation-based monitoring studies confirm that multi-step predictive control yields significant improvements in fire front localization accuracy over myopic strategies, and that particle filtering in nonlinear stochastic growth settings enables robust uncertainty quantification (Papaioannou et al., 16 Jan 2026).

7. Applications and Operational Significance

Stochastic nonlinear elliptical-growth fire front models underpin a range of operational and analytical use-cases:

• Risk-Informed Fire Planning: Probabilistic burn scar maps provide quantitative input for evacuation, resource allocation, and landscape management.
• Strategic Sensor Deployment: Integration with mobile agent control policies supports adaptive surveillance aimed at minimizing risk-weighted estimation uncertainty (Papaioannou et al., 16 Jan 2026).
• Real-Time Forecasting: Efficient, parallelized simulation enables rapid scenario exploration and near-real-time projection of evolving fire fronts under varying meteorological realizations (Pais et al., 2019).
• Comparative Model Assessment: Metrics such as MSE, SSIM, and Frobenius norms provide quantitative benchmarks for comparing stochastic elliptical-growth models against industry-standard simulators and empirical fire records.

These models facilitate both high-fidelity deterministic forecasting and statistically robust risk assessment in the face of inherent environmental variability and data uncertainty.

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Key references:

• Cell2Fire: "Cell2Fire: A Cell Based Forest Fire Growth Model" (Pais et al., 2019)
• Adaptive Monitoring: "Adaptive Monitoring of Stochastic Fire Front Processes via Information-seeking Predictive Control" (Papaioannou et al., 16 Jan 2026)