Optimal design for linear models via gradient flow
Ruhui Jin, Martin Guerra, Qin Li, Stephen Wright
TL;DR
This work tackles optimal experimental design when the design space is continuous by reframing the problem as optimization over probability measures endowed with the Wasserstein metric. It introduces a Wasserstein gradient-flow framework, approximated via Monte Carlo particles, to minimize or maximize standard OED criteria (A-optimal and D-optimal) in the continuous setting. The paper establishes first-order criticality conditions, proves convexity in the L2 sense, and analyzes particle-flow simulation error, providing a groundwork for convergence and stability. The authors demonstrate the approach on linearized inverse problems, notably electrical impedance tomography and one-dimensional Darcy flow, revealing informative design structures and highlighting initialization-dependent nonconvexities in Wasserstein space. The methodology offers a scalable, measure-theoretic path to continuous OED with potential impact across PDE-constrained inference and sensor-placement problems.
Abstract
Optimal experimental design (OED) aims to choose the observations in an experiment to be as informative as possible, according to certain statistical criteria. In the linear case (when the observations depend linearly on the unknown parameters), it seeks the optimal weights over rows of the design matrix A under certain criteria. Classical OED assumes a discrete design space and thus a design matrix with finite dimensions. In many practical situations, however, the design space is continuous-valued, so that the OED problem is one of optimizing over a continuous-valued design space. The objective becomes a functional over the probability measure, instead of over a finite dimensional vector. This change of perspective requires a new set of techniques that can handle optimizing over probability measures, and Wasserstein gradient flow becomes a natural candidate. Both the first-order criticality and the convexity properties of the OED objective are presented. Computationally Monte Carlo particle simulation is deployed to formulate the main algorithm. This algorithm is applied to two elliptic inverse problems.