Localization of sources in weakly nonlinear fluid systems using linear and quadratic sensitivity analysis

TL;DR

This work introduces a one-shot source localization framework for weakly nonlinear PDEs that extends classical adjoint sensitivity with a quadratic embedding derived from a low-rank Hessian expansion. By constructing linear and quadratic positional embeddings from adjoint fields and Hessian modes, the measurement vector is projected onto a higher-dimensional subspace, and source locations are inferred via a principal-angle (MAP) criterion with a tunable angular tolerance $\gamma$. The method is demonstrated on a 1D viscous Burgers problem and a 2D stratified channel flow, where quadratic embeddings significantly improve localization accuracy, especially in regions where linear sensitivity vanishes and nonlinear interactions dominate. The results show that the approach can disambiguate multiple sources and operate without iterative candidate updates, offering a scalable tool for real-time or large-scale flow diagnostics while pointing to future work on uncertainty quantification and sensor-placement optimization.

Abstract

We develop a framework for localized source detection in dynamical systems governed by nonlinear partial differential equations based on first and second-order sensitivity analysis. Building on the standard adjoint formulation, which relates multiple measurements to external sources through a linear duality relation, we first introduce a linear positional embedding that identifies the source location by aligning the measurement vector with the embedding. To capture weakly nonlinear effects that arise when the source intensity is finite, we then incorporate a quadratic correction represented as a symmetric bilinear operator and approximated via a truncated eigen-expansion obtained with Krylov subspace iterations. This yields quadratic positional embeddings that augment the linear adjoint field, enabling measurement data to be projected onto a higher-dimensional hyperplane, spanned by the linear and quadratic embeddings. A source search algorithm is formulated based on principal angle minimization between this hyperplane and the observation vector, providing a natural probabilistic interpretation of source location. The method operates in a one-shot fashion without iterative updates of candidate source positions, and it can be readily extended to scenarios involving multiple sources. Demonstrations on benchmark inverse problems include perturbation-source identification in the viscous Burgers equation and heat-source detection in a two-dimensional laminar stratified channel. The results with quadratic embeddings show significant improvements in localization accuracy compared with linear adjoint-based sensitivity methods, especially in the region where linear adjoint sensitivity vanishes.

Figures (10)

• Figure 1: Schematic of the linear and quadratic positional embedding for source localization.
• Figure 2: (a) The surface plot of $q$, colored by the evolution of the total forward field $\tilde{u}$, with the resulting probability distribution of the source, $P(x_s)$ using linear embedding (equation \ref{['eqn:linear_angle']}), shown in blue patch, and quadratic embedding (equation \ref{['eqn:quad_angle']}) in red patch. (b.i) Scaled adjoint field $200 s^{\dagger}_1$ (red) and five leading eigenmodes of the Hessian, $\psi_{1k}$ (black) from the first sensor. The black dashed lines mark the base state $u$ at the measurement time. (b.ii) Scaled adjoint field $200 s^{\dagger}_5$ (red) and five leading eigenmodes of the Hessian, $\psi_{5k}$ (black) from the last sensor. (c.i) Leading eigenvalues ($\{\lambda_{1k}\}_1^{10}$) of the Hessian for the first sensor. (c.ii) Leading eigenvalues ($\{\lambda_{5k}\}_1^{10}$) of the Hessian for the last sensor.
• Figure 3: Taylor test for the accuracy of linear and quadratic embeddings, showing (a) the relative difference in the measurement and its approximation using duality relations, and (b) angle between the embedding and the measurement, as a function of the source intensity $I_s$. The linear embedding is shown by the black line, while quadratic embeddings with five and ten eigenmodes are shown by the thin and thick red lines, respectively.
• Figure 4: (a.i) Predicted probability distribution $P(x_s)$ for selected source locations (black dashed lines) with moderate intensity $I_s=0.1$. Blue and red shades show results from linear and quadratic embeddings. black dashed lines mark the true source locations. (a.ii) As in (a.i), but for a smaller source intensity $I_s=0.01$. (b.i) Log-MAP distance as a function of source location $x_s$ and source intensity $I_s$ using linear embedding. (b.ii) As in (b.i), but for quadratic embedding with five Hessian eigenmodes.
• Figure 5: (a-c) Forward temperature field $c(\boldsymbol{x},T)$, first-order sensitivity $s^{\dagger}$, and leading eigenvector of the second-order sensitivity $\psi_1$ for time horizons $T=\{1,4,8\}$. Contours indicate the zero-level curves of the respective fields. (d) The eigenvalues of the Hessian matrix $\mathcal{H}$ for the same sensor but with different time horizons $T=\{1,2,4,8\}$, while we show the data with absolute values, the negative ones are marked with crosses, while the positive ones are marked with circles.