Statistical tools of uncertainty quantification can be used to assess the information content of measured observables with respect to present-day theoretical models, to estimate model errors and ...thereby improve predictive capability, to extrapolate beyond the regions reached by experiment, and to provide meaningful input to applications and planned measurements. To showcase new opportunities offered by such tools, we make a rigorous analysis of theoretical statistical uncertainties in nuclear density functional theory using Bayesian inference methods. By considering the recent mass measurements from the Canadian Penning Trap at Argonne National Laboratory, we demonstrate how the Bayesian analysis and a direct least-squares optimization, combined with high-performance computing, can be used to assess the information content of the new data with respect to a model based on the Skyrme energy density functional approach. Employing the posterior probability distribution computed with a Gaussian process emulator, we apply the Bayesian framework to propagate theoretical statistical uncertainties in predictions of nuclear masses, two-neutron dripline, and fission barriers. Overall, we find that the new mass measurements do not impose a constraint that is strong enough to lead to significant changes in the model parameters. The example discussed in this study sets the stage for quantifying and maximizing the impact of new measurements with respect to current modeling and guiding future experimental efforts, thus enhancing the experiment-theory cycle in the scientific method.
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We optimize the nucleon-nucleon interaction from chiral effective field theory at next-to-next-to-leading order (NNLO). The resulting new chiral force NNLO(opt) yields χ(2)≈1 per degree of freedom ...for laboratory energies below approximately 125 MeV. In the A=3, 4 nucleon systems, the contributions of three-nucleon forces are smaller than for previous parametrizations of chiral interactions. We use NNLO(opt) to study properties of key nuclei and neutron matter, and we demonstrate that many aspects of nuclear structure can be understood in terms of this nucleon-nucleon interaction, without explicitly invoking three-nucleon forces.
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We describe the new version 3.00 of the code hfbtho that solves the nuclear Hartree–Fock (HF) or Hartree–Fock–Bogolyubov (HFB) problem by using the cylindrical transformed deformed harmonic ...oscillator basis. In the new version, we have implemented the following features: (i) the full Gogny force in both particle–hole and particle–particle channels, (ii) the calculation of the nuclear collective inertia at the perturbative cranking approximation, (iii) the calculation of fission fragment charge, mass and deformations based on the determination of the neck, (iv) the regularization of zero-range pairing forces, (v) the calculation of localization functions, (vi) a MPI interface for large-scale mass table calculations.
Program title:hfbtho v3.00
Program Files doi:http://dx.doi.org/10.17632/c5g2f92by3.1
Licensing provisions: GPL v3
Programming language: FORTRAN-95
Journal reference of previous version: M.V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, and S. Wild, Comput. Phys. Commun. 184 (2013).
Does the new version supersede the previous one: Yes
Summary of revisions:
1. the Gogny force in both particle–hole and particle–particle channels was implemented;
2. the nuclear collective inertia at the perturbative cranking approximation was implemented;
3. fission fragment charge, mass and deformations were implemented based on the determination of the position of the neck between nascent fragments;
4. the regularization method of zero-range pairing forces was implemented;
5. the localization functions of the HFB solution were implemented;
6. a MPI interface for large-scale mass table calculations was implemented.
Nature of problem:hfbtho is a physics computer code that is used to model the structure of the nucleus. It is an implementation of the energy density functional (EDF) approach to atomic nuclei, where the energy of the nucleus is obtained by integration over space of some phenomenological energy density, which is itself a functional of the neutron and proton intrinsic densities. In the present version of hfbtho, the energy density derives either from the zero-range Skyrme or the finite-range Gogny effective two-body interaction between nucleons. Nuclear super-fluidity is treated at the Hartree–Fock–Bogolyubov (HFB) approximation. Constraints on the nuclear shape allows probing the potential energy surface of the nucleus as needed e.g., for the description of shape isomers or fission. The implementation of a local scale transformation of the single-particle basis in which the HFB solutions are expanded provide a tool to properly compute the structure of weakly-bound nuclei.
Solution method: The program uses the axial Transformed Harmonic Oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogolyubov Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions or the finite-range Gogny force until a self-consistent solution is found. A previous version of the program was presented in M.V. Stoitsov, N. Schunck, M. Kortelainen, N. Michel, H. Nam, E. Olsen, J. Sarich, and S. Wild, Comput. Phys. Commun. 184 (2013) 1592–1604 with much of the formalism presented in the original paper M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comput. Phys. Commun. 167 (2005) 43–63.
Additional comments: The user must have access to (i) the LAPACK subroutines dsyeevr, dsyevd, dsytrf and dsytri, and their dependencies, which compute eigenvalues and eigenfunctions of real symmetric matrices, (ii) the LAPACK subroutines dgetri and dgetrf, which invert arbitrary real matrices, and (iii) the BLAS routines dcopy, dscal, dgemm and dgemv for double-precision linear algebra (or provide another set of subroutines that can perform such tasks). The BLAS and LAPACK subroutines can be obtained from the Netlib Repository at the University of Tennessee, Knoxville: http://netlib2.cs.utk.edu/.
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We describe the new version 2.00d of the code hfbtho that solves the nuclear Skyrme-Hartree–Fock (HF) or Skyrme-Hartree–Fock–Bogoliubov (HFB) problem by using the cylindrical transformed deformed ...harmonic oscillator basis. In the new version, we have implemented the following features: (i) the modified Broyden method for non-linear problems, (ii) optional breaking of reflection symmetry, (iii) calculation of axial multipole moments, (iv) finite temperature formalism for the HFB method, (v) linear constraint method based on the approximation of the Random Phase Approximation (RPA) matrix for multi-constraint calculations, (vi) blocking of quasi-particles in the Equal Filling Approximation (EFA), (vii) framework for generalized energy density with arbitrary density-dependences, and (viii) shared memory parallelism via OpenMP pragmas.
Program title: HFBTHO v2.00d
Catalog identifier: ADUI_v2_0
Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUI_v2_0.html
Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland
Licensing provisions: GNU General Public License version 3
No. of lines in distributed program, including test data, etc.: 167228
No. of bytes in distributed program, including test data, etc.: 2672156
Distribution format: tar.gz
Programming language: FORTRAN-95.
Computer: Intel Pentium-III, Intel Xeon, AMD-Athlon, AMD-Opteron, Cray XT5, Cray XE6.
Operating system: UNIX, LINUX, WindowsXP.
RAM: 200 Mwords
Word size: 8 bits
Classification: 17.22.
Does the new version supercede the previous version?: Yes
Catalog identifier of previous version: ADUI_v1_0
Journal reference of previous version: Comput. Phys. Comm. 167 (2005) 43
Nature of problem:
The solution of self-consistent mean-field equations for weakly-bound paired nuclei requires a correct description of the asymptotic properties of nuclear quasi-particle wave functions. In the present implementation, this is achieved by using the single-particle wave functions of the transformed harmonic oscillator, which allows for an accurate description of deformation effects and pairing correlations in nuclei arbitrarily close to the particle drip lines.
Solution method:
The program uses the axial Transformed Harmonic Oscillator (THO) single- particle basis to expand quasi-particle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogoliubov Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions until a self-consistent solution is found. A previous version of the program was presented in: M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comput. Phys. Commun. 167 (2005) 43–63.
Reasons for new version:
Version 2.00d of HFBTHO provides a number of new options such as the optional breaking of reflection symmetry, the calculation of axial multipole moments, the finite temperature formalism for the HFB method, optimized multi-constraint calculations, the treatment of odd–even and odd–odd nuclei in the blocking approximation, and the framework for generalized energy density with arbitrary density-dependences. It is also the first version of HFBTHO to contain threading capabilities.
Summary of revisions:1.The modified Broyden method has been implemented,2.Optional breaking of reflection symmetry has been implemented,3.The calculation of all axial multipole moments up to λ=8 has been implemented,4.The finite temperature formalism for the HFB method has been implemented,5.The linear constraint method based on the approximation of the Random Phase Approximation (RPA) matrix for multi-constraint calculations has been implemented,6.The blocking of quasi-particles in the Equal Filling Approximation (EFA) has been implemented,7.The framework for generalized energy density functionals with arbitrary density-dependence has been implemented,8.Shared memory parallelism via OpenMP pragmas has been implemented.
Restrictions:
Axial- and time-reversal symmetries are assumed.
Unusual features:
The user must have access to (i)the LAPACK subroutines DSYEVD, DSYTRF and DSYTRI, and their dependences, which compute eigenvalues and eigenfunctions of real symmetric matrices,(ii)the LAPACK subroutines DGETRI and DGETRF, which invert arbitrary real matrices, and(iii)the BLAS routines DCOPY, DSCAL, DGEMM and DGEMV for double-precision linear algebra (or provide another set of subroutines that can perform such tasks). The BLAS and LAPACK subroutines can be obtained from the Netlib Repository at the University of Tennessee, Knoxville: http://netlib2.cs.utk.edu/.
Running time:
Highly variable, as it depends on the nucleus, size of the basis, requested accuracy, requested configuration, compiler and libraries, and hardware architecture. An order of magnitude would be a few seconds for ground-state configurations in small bases Nmax≈8−12, to a few minutes in very deformed configuration of a heavy nucleus with a large basis Nmax>20.
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.
Nuclear density functional theory (DFT) is one of the main theoretical tools used to study the properties of heavy and superheavy elements, or to describe the structure of nuclei far from ...stability. While on-going efforts seek to better root nuclear DFT in the theory of nuclear forces (see Duguet
et al.
, this Topical Issue), energy functionals remain semi-phenomenological constructions that depend on a set of parameters adjusted to experimental data in finite nuclei. In this paper, we review recent efforts to quantify the related uncertainties, and propagate them to model predictions. In particular, we cover the topics of parameter estimation for inverse problems, statistical analysis of model uncertainties and Bayesian inference methods. Illustrative examples are taken from the literature.
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Nuclear density functional theory is the only microscopical theory that can be applied throughout the entire nuclear landscape. Its key ingredient is the energy density functional. Here, we propose a ...new parametrization UNEDF2 of the Skyrme energy density functional. The functional optimization is carried out using the POUNDERS optimization algorithm within the framework of the Skyrme Hartree-Fock-Bogoliubov theory. Compared to the previous parametrization UNEDF1, restrictions on the tensor term of the energy density have been lifted, yielding a very general form of the energy density functional up to second order in derivatives of the one-body density matrix. In order to impose constraints on all the parameters of the functional, selected data on single-particle splittings in spherical doubly-magic nuclei have been included into the experimental dataset. The agreement with both bulk and spectroscopic nuclear properties achieved by the resulting UNEDF2 parametrization is comparable with UNEDF1. While there is a small improvement on single-particle spectra and binding energies of closed shell nuclei, the reproduction of fission barriers and fission isomer excitation energies has degraded. As compared to previous UNEDF parametrizations, the parameter confidence interval for UNEDF2 is narrower. In particular, our results overlap well with those obtained in previous systematic studies of the spin-orbit and tensor terms. UNEDF2 can be viewed as an all-around Skyrme EDF that performs reasonably well for both global nuclear properties and shell structure. However, after adding new data aiming to better constrain the nuclear functional, its quality has improved only marginally. These results suggest that the standard Skyrme energy density has reached its limits, and significant changes to the form of the functional are needed.
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Background: Although uncertainty quantification has been making its way into nuclear theory, these methods have yet to be explored in the context of reaction theory. For example, it is well known ...that different parameterizations of the optical potential can result in different cross sections, but these differences have not been systematically studied and quantified. Purpose: The purpose of this work is to investigate the uncertainties in nuclear reactions that result from fitting a given model to elastic-scattering data, as well as to study how these uncertainties propagate to the inelastic and transfer channels. Method: We use statistical methods to determine a best fit and create corresponding 95% confidence bands. A simple model of the process is fit to elastic-scattering data and used to predict either inelastic or transfer cross sections. In this initial work, we assume that our model is correct, and the only uncertainties come from the variation of the fit parameters. Results: We study a number of reactions involving neutron and deuteron projectiles with energies in the range of 5-25 MeV/u, on targets with mass A = 12-208. We investigate the correlations between the parameters in the fit. The case of deuterons on C-12 is discussed in detail: the elastic-scattering fit and the prediction of C-12(d, p) C-13 transfer angular distributions, using both uncorrelated and correlated. 2 minimization functions. The general features for all cases are compiled in a systematic manner to identify trends. Conclusions: Our work shows that, in many cases, the correlated chi(2) functions (in comparison to the uncorrelated chi(2) functions) provide a more natural parameterization of the process. These correlated functions do, however, produce broader confidence bands. Further optimization may require improvement in the models themselves and/or more information included in the fit.
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The UNEDF SciDAC project to develop and optimize the energy density functional for atomic nuclei using state-of-the art computational infrastructure is briefly described. The ultimate goal is to ...replace current phenomenological models of the nucleus with a well-founded microscopic theory with minimal uncertainties, capable of describing nuclear data and extrapolating to unknown regions.
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