Uncertainty quantification in breakup reactions

Breakup reactions are one of the favored probes to study loosely bound nuclei, particularly those in the limit of stability forming a halo. In order to interpret such breakup experiments, the continuum discretized coupled channel method is typically used. In this study, the first Bayesian analysis of a breakup reaction model is performed. We use a combination of statistical methods together with a three-body reaction model (the continuum discretized coupled channel method) to quantify the uncertainties on the breakup observables due to the parameters in the effective potential describing the loosely bound projectile of interest. The combination of tools we develop opens the path for a Bayesian analysis of not only breakup processes, but also a wide array of complex processes that require computationally intensive reaction models.

Figure 1

Space-filling Latin hypercube sample (LHS) used to specify the set of breakup simulation runs to train the GP emulator (black) and to test the emulator (red). The two-dimensional projections of this design are shown for each pair of parameters.

Figure 2

The individual and cumulative contributions of principal components to the complete data set (angular distribution).

Figure 3

The individual and cumulative contributions of principal components to the complete data set (energy distribution).

Figure 4

Diagnostic plots to check the quality of the emulator using test parameters ${\rho }_1^{\mathrm{test}},...,{\rho }_l^{\mathrm{test}}$ and their corresponding simulation outputs $\sigma \left({\rho }_1^{\mathrm{test}}\right),...,\sigma \left({\rho }_l^{\mathrm{test}}\right)$ for the emulator based on combined angular and energy distributions with $r^2=0.99$: (a) standardized errors and (b) relative errors. Blue line shows the density of a standard normal random variable.

Figure 5

Univariate marginal estimates of the posterior distribution for the 4-dimensional parameter vector of the breakup reactions for the emulator based on both angular and energy distributions: prior distributions (light grey) and posterior distributions (dark grey) for the Coulomb radius parameter (a), the Woods-Saxon radius (b), the Woods-Saxon diffuseness (c), and the spin-orbit depth (d). The red dashed line corresponds to the true parameter values.

Figure 6

Pair plots for the posterior samples of four parameters obtained using MCMC with angular and energy distribution data.

Figure 7

Cross section for ${}^8\mathrm{B}+{}^{208}\mathrm{Pb}{\to }^7\mathrm{Be}+p+{}^{208}\mathrm{Pb}$ at 80 MeV/A (a) angular distribution and (b) relative ${}^7\mathrm{Be}+p$ energy distribution: prediction mean (dashed black line) and the 95% credible interval (shaded gray area) obtained from the Bayesian analysis, compared with mock data. Results correspond to the emulator which uses combined angular and energy distribution data.

Figure 8

The estimate of the posterior distribution (dark grey bins) of ANC parameters as compared to its prior distribution obtained from training parameters (light grey bins).

Figure 9

CDCC angular distributions for the breakup of ${}^8\mathrm{B}$ on ${}^{208}\mathrm{Pb}$ at 80 MeV/A (grey lines) compared to the CDCC results corresponding to the mock data obtained with $R_C=R_{\mathrm{WS}}=2.391$ fm, $a_{\mathrm{WS}}=0.52$ fm, and $V_{so}=4.898$ MeV (red circles).

Figure 10

Same as Fig. 9 for the energy distributions: (a) before filtering and (b) after filtering.

Figure 11

Diagnostic plots to check the quality of the emulator using test parameters ${\rho }_1^{\mathrm{test}},...,{\rho }_l^{\mathrm{test}}$ and their corresponding simulation outputs $\sigma \left({\rho }_1^{\mathrm{test}}\right),...,\sigma \left({\rho }_l^{\mathrm{test}}\right)$ for the emulator based on angular distribution with $r^2=1$: (a) standardized errors and (b) relative errors. Blue line shows the density of a standard normal random variable.

Figure 12

Diagnostic plots to check the quality of the emulator using test parameters ${\rho }_1^{\mathrm{test}},...,{\rho }_l^{\mathrm{test}}$ and their corresponding simulation outputs $\sigma \left({\rho }_1^{\mathrm{test}}\right),...,\sigma \left({\rho }_l^{\mathrm{test}}\right)$ for the emulator based on energy distribution with $r^2=0.96$: (a) standardized errors and (b) relative errors. Blue line shows the density of a standard normal random variable.

Figure 13

Univariate marginal estimates of the posterior distribution for the four-dimensional parameter vector of the breakup reactions for the emulator based on angular distributions: prior distributions (light grey) and posterior distributions (dark grey) for the Coulomb radius parameter (a), the Woods-Saxon radius (b), the Woods-Saxon diffuseness (c), and the spin-orbit depth (d). The red dashed line corresponds to the true parameter values.

Figure 14

Univariate marginal estimates of the posterior distribution for the four-dimensional parameter vector of the breakup reactions for the emulator based on energy distributions: prior distributions (light grey) and posterior distributions (dark grey) for the Coulomb radius parameter (a), the Woods-Saxon radius (b), the Woods-Saxon diffuseness (c), and the spin-orbit depth (d). The red dashed line corresponds to the true parameter values.

Figure 15

Cross section angular distributions for ${}^8\mathrm{B}+{}^{208}\mathrm{Pb}{\to }^7\mathrm{Be}+p+{}^{208}\mathrm{Pb}$ at 80 MeV/A using the emulator trained on angular distributions alone: prediction mean (dashed black line) and the 95% credible interval (shaded gray area) obtained from the Bayesian analysis, compared with mock data.

Figure 16

Cross section energy distributions for ${}^8\mathrm{B}+{}^{208}\mathrm{Pb}{\to }^7\mathrm{Be}+p+{}^{208}\mathrm{Pb}$ at 80 MeV/A using the emulator trained on energy distributions alone: prediction mean (dashed black line) and the 95% credible interval (shaded gray area) obtained from the Bayesian analysis, compared with mock data.