Optimizing Nuclear Reaction Analysis (NRA) using Bayesian Experimental Design
Nuclear Reaction Analysis with ${}^{3}$He holds the promise to measure Deuterium depth profiles up to large depths. However, the extraction of the depth profile from the measured data is an ill-posed inversion problem. Here we demonstrate how Bayesian Experimental Design can be used to optimize the number of measurements as well as the measurement energies to maximize the information gain. Comparison of the inversion properties of the optimized design with standard settings reveals huge possible gains. Application of the posterior sampling method allows to optimize the experimental settings interactively during the measurement process.
💡 Research Summary
The paper addresses the long‑standing challenge of extracting accurate deuterium depth profiles from Nuclear Reaction Analysis (NRA) measurements that use ${}^{3}$He projectiles. In conventional NRA, a series of measurements are performed at equally spaced incident energies, and the resulting spectra are inverted to obtain the concentration versus depth. This inversion is severely ill‑posed because the forward model depends on uncertain reaction cross‑sections, stopping powers, detector efficiencies, and measurement noise. Small variations in the data can therefore produce large errors in the reconstructed profile.
To overcome these difficulties, the authors embed the problem in a Bayesian framework. The forward model $y = f(E,\theta) + \epsilon$ predicts the measured signal $y$ for a given incident energy $E$ and a depth‑dependent deuterium concentration vector $\theta$. The noise term $\epsilon$ is modeled as Gaussian. A prior distribution $p(\theta)$ encodes physical constraints such as non‑negativity, smoothness, and plausible concentration ranges; the authors explore both Beta‑type priors and Gaussian‑process priors. Posterior inference $p(\theta|y)$ is performed with Markov‑Chain Monte Carlo (MCMC) sampling, providing full uncertainty quantification (credible intervals and parameter correlations).
The central methodological contribution is the application of Bayesian Experimental Design (BED) to select the optimal set of measurement energies and the total number of measurements. The design criterion is the Expected Information Gain (EIG), also known as the expected Kullback‑Leibler divergence between the prior and the posterior after a hypothetical measurement at energy $E$:
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