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Batch-based Bayesian Optimal Experimental Design in Linear Inverse Problems

Sofia Mäkinen, Andrew B. Duncan, Tapio Helin

TL;DR

The paper addresses batch-based Bayesian OED for linear inverse problems on continuous domains by relaxing the discrete sensor design to finite positive measures ${\mathcal{P}}({\mathcal{X}})$. It derives a non-parametric Bayesian inference task corresponding to this relaxed A-optimal design, establishes the concavity of the resulting expected utility, and develops a Wasserstein gradient-flow method to optimize the utility, including first-variation formulas. A practical, regularized tensorized formulation is introduced to preserve feasibility constraints, featuring variance-based concentration ($\mathcal{R}_v$) and inter-sensor repulsion ($\mathcal{R}_r$) terms. Numerical experiments on Poisson and Schrödinger inverse problems demonstrate convergence to empirical sensor configurations and illustrate the role of regularization in shaping sensor layouts. The approach provides a principled, scalable framework for batch sensor placement in continuous domains with potential extensions to other optimality criteria and high-dimensional inverse problems.

Abstract

Experimental design is central to science and engineering. A ubiquitous challenge is how to maximize the value of information obtained from expensive or constrained experimental settings. Bayesian optimal experimental design (OED) provides a principled framework for addressing such questions. In this paper, we study experimental design problems such as the optimization of sensor locations over a continuous domain in the context of linear Bayesian inverse problems. We focus in particular on batch design, that is, the simultaneous optimization of multiple design variables, which leads to a notoriously difficult non-convex optimization problem. We tackle this challenge using a promising strategy recently proposed in the frequentist setting, which relaxes A-optimal design to the space of finite positive measures. Our main contribution is the rigorous identification of the Bayesian inference problem corresponding to this relaxed A-optimal OED formulation. Moreover, building on recent work, we develop a Wasserstein gradient-flow -based optimization algorithm for the expected utility and introduce novel regularization schemes that guarantee convergence to an empirical measure. These theoretical results are supported by numerical experiments demonstrating both convergence and the effectiveness of the proposed regularization strategy.

Batch-based Bayesian Optimal Experimental Design in Linear Inverse Problems

TL;DR

The paper addresses batch-based Bayesian OED for linear inverse problems on continuous domains by relaxing the discrete sensor design to finite positive measures . It derives a non-parametric Bayesian inference task corresponding to this relaxed A-optimal design, establishes the concavity of the resulting expected utility, and develops a Wasserstein gradient-flow method to optimize the utility, including first-variation formulas. A practical, regularized tensorized formulation is introduced to preserve feasibility constraints, featuring variance-based concentration () and inter-sensor repulsion () terms. Numerical experiments on Poisson and Schrödinger inverse problems demonstrate convergence to empirical sensor configurations and illustrate the role of regularization in shaping sensor layouts. The approach provides a principled, scalable framework for batch sensor placement in continuous domains with potential extensions to other optimality criteria and high-dimensional inverse problems.

Abstract

Experimental design is central to science and engineering. A ubiquitous challenge is how to maximize the value of information obtained from expensive or constrained experimental settings. Bayesian optimal experimental design (OED) provides a principled framework for addressing such questions. In this paper, we study experimental design problems such as the optimization of sensor locations over a continuous domain in the context of linear Bayesian inverse problems. We focus in particular on batch design, that is, the simultaneous optimization of multiple design variables, which leads to a notoriously difficult non-convex optimization problem. We tackle this challenge using a promising strategy recently proposed in the frequentist setting, which relaxes A-optimal design to the space of finite positive measures. Our main contribution is the rigorous identification of the Bayesian inference problem corresponding to this relaxed A-optimal OED formulation. Moreover, building on recent work, we develop a Wasserstein gradient-flow -based optimization algorithm for the expected utility and introduce novel regularization schemes that guarantee convergence to an empirical measure. These theoretical results are supported by numerical experiments demonstrating both convergence and the effectiveness of the proposed regularization strategy.
Paper Structure (18 sections, 12 theorems, 115 equations, 5 figures, 2 algorithms)

This paper contains 18 sections, 12 theorems, 115 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Literature overview
  4. Relaxation of the Bayesian inference problem
  5. Bayesian OED in the relaxed setting
  6. A-optimality criterion
  7. Differentiability of the expected utility
  8. Optimization by Wasserstein gradient flow
  9. Preliminaries
  10. First Order Optimality Condition
  11. Particle Approximation and Regularisation
  12. Numerical Simulations
  13. Inverse source problem for one-dimensional Poisson's equation
  14. Time-harmonic Schrödinger equation
  15. Sensitivity of regularization
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose the random variable $(g_\mu, f) \in (Z_{-s}^\mu, U)$ has the joint distribution induced by the marginal $f\in {\mathcal{N}}(0, C_f)$ on $U$ and the likelihood induced by eq:relaxed_inference. Then the conditional distribution of $f$ given $g_\mu$ has a regular Gaussian version with a trace-c

Figures (5)

  • Figure 1: Expected utility w.r.t. to the design variables $x_1$ and $x_2$ in Section \ref{['subsec:poisson']} is illustrated. The black cross represents the optimal configuration $x \approx (0.1614,0.8386)$ found. The red dots mark the true optimal configurations.
  • Figure 2: Trajectories of the particles through iterations on the left panel and the resulting empirical distribution of $\mu$ on the right panel is illustrated in Section \ref{['subsec:poisson']}.
  • Figure 3: The expected utility for single observation and the evolution of the particle ensembles in Section \ref{['subsec:schrodinger']} is illustrated. The black markers on the left plot show the optimal configuration found. The plot on the right visualizes the evolution of the particles from the initial set of particles to the optimal locations.
  • Figure 4: Effect of $\mathcal{R}_v$ illustrated in Section \ref{['subsec:alpha sensitivity']}. The empirical variances of the regularizer $\mathcal{R}_v$ on a log-scale are plotted on the left panel. On the right panel the resulting empirical distributions are plotted.
  • Figure 5: Effect of $\mathcal{R}_r$ illustrated in Section \ref{['subsec:beta sensitivity']}. The distances $d_1(\mu_N^1,\mu_N^2),...,d_7(\mu_N^7,\mu_N^8)$ between particle clouds on a log-scale are plotted on the left panel. On the right panel the comparison of two resulting empirical distributions are plotted.

Theorems & Definitions (29)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Example 2.3: An empirical measure
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 19 more