Optimal Experimental Design of a Moving Sensor for Linear Bayesian Inverse Problems

2025-09-19

AI Summary

This paper addresses the trajectory optimization of mobile sensors for Bayesian inverse problems, aiming to minimize the posterior uncertainty of parameters in linear partial differential equation (PDE) models. Method: We propose a continuous-observation-based optimal experimental design framework: (i) deriving a closed-form gradient expression of the posterior covariance matrix with respect to the sensor trajectory; (ii) constructing a differentiable objective functional that converges uniformly under temporal discretization refinement; and (iii) incorporating obstacle-avoidance constraints and control parameterization to ensure physical realizability. The method integrates Bayesian inference, linear PDE modeling, and optimal experimental design theory, and solves the resulting constrained nonconvex optimization problem efficiently via an interior-point method. Results: Evaluated on initial condition inversion for a convection–diffusion equation, the optimized trajectories significantly reduce parameter estimation uncertainty while demonstrating computational robustness and engineering practicality.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We optimize the path of a mobile sensor to minimize the posterior uncertainty of a Bayesian inverse problem. Along its path, the sensor continuously takes measurements of the state, which is a physical quantity modeled as the solution of a partial differential equation (PDE) with uncertain parameters. Considering linear PDEs specifically, we derive the closed-form expression of the posterior covariance matrix of the model parameters as a function of the path, and formulate the optimal experimental design problem for minimizing the posterior's uncertainty. We discretize the problem such that the cost function remains consistent under temporal refinement. Additional constraints ensure that the path avoids obstacles and remains physically interpretable through a control parameterization. The constrained optimization problem is solved using an interior-point method. We present computational results for a convection-diffusion equation with unknown initial condition.

Problem

• Optimizing mobile sensor path to minimize Bayesian posterior uncertainty
• Modeling sensor measurements of PDE state with uncertain parameters
• Solving constrained optimization for obstacle avoidance and interpretable paths

Innovation

• Optimizing mobile sensor path for Bayesian inference
• Closed-form posterior covariance with path function
• Interior-point method for constrained path optimization