# A Fast and Scalable Method for A-Optimal Design of Experiments for Infinite-dimensional Bayesian Nonlinear Inverse Problems

arXiv: https://arxiv.org/abs/1410.5899 · HTML: https://mlsanity.com/paper/1410.5899/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2014.

## Built on by (most cited first, 14 of 14)

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- [Robust A-Optimal Experimental Design for Bayesian Inverse Problems](https://mlsanity.com/paper/2305.03855/) (2023)
- [Probabilistic Approach to Black-Box Binary Optimization with Budget Constraints: Application to Sensor Placement](https://mlsanity.com/paper/2406.05830/) (2024)
- [Sequential Bayesian Optimal Experimental Design in Infinite Dimensions via Policy Gradient Reinforcement Learning](https://mlsanity.com/paper/2601.05868/) (2026)
- [Stochastic Learning Approach to Binary Optimization for Optimal Design of Experiments](https://mlsanity.com/paper/2101.05958/) (2021)
- [Scalable method for Bayesian experimental design without integrating over posterior distribution](https://mlsanity.com/paper/2306.17615/) (2023)
- [From Displacements to Distributions: A Machine-Learning Enabled Framework for Quantifying Uncertainties in Parameters of Computational Models](https://mlsanity.com/paper/2403.03233/) (2024)
- [Beyond Expected Information Gain: Stable Bayesian Optimal Experimental Design with Integral Probability Metrics and Plug-and-Play Extensions](https://mlsanity.com/paper/2604.21849/) (2026)

## Abstract (arXiv)

We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by PDEs. The goal is to find a placement of sensors, at which experimental data are collected, so as to minimize the uncertainty in the inferred parameter field. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. We seek a method that solves the OED problem at a cost (measured in the number of forward PDE solves) that is independent of both the parameter and sensor dimensions. To facilitate this, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this (implicitly defined) covariance operator. The resulting OED problem includes as constraints the PDEs characterizing the MAP point, and the PDEs describing the action of the covariance operator to vectors. The sparsity of the sensor configurations is controlled using sparsifying penalty functions. We elaborate our OED method for the problem of determining the sensor placement to best infer the coefficient of an elliptic PDE. Adjoint methods are used to compute the gradient of the PDE-constrained OED objective function. We provide numerical results for inference of the permeability field in a porous medium flow problem, and demonstrate that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter and sensor dimensions. The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.