Stability of the manifold boundary approximation method for reductions of nuclear structure models

The framework of nuclear energy density functionals has been employed to describe nuclear structure phenomena for a wide range of nuclei. Recently, statistical properties of a given nuclear model, such as parameter confidence intervals and correlations, have received much attention, particularly when one tries to fit complex models. We apply information-theoretic methods to investigate stability of model reductions by the manifold boundary approximation method (MBAM). In an illustrative example of the density-dependent point-coupling model of the relativistic energy density functional, utilizing Monte Carlo simulations, it is found that main conclusions obtained from the MBAM procedure are stable under variation of the model parameters. Furthermore, we find that the end of the geodesic occurs when the determinant of the Fisher information metric vanishes, thus effectively separating the parameter space into two disconnected regions.

Figure 1

Results of extrapolating the geodesic after the $detg=0$ point. Shown are (a) the behavior of the evaluated model for different $\tau$'s along the geodesic, (b) the model parameters, (c) the FIM eigenvalues as functions of $\tau$, (d) the squares of the FIM eigenvector $v^0$ components, (e) the Ricci scalar, and (f) the FIM determinant along the geodesic. Solid, dashed, and dotted lines stand for, respectively, the initial odeint solutions, the linear interpolation, and the values derived using odeint starting from the endpoint of the interpolated solutions.

Figure 2

Monte Carlo simulated sample parameters using the best-fitting covariance matrix (black symbols and contours) and its propagation towards $\tau =1.3$ along the geodesic using the Jacobi equation (15) (red symbols and contours).

Figure 3

Monte Carlo simulations of uncertainty propagation using the Jacobi equation (15). Shown are the median and its uncertainty derived using 1300 simulated samples starting from the best-fitting point. The figure shows (a) the simulated FIM $v^0$ eigenvector components squared, (b) FIM eigenvalues, (c) FIM determinant, and (d) scalar curvature. The shaded areas correspond to the $1\sigma$ percentile interval, while the dotted lines in panels (c) and (d) additionally show the 5th and the 95th percentiles, respectively. Solid orange lines in panels (c) and (d) stand for the respective quantities computed along the path of the MBAM geodesic.

Figure 4

Same as Fig. 2, but for Monte Carlo simulated sample (base 10) logarithms of the eigenvalues of the FIM.

Figure 5

Same as Fig. 2, but for Monte Carlo simulated sample components of the FIM $v^0$ eigenvector.

Figure 6

Same as Fig. 3, but for the reparametrized model described in Sec. 4c.

Figure 7

Monte Carlo simulations of posterior distributions of the error estimates for the reparametrized model, based on the MCMC algorithm. The figure shows the $1\sigma , 2\sigma$, and $3\sigma$ covariance ellipses in red, as estimated from the FIM inverse, and the estimates of the covariance ellipses based on the MCMC sample points in blue.