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A global optimum-informed greedy algorithm for A-optimal experimental design

Christian Aarset

TL;DR

This work tackles A-optimal experimental design for sensor placement in PDE-based Bayesian inverse problems. It combines a classical greedy sensor-selection approach with a global optimum-informed correction derived from a convex relaxation’s first-order optimality, enabling explicit identification of sensor indices that should be fixed to 0 or 1. The authors employ a gradient-based criterion to guide removal of underperforming sensors and re-optimization, demonstrating in a Helmholtz PDE setting that the modified greedy approach yields consistent improvements in the A-optimality objective, at the cost of additional computation. The results offer a principled, practically tractable path to more effective OED by bridging the gap between greedy robustness and global optimality, with potential for efficiency-focused refinements.

Abstract

Optimal experimental design (OED) concerns itself with identifying ideal methods of data collection, e.g.~via sensor placement. The \emph{greedy algorithm}, that is, placing one sensor at a time, in an iteratively optimal manner, stands as an extremely robust and easily executed algorithm for this purpose. However, it is a priori unclear whether this algorithm leads to sub-optimal regimes. Taking advantage of the author's recent work on non-smooth convex optimality criteria for OED, we here present a framework for rejection of sub-optimal greedy indices, and study the numerical benefits this offers.

A global optimum-informed greedy algorithm for A-optimal experimental design

TL;DR

This work tackles A-optimal experimental design for sensor placement in PDE-based Bayesian inverse problems. It combines a classical greedy sensor-selection approach with a global optimum-informed correction derived from a convex relaxation’s first-order optimality, enabling explicit identification of sensor indices that should be fixed to 0 or 1. The authors employ a gradient-based criterion to guide removal of underperforming sensors and re-optimization, demonstrating in a Helmholtz PDE setting that the modified greedy approach yields consistent improvements in the A-optimality objective, at the cost of additional computation. The results offer a principled, practically tractable path to more effective OED by bridging the gap between greedy robustness and global optimality, with potential for efficiency-focused refinements.

Abstract

Optimal experimental design (OED) concerns itself with identifying ideal methods of data collection, e.g.~via sensor placement. The \emph{greedy algorithm}, that is, placing one sensor at a time, in an iteratively optimal manner, stands as an extremely robust and easily executed algorithm for this purpose. However, it is a priori unclear whether this algorithm leads to sub-optimal regimes. Taking advantage of the author's recent work on non-smooth convex optimality criteria for OED, we here present a framework for rejection of sub-optimal greedy indices, and study the numerical benefits this offers.
Paper Structure (6 sections, 1 theorem, 5 equations, 3 figures, 2 algorithms)

This paper contains 6 sections, 1 theorem, 5 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A-optimal designs
  3. Optimality for sensor placement
  4. Greedy algorithms
  5. Numerical experiments
  6. Conclusion and outlook

Key Result

theorem 1

Given $m_0\leq m$ and $w\in K_1$, assume (reordering if necessary) that the indices $k$ of $w$ are ordered so that Then $w=w^*$ if and only $w_k=1$ for all $k$ satisfying $\nabla J(w)_k < \nabla J(w)_{m_0+1}$, $w_k=0$ for all $k$ satisfying $\nabla J(w)_k > \nabla J(w)_{m_0}$ and $\sum_{k=1}^mw_k=m_0$.

Figures (3)

  • Figure 1: Left to right: Globally optimal designs $w^*$, global optimum-informed greedy designs $w^{m_0}_*$, greedy designs $w^{m_0}$. Top to bottom: Designs for $m_0=12$, $24$, $36$.
  • Figure 2: A-optimal objectives $\Jc$ for the standard greedy sequence (red), global optimum-informed greedy sequence (green) and non-binary global optimum sequence (blue).
  • Figure 3: Left: $f$. Middle column: Posterior mean, reconstruction error (global optimum-informed design, $m_0=12$). Right column: Posterior mean, reconstruction error (greedy design, $m_0=12$).

Theorems & Definitions (1)

  • theorem 1: Aarset