arXiv subjects

Steven Maio

Publications and source records attributed to Steven Maio.

4 recordsLinked to original sources

arXiv2026

Multi-type Sensor Placement for PDE-based Bayesian Inverse Problems

We address optimal placement of multi-type sensors for Bayesian inverse problems governed by partial differential equations (PDEs). The proposed framework allows for sensors with different accuracies and observation types. We formulate the optimal experimental design (OED) problem as a knapsack-constrained binary optimization problem for maximizing expected information gain (EIG). To approximately solve the resulting optimization problems, we propose a stochastic cost-benefit greedy algorithm, which admits theoretical guarantees for monotone submodular set functions. Specifically, these guarantees apply in the case of linear Gaussian inverse problems with uncorrelated measurement errors, where the EIG admits a convenient closed-form expression. For nonlinear inverse problems, we develop a non-intrusive approach that uses the Bayesian approximation error framework to define an observation model with an error-corrected global linear model. We show that the corresponding approximate EIG is a lower bound for the exact EIG and thus provides a principled surrogate objective for the OED problem. The effectiveness of the proposed methods is demonstrated in two model inverse problems governed by PDEs.

math.NA↗

arXiv2026

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

We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated measurement errors and focus on the problem of selecting 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, the 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 efficient implementations of the greedy procedure.

math.OC↗

arXiv2025

On submodularity of the expected information gain

We consider finite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor measurements. In this setting, it is known that the expected information gain, quantified by the expected Kullback-Leibler divergence from the posterior measure to the prior measure, is submodular. We present a simple alternative proof of this fact tailored to a weighted inner product space setting arising from discretization of infinite-dimensional inverse problems constrained by partial differential equations (PDEs).

math.OC↗

arXiv2020

On the Hausdorff dimension of the residual set of a packing by smooth curves

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $\sigma$-finite Hausdorff 1-measure.

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