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5 Formulation and Implementation of the Gamow Shell Model
On the contrary, errors can differ by up to two orders of magnitude for poorly
defined parameters. In this case, the Bayesian inference might be necessary to
precisely analyze the causes of the weak dependence of these parameters on
observables in order to better constrain the model. Nevertheless, linear regression
is qualitatively correct for that matter, as the parameters which are sloppy therein
are also poorly defined using Bayesian inference.
Undeniably, a full statistical study of fitted interactions in the Gamow shell
model will have to be effected with Bayesian statistics, as it will allow to precisely
determine the quality of fitted parameters. One can consider adding additional
observables for the fit of interaction parameters to better constrain them. Alternatively, as linear regression can both quantify errors made on well constrained
parameters and identify poorly constrained parameters, it is a very reliable tool to
test fitted interactions despite the simplicity of linear approximation. It can then be
used to quantitatively analyze modeling errors in the Gamow shell model.
Solutions to Exercises 1
Exercise I.
A. The Berggren basis contours are discretized with the Gauss-Legendre quadrature. This method is very precise even with a small number of points as
integrated functions in the k-space are smooth. Multiplying the number of points
by two only slightly increases the numerical precision.
B. The calculation involving the Berggren basis contours either too close to the
resonance in a basis or too deep in the complex plane are not precise, even with
a large number of discretized scattering states. On the contrary, using a Berggren
basis contour in between the latter contours lead to precise results, even if the
number of discretized scattering states remains rather small.
C. If a Berggren basis contour is too close to a resonance of the basis, the
phase shifts of scattering states vary very much in the vicinity of resonances.
Consequently, the Gauss-Legendre discretization becomes imprecise. When
basis scattering states are far away from the real axis, the imaginary parts
of two-body matrix elements become very large, so that important numerical
cancellations occur between Hamiltonian matrix elements involving scattering
states. Thus, both these effects generate numerical inaccuracies.
Increasing the number of discretized scattering states when the p 3/2 contour
is close from the 0p 3/2 resonance does not ameliorate the situation as the
change of phase shifts close to the 0p 3/2 resonance is large for all discretization
schemes. The numerical cancellations between large two-body matrix elements
still exist even if one increases the number of discretized scattering states when
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.
5 Formulation and Implementation of the Gamow Shell Model
On the contrary, errors can differ by up to two orders of magnitude for poorly
defined parameters. In this case, the Bayesian inference might be necessary to
precisely analyze the causes of the weak dependence of these parameters on
observables in order to better constrain the model. Nevertheless, linear regression
is qualitatively correct for that matter, as the parameters which are sloppy therein
are also poorly defined using Bayesian inference.
Undeniably, a full statistical study of fitted interactions in the Gamow shell
model will have to be effected with Bayesian statistics, as it will allow to precisely
determine the quality of fitted parameters. One can consider adding additional
observables for the fit of interaction parameters to better constrain them. Alternatively, as linear regression can both quantify errors made on well constrained
parameters and identify poorly constrained parameters, it is a very reliable tool to
test fitted interactions despite the simplicity of linear approximation. It can then be
used to quantitatively analyze modeling errors in the Gamow shell model.
Solutions to Exercises 1
Exercise I.
A. The Berggren basis contours are discretized with the Gauss-Legendre quadrature. This method is very precise even with a small number of points as
integrated functions in the k-space are smooth. Multiplying the number of points
by two only slightly increases the numerical precision.
B. The calculation involving the Berggren basis contours either too close to the
resonance in a basis or too deep in the complex plane are not precise, even with
a large number of discretized scattering states. On the contrary, using a Berggren
basis contour in between the latter contours lead to precise results, even if the
number of discretized scattering states remains rather small.
C. If a Berggren basis contour is too close to a resonance of the basis, the
phase shifts of scattering states vary very much in the vicinity of resonances.
Consequently, the Gauss-Legendre discretization becomes imprecise. When
basis scattering states are far away from the real axis, the imaginary parts
of two-body matrix elements become very large, so that important numerical
cancellations occur between Hamiltonian matrix elements involving scattering
states. Thus, both these effects generate numerical inaccuracies.
Increasing the number of discretized scattering states when the p 3/2 contour
is close from the 0p 3/2 resonance does not ameliorate the situation as the
change of phase shifts close to the 0p 3/2 resonance is large for all discretization
schemes. The numerical cancellations between large two-body matrix elements
still exist even if one increases the number of discretized scattering states when
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.