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Submodularity of the expected information gain in infinite-dimensional linear inverse problems

Alen Alexanderian, Steven Maio

TL;DR

This work addresses optimal sensor placement for infinite-dimensional Bayesian linear inverse problems constrained by PDEs by maximizing the expected information gain (EIG). It proves that EIG is monotone and submodular in a Hilbert-space setting, ensuring that a greedy sensor-selection strategy achieves a near-optimal solution with a $(1-1/e)$ guarantee, independent of discretization. The paper derives a posterior-covariance representation using a prior-preconditioned data-misfit Hessian and develops a Hilbert-space Sherman–Morrison–Woodbury update for rank-one sensor additions, enabling efficient updates. It also presents computational strategies, including a measurement-space formulation and lazy greedy optimization, to exploit problem structure and achieve scalable OED for PDE-based inverse problems, with implications for real-time decision-making in digital twins and related frameworks.

Abstract

We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor data, and focus on the problem of finding sensor placements that maximize the expected information gain (EIG). This study is motivated by optimal sensor placement for linear inverse problems constrained by partial differential equations (PDEs). We consider measurement models where each sensor collects a single-snapshot measurement. This covers sensor placement for inverse problems governed by linear steady PDEs or evolution equations with final-in-time observations. It is well-known that in the finite-dimensional (discretized) formulations of such inverse problems, EIG is a monotone submodular function. This also entails a theoretical guarantee for greedy sensor placement in the discretized setting. We extend the result on submodularity of the EIG to the infinite-dimensional setting, proving that the approximation guarantee of greedy sensor placement remains valid in the infinite-dimensional limit. We also discuss computational considerations and present strategies that exploit problem structure and submodularity to yield an efficient implementation of the greedy procedure.

Submodularity of the expected information gain in infinite-dimensional linear inverse problems

TL;DR

This work addresses optimal sensor placement for infinite-dimensional Bayesian linear inverse problems constrained by PDEs by maximizing the expected information gain (EIG). It proves that EIG is monotone and submodular in a Hilbert-space setting, ensuring that a greedy sensor-selection strategy achieves a near-optimal solution with a guarantee, independent of discretization. The paper derives a posterior-covariance representation using a prior-preconditioned data-misfit Hessian and develops a Hilbert-space Sherman–Morrison–Woodbury update for rank-one sensor additions, enabling efficient updates. It also presents computational strategies, including a measurement-space formulation and lazy greedy optimization, to exploit problem structure and achieve scalable OED for PDE-based inverse problems, with implications for real-time decision-making in digital twins and related frameworks.

Abstract

We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor data, and focus on the problem of finding sensor placements that maximize the expected information gain (EIG). This study is motivated by optimal sensor placement for linear inverse problems constrained by partial differential equations (PDEs). We consider measurement models where each sensor collects a single-snapshot measurement. This covers sensor placement for inverse problems governed by linear steady PDEs or evolution equations with final-in-time observations. It is well-known that in the finite-dimensional (discretized) formulations of such inverse problems, EIG is a monotone submodular function. This also entails a theoretical guarantee for greedy sensor placement in the discretized setting. We extend the result on submodularity of the EIG to the infinite-dimensional setting, proving that the approximation guarantee of greedy sensor placement remains valid in the infinite-dimensional limit. We also discuss computational considerations and present strategies that exploit problem structure and submodularity to yield an efficient implementation of the greedy procedure.
Paper Structure (15 sections, 6 theorems, 56 equations, 1 figure, 1 algorithm)

This paper contains 15 sections, 6 theorems, 56 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation and definitions
  4. Bayesian linear inverse problems and optimal sensor placement
  5. Submodular function optimization
  6. Incorporating the experimental design in the inverse problem
  7. Design-dependent posterior covariance operator
  8. An update formula and monotonicity of $\mathrm{C}_\text{post}(S)$
  9. The OED problem
  10. The expected information gain and its submodularity
  11. Some computational considerations
  12. A measurement space approach to computing the marginal gains
  13. Exploiting submodularity via a lazy greedy procedure
  14. Conclusion
  15. Proof of Lemma \ref{['lem:basic']}

Key Result

Lemma 3.1

The operator $\mathrm{H}$ admits the following representation: where $\mathrm{f}_i = \sigma_i^{-1} \mathrm{F}^* e_i$, with $\{e_i\}_{i=1}^d$ the standard basis in $\mathbb{R}^d$ and $\mathrm{F}^*$ the adjoint of $\mathrm{F}$.

Figures (1)

  • Figure 1: One step of the lazy greedy algorithm.

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 4 more