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.

翻译：最优实验设计旨在根据特定统计准则，在实验中选择最具信息量的观测值。在线性情形下（当观测值线性依赖于未知参数时），该问题寻求在特定准则下对设计矩阵A各行权值的最优分配。经典最优实验设计假设设计空间是离散的，因此设计矩阵具有有限维度。然而在许多实际场景中，设计空间是连续取值的，这使得最优实验设计问题转化为在连续值设计空间上的优化问题。目标函数变为定义在概率测度上的泛函，而非有限维向量上的函数。这种视角的转变需要一套能够处理概率测度优化的新方法，而Wasserstein梯度流自然成为合适的选择。本文阐述了最优实验设计目标函数的一阶临界性及凸性性质。在计算方面，采用蒙特卡洛粒子模拟来构建核心算法。该算法被应用于两个椭圆型反问题。