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Non-intrusive optimal experimental design for large-scale nonlinear Bayesian inverse problems using a Bayesian approximation error approach

Karina Koval, Ruanui Nicholson

TL;DR

This work tackles the challenge of designing informative data acquisitions for large-scale nonlinear Bayesian inverse problems. It introduces a linearize-then-design approach that leverages the Bayesian approximation error (BAE) to account for model discrepancies, achieving an uncertainty-aware A-optimal design with a derivative-free, zero-map surrogate. The authors prove that, asymptotically, the resulting OED objective is independent of the chosen linearization and extend the framework to marginalized designs that include nuisance parameters. Two numerical experiments—a subsurface flow inversion and a tsunami source detection problem—demonstrate that the proposed method yields superior sensor layouts and reliable uncertainty quantification compared to random or ignorance-based designs, highlighting its practical utility for complex PDE-constrained inference.

Abstract

We consider optimal experimental design (OED) for nonlinear inverse problems within the Bayesian framework. Optimizing the data acquisition process for large-scale nonlinear Bayesian inverse problems is a computationally challenging task since the posterior is typically intractable and commonly-encountered optimality criteria depend on the observed data. Since these challenges are not present in OED for linear Bayesian inverse problems, we propose an approach based on first linearizing the associated forward problem and then optimizing the experimental design. Replacing an accurate but costly model with some linear surrogate, while justified for certain problems, can lead to incorrect posteriors and sub-optimal designs if model discrepancy is ignored. To avoid this, we use the Bayesian approximation error (BAE) approach to formulate an A-optimal design objective for sensor selection that is aware of the model error. In line with recent developments, we prove that this uncertainty-aware objective is independent of the exact choice of linearization. This key observation facilitates the formulation of an uncertainty-aware OED objective function using a completely trivial linear map, the zero map, as a surrogate to the forward dynamics. The base methodology is also extended to marginalized OED problems, accommodating uncertainties arising from both linear approximations and unknown auxiliary parameters. Our approach only requires parameter and data sample pairs, hence it is particularly well-suited for black box forward models. We demonstrate the effectiveness of our method for finding optimal designs in an idealized subsurface flow inverse problem and for tsunami detection.

Non-intrusive optimal experimental design for large-scale nonlinear Bayesian inverse problems using a Bayesian approximation error approach

TL;DR

This work tackles the challenge of designing informative data acquisitions for large-scale nonlinear Bayesian inverse problems. It introduces a linearize-then-design approach that leverages the Bayesian approximation error (BAE) to account for model discrepancies, achieving an uncertainty-aware A-optimal design with a derivative-free, zero-map surrogate. The authors prove that, asymptotically, the resulting OED objective is independent of the chosen linearization and extend the framework to marginalized designs that include nuisance parameters. Two numerical experiments—a subsurface flow inversion and a tsunami source detection problem—demonstrate that the proposed method yields superior sensor layouts and reliable uncertainty quantification compared to random or ignorance-based designs, highlighting its practical utility for complex PDE-constrained inference.

Abstract

We consider optimal experimental design (OED) for nonlinear inverse problems within the Bayesian framework. Optimizing the data acquisition process for large-scale nonlinear Bayesian inverse problems is a computationally challenging task since the posterior is typically intractable and commonly-encountered optimality criteria depend on the observed data. Since these challenges are not present in OED for linear Bayesian inverse problems, we propose an approach based on first linearizing the associated forward problem and then optimizing the experimental design. Replacing an accurate but costly model with some linear surrogate, while justified for certain problems, can lead to incorrect posteriors and sub-optimal designs if model discrepancy is ignored. To avoid this, we use the Bayesian approximation error (BAE) approach to formulate an A-optimal design objective for sensor selection that is aware of the model error. In line with recent developments, we prove that this uncertainty-aware objective is independent of the exact choice of linearization. This key observation facilitates the formulation of an uncertainty-aware OED objective function using a completely trivial linear map, the zero map, as a surrogate to the forward dynamics. The base methodology is also extended to marginalized OED problems, accommodating uncertainties arising from both linear approximations and unknown auxiliary parameters. Our approach only requires parameter and data sample pairs, hence it is particularly well-suited for black box forward models. We demonstrate the effectiveness of our method for finding optimal designs in an idealized subsurface flow inverse problem and for tsunami detection.
Paper Structure (31 sections, 4 theorems, 45 equations, 12 figures, 1 algorithm)

This paper contains 31 sections, 4 theorems, 45 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Limitations
  5. Outline of the paper
  6. Background
  7. Preliminaries
  8. Bayesian inverse problems
  9. The linear Gaussian case
  10. The Bayesian approximation error approach
  11. Computing the approximation error statistics
  12. A data-driven approach to optimal experimental designs
  13. Optimal sensor placement for Bayesian inverse problems
  14. The linear Gaussian case
  15. Invariance of the uncertainty-aware optimal design to linearization choice
  16. ...and 16 more sections

Key Result

Theorem 1

\newlabelthm:10 Let $\mathscr{H}$ be a Hilbert space with $v\in\mathscr{H}$ and assume $v$ has prior measure $\mu_v$ with mean and (trace-class) covariance operator given by $v_0$ and $\mathcal{C}_{vv}$, respectively. Suppose further that where $\mathcal{G}:\mathscr{H}\rightarrow\mathbb{R}^{d}$ is the bounded PTO map, and $\mathbf{e}\in\mathbb{R}^{d}$ has mean $\mathbf{e}_0$ and covariance matri

Figures (12)

  • Figure 1: The true underlying level set field $\psi_{\rm true}$ (far left), the induced true ($\log$-) permeability $\xi_{\rm true}=\Phi(\psi_{\rm true})$ (center left), the true ($\log$-) boundary flux $m_{\rm true}$ through $\Gamma_{\rm T}$ (center right), and the resulting pressure $u_{\rm true}$ (far right).
  • Figure 2: Samples from the prior. Top row are samples of $\psi$, bottom row are corresponding samples of $\xi$. Samples of $m$ are visualized in the right-most column where the grey-scale represents the prior density and the black dotted lines the plus/minus (marginal) one standard deviation.
  • Figure 3: Comparison of the marginal distributions of $m$ for the subsurface flow problem. On the left we show samples of $m$ from the (approximate) posterior found using the linear(-ized) approach while on the right we show samples of $m$ from the posterior found using pCN MCMC. In both figures the grey-scale represents the respective posterior density and the black dotted lines the respective posterior plus/minus (marginal) one standard deviation.
  • Figure 4: Comparison of the marginal distributions of $\psi$ and $\xi$ for the subsurface flow problem. On the top row from left to right are shown the the marginal posterior standard deviation of $\psi$, the conditional mean estimate for $\psi$, the push-forward of conditional mean of $\psi$, i.e., $\phi(w_{\ast\vert\mathbf{d}})$, and the marginal posterior standard deviation of $\xi$ computing using the linear(-ized) approach, while on the bottom we show the respective quantities computed using pCN MCMC.
  • Figure 5: Optimal experimental design results for the the subsurface flow problem treating $m$ and $\xi$ as inversion parameters. On the left we show the optimal sensor placements using the linearize-then-optimize approach. In the center we compare the trace of the uncertainty-aware approximate posterior covariance operator found using the optimal sensor (green stars) to 100 randomly chosen designs (black boxplot). On the right we compare the trace of the posterior covariance operator for 20 sensors using the linearization and pCN MCMC for the optimal sensor placements (green and magenta stars, respectively) and for randomly chosen designs (black and blue boxplots, respectively) and the trace of the prior covariance operator (black dotted line).
  • ...and 7 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Corollary 2
  • Corollary 3
  • Proof 1
  • Remark 2