Path-OED for infinite-dimensional Bayesian linear inverse problems governed by PDEs

TL;DR

This work develops a rigorous path-optimized experimental design (path-OED) framework for infinite-dimensional Bayesian linear inverse problems governed by time-dependent PDEs. It introduces a function-space formulation with Bayesian $c$-optimality, discretizes in weighted time-space inner products, and derives efficient, gradient-based methods leveraging adjoint and low-rank updates to compute optimal mobile sensor trajectories. The methodology is instantiated with Bézier and Fourier path parameterizations, including practical constraints such as confinement, acceleration penalties, and obscured regions, and is demonstrated on an advection–diffusion model showing substantial posterior variance reduction for goal functionals. The framework is scalable, adaptable to various linear inverse problems and design criteria, and sets the stage for extensions to multiple sensors, nonlinear models, and online/interactive path planning.

Abstract

We consider infinite-dimensional Bayesian linear inverse problems governed by time-dependent partial differential equations (PDEs) and develop a mathematical and computational framework for optimal design of mobile sensor paths in this setting. The proposed path optimal experimental design (path-OED) framework is established rigorously in a function space setting and elaborated for the case of Bayesian c-optimality, which quantifies the posterior variance in a linear functional of the inverse parameter. The latter is motivated by goal-oriented formulations, where we seek to minimize the uncertainty in a scalar prediction of interest. To facilitate computations, we complement the proposed infinite-dimensional framework with discretized formulations, in suitably weighted finite-dimensional inner product spaces, and derive efficient methods for finding optimal sensor paths. The resulting computational framework is flexible, scalable, and can be adapted to a broad range of linear inverse problems and design criteria. We also present extensive computational experiments, for a model inverse problem constrained by an advection-diffusion equation, to demonstrate the effectiveness of the proposed approach.

Figures (11)

• Figure 1: Left and Middle: Bezier curves of degree $N_{\textnormal{b}} = 5$ with control points plotted as dots. Right: A Fourier path with $N_{\textnormal{f}} = 3$ modes.
• Figure 2: Contours of the RBF $p$ with $\boldsymbol{x}_{D} = (0.5, 0.5)$, $R_{D} = 0.16$, and $\beta \in \{1, 0.1, 0.01, 0.001\}$. The dotted blue circle represents the boundary of the obscured region.
• Figure 3: The velocity field $\boldsymbol{F}(\boldsymbol{x})$ (left) and amplitude $a(t)$ (right) used for the Bezier experiment.
• Figure 4: Time-series of the solution $u_{\text{true}} = \mathcal{S}m_{\text{true}}$ and nominal Bézier path $\boldsymbol{r}_{\textnormal{b}}$ at four times. The red dot is the current position of the sensor and the black dots represent measurement locations.
• Figure 5: The true parameter (left) and three posterior samples corresponding to the Bézier inverse problem.

Theorems & Definitions (23)

• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Corollary 5