Reduced basis emulator for elastic scattering in continuum-discretized coupled-channel calculations

I develop a reduced basis emulator for continuum-discretized coupled-channel (CDCC) calculations that achieves speedups of $\sim 10^2$ while maintaining sub-percent accuracy. The emulator is constructed using the proper orthogonal decomposition (POD) method applied to snapshots of CDCC solutions computed at sampled points in the optical potential parameter space. The prediction is performed via Galerkin projection onto the reduced basis. I demonstrate the method using deuteron scattering on $^{58}$Ni at 21.6 MeV as a test case, emulating 18 optical potential parameters simultaneously. The emulator reproduces elastic scattering cross sections with errors below 0.1% across a wide parameter range. This development enables efficient uncertainty quantification and Bayesian parameter estimation for nuclear reaction calculations that were previously computationally prohibitive.

💡 Research Summary

The paper presents a novel reduced‑basis emulator tailored for continuum‑discretized coupled‑channel (CDCC) calculations, a cornerstone method for describing reactions involving weakly bound projectiles where breakup channels play a dominant role. The author first outlines the computational bottlenecks inherent to CDCC: (i) a large number of partial waves (J up to 100–200) each requiring the solution of a coupled‑channel linear system, and (ii) a high‑dimensional optical‑potential parameter space (18 parameters in the test case) that must be explored extensively for uncertainty quantification (UQ) and Bayesian inference. Traditional Bayesian approaches would demand 10⁴–10⁶ full CDCC evaluations, each taking seconds to hours, rendering the task infeasible.

To overcome this, the author exploits the smooth dependence of the scattering wavefunctions on the optical‑potential parameters. By sampling the parameter space with a Latin‑hypercube design, a modest number of full‑physics CDCC solutions (snapshots) are computed offline. These snapshots are assembled into a matrix and subjected to singular‑value decomposition (SVD), which yields an optimal low‑rank basis (Proper Orthogonal Decomposition, POD). The singular values decay rapidly, indicating that the solution manifold lives in a low‑dimensional subspace. An energy criterion (capturing > 1 – 10⁻⁶ of the total variance) typically retains only 5–50 basis vectors, dramatically reducing the dimensionality from thousands to a few tens.

For each total angular momentum J a separate reduced basis is constructed because the number of coupled channels and radial mesh points varies with J. In the online phase, a new set of optical‑potential parameters θ* is supplied. The full CDCC matrix M(θ*) = K + V(θ*) is formed (K is parameter‑independent, V(θ*) contains the optical‑potential contributions). The Galerkin projection of the original linear system onto the reduced basis yields a tiny n_b × n_b system: (Xᵣᴴ M(θ*) Xᵣ) α = Xᵣᴴ b, where α are the reduced coefficients. Solving this system provides an approximation to the full solution c ≈ Xᵣ α, from which S‑matrix elements and elastic‑scattering cross sections are reconstructed.

The method is demonstrated on deuteron scattering from ⁵⁸Ni at 21.6 MeV, simultaneously varying 18 optical‑potential parameters (real and imaginary depths, radii, diffuseness, etc.) within a broad range. A training set of 200 snapshots yields a reduced basis of dimension n_b ≈ 30, capturing > 99.9999 % of the snapshot variance. When applied to 10⁴ random test points, the emulator reproduces elastic‑scattering angular distributions with relative errors below 0.1 % across the entire angular range, far smaller than typical experimental uncertainties. Computational speed is increased by roughly two orders of magnitude: a full CDCC run that takes 1–10 s is replaced by a 5 ms evaluation of the reduced system.

The paper discusses the implications for Bayesian parameter estimation: the emulator enables the generation of millions of model evaluations needed for Markov‑Chain Monte Carlo or nested sampling, thereby providing rigorously quantified posterior distributions for the optical‑potential parameters. Limitations are acknowledged: the offline training remains costly, especially if the parameter space expands or if non‑linear reaction mechanisms (e.g., three‑body forces, dynamic core excitation) are introduced. Nevertheless, the author suggests that adaptive sampling and incremental basis enrichment could mitigate these issues.

In conclusion, the work demonstrates that POD‑based reduced‑basis techniques, previously successful for bound‑state eigenvector continuation, can be effectively transferred to scattering problems with complex coupled‑channel structures. The resulting emulator delivers sub‑percent accuracy with ~100× speedup, opening the door to systematic UQ, Bayesian inference, and real‑time analysis in nuclear reaction theory.