Subspace accelerated measure transport methods for fast and scalable sequential experimental design, with application to photoacoustic imaging

TL;DR

This work addresses sequential Bayesian experimental design for expensive, high-dimensional inverse problems by deriving a derivative-based upper bound on the incremental information gain (iEIG) and using likelihood-informed subspaces to reduce parameter dimensionality. It integrates Knothe–Rosenblatt measure transport with tensor-train density representations and a restart strategy to enable scalable, amortized inference via conditional transport maps, allowing real-time posterior updates. The approach is demonstrated on PDE-based problems (diffusivity field and photoacoustic imaging), showing that designs maximizing the iEIG upper bound yield accurate posteriors with substantial computational savings compared to nested Monte Carlo and Gaussian-based methods. The framework provides a principled, scalable path for fast, high-fidelity sOED in large- and infinite-dimensional settings, with practical impact for imaging and sensing applications.

Abstract

We propose a novel approach for sequential optimal experimental design (sOED) for Bayesian inverse problems involving expensive models with large-dimensional unknown parameters. The focus of this work is on designs that maximize the expected information gain (EIG) from prior to posterior, which is a computationally challenging task in the non-Gaussian setting. This challenge is amplified in sOED, as the incremental expected information gain (iEIG) must be approximated multiple times in distinct stages, with both prior and posterior distributions often being intractable. To address this, we derive a derivative-based upper bound for the iEIG, which not only guides design placement but also enables the construction of projectors onto likelihood-informed subspaces, facilitating parameter dimension reduction. By combining this approach with conditional measure transport maps for the sequence of posteriors, we develop a unified framework for sOED, together with amortized inference, scalable to high- and infinite-dimensional problems. Numerical experiments for two inverse problems governed by partial differential equations (PDEs) demonstrate the effectiveness of designs that maximize our proposed upper bound.

Figures (4)

• Figure 1: The "true" diffusivity field used to synthesize data (left) and the corresponding pressure field $u$ (right). The $n_{y} = 121$ candidate locations for the design problem are visualized as black dots in the right figure.
• Figure 2: The upper bound on the incremental EIG for stages $k = 1,2,3,4$ (top row, from left to right) compared with a nested Monte Carlo estimator (middle row) and a linearization-based estimator (bottom row). The upper bound was computed using \ref{['Alg:DLIS']} with $N = 100$ samples to approximate $\mathcal{H}_{I}^k$ at each stage. The nested Monte Carlo estimates were computed using $N = 10000.0$ samples from the joint $\mathcal{L}(\textbf{y}_k\left.\space\middle|\space\right.\textbf{m})\,\widehat{\pi}(\textbf{m} \left.\space\middle|\space\right. \textbf{H}_{k-1})$.
• Figure 4: Two sample absorption coefficients from the prior (left), the corresponding light fluence $\phi$ solving \ref{['eq:optic_RB']} (middle) with light source on the top and bottom, respectively, and the initial pressure $p_0$ (right).
• Figure 6: A comparison of the iEIG for stages $1-5$ using the designs maximizing the iEIG upper bound (pink diamonds) as well as $50$ randomly chosen designs at each stage. The left figure corresponds to "true" absorption coeffient $\mu_a^1$, and the right to "true" absorption coefficient $\mu_a^2$.