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Register a new userNon-parametric Bayesian approach to extrapolation problems in configuration interaction methods
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Sota Yoshida
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We propose a non-parametric extrapolation method based on constrained Gaussian processes for configuration interaction methods. Our method has many advantages: (i) applicability to small data sets such as results of {it ab initio} methods, (ii) flexibility to incorporate constraints, which are guided by physics, into the extrapolation model, (iii) providing predictions with quantified extrapolation uncertainty, etc. In the present study, we show an application to the extrapolation needed in full configuration interaction method as an example.
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The no-core configuration-interaction model based on the isospin- and angular-momentum projected density functional formalism is introduced. Two applications of the model are presented: (i) determination of spectra of 0+ states in 62Zn and (ii) determination of isospin-symmetry-breaking corrections to superallowed beta-decay between isobaric-analogue 0+ states in 38Ca and 38K. It is shown that, without readjusting a single parameter of the underlying Skyrme interaction, in all three nuclei, the model reproduces the 0+ spectra surprisingly well.
[Background] Single-reference density functional theory is very successful in reproducing bulk nuclear properties like binding energies, radii, or quadrupole moments throughout the entire periodic table. Its extension to the multi-reference level allows for restoring symmetries and, in turn, for calculating transition rates. [Purpose] We propose a new no-core-configuration-interaction (NCCI) model treating properly isospin and rotational symmetries. The model is applicable to any nucleus irrespective of its mass and neutron- and proton-number parity. It properly includes polarization effects caused by an interplay between the long- and short-range forces acting in the atomic nucleus. [Methods] The method is based on solving the Hill-Wheeler-Griffin equation within a model space built of linearly-dependent states having good angular momentum and properly treated isobaric spin. The states are generated by means of the isospin and angular-momentum projection applied to a set of low-lying (multi)particle-(multi)hole deformed Slater determinants calculated using the self-consistent Skyrme-Hartree-Fock approach. [Results] The theory is applied to calculate energy spectra in N~Z nuclei that are relevant from the point of view of a study of superallowed Fermi beta-decays. In particular, a new set of the isospin-symmetry-breaking corrections to these decays is given. [Conclusions] It is demonstrated that the NCCI model is capable to capture main features of low-lying energy spectra in light and medium-mass nuclei using relatively small model space and without any local readjustment of its low-energy coupling constants. Its flexibility and a range of applicability makes it an interesting alternative to the conventional nuclear shell model.
The energy variance extrapolation method consists in relating the approximate energies in many-body calculations to the corresponding energy variances and inferring eigenvalues by extrapolating to zero variance. The method needs a fast evaluation of the energy variances. For many-body methods that expand the nuclear wave functions in terms of deformed Slater determinants, the best available method for the evaluation of energy variances scales with the sixth power of the number of single-particle states. We propose a new method which depends on the number of single-particle orbits and the number of particles rather than the number of single-particle states. We discuss as an example the case of ${}^4He$ using the chiral N3LO interaction in a basis consisting up to 184 single-particle states.
An ab initio quantum Monte Carlo method is introduced for calculating total rates of muon weak capture in light nuclei with mass number $A leq 12$. As a first application of the method, we perform a calculation of the rate in $^4$He in a dynamical framework based on realistic two- and three-nucleon interactions and realistic nuclear charge-changing weak currents. The currents include one- and two-body terms induced by $pi$- and $rho$-meson exchange, and $N$-to-$Delta$ excitation, and are constrained to reproduce the empirical value of the Gamow-Teller matrix element in tritium. We investigate the sensitivity of theoretical predictions to current parametrizations of the nucleon axial and induced pseudoscalar form factors as well as to two-body contributions in the weak currents. The large uncertainties in the measured values obtained from bubble-chamber experiments (carried out over 50 years ago) prevent us from drawing any definite conclusions.
Population pharmacokinetic (PK) modeling methods can be statistically classified as either parametric or nonparametric (NP). Each classification can be divided into maximum likelihood (ML) or Bayesian (B) approaches. In this paper we discuss the nonparametric case using both maximum likelihood and Bayesian approaches. We present two nonparametric methods for estimating the unknown joint population distribution of model parameter values in a pharmacokinetic/pharmacodynamic (PK/PD) dataset. The first method is the NP Adaptive Grid (NPAG). The second is the NP Bayesian (NPB) algorithm with a stick-breaking process to construct a Dirichlet prior. Our objective is to compare the performance of these two methods using a simulated PK/PD dataset. Our results showed excellent performance of NPAG and NPB in a realistically simulated PK study. This simulation allowed us to have benchmarks in the form of the true population parameters to compare with the estimates produced by the two methods, while incorporating challenges like unbalanced sample times and sample numbers as well as the ability to include the covariate of patient weight. We conclude that both NPML and NPB can be used in realistic PK/PD population analysis problems. The advantages of one versus the other are discussed in the paper. NPAG and NPB are implemented in R and freely available for download within the Pmetrics package from www.lapk.org.
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