Recent developments in theoretical Nuclear Physics offer new effective tools that improve the predictive power of modern low-energy Nuclear Structure. Specifically, the high-quality results achievable with ab initio methods allow us to test our knowledge of the strong interaction and to quantify the uncertainty in that knowledge. We review the progress and challenges of Nuclear Structure up to the present and perform uncertainty studies for several modern methods being developed to improve current solutions to the nuclear many-body problem. The ab initio No-Core Shell Model (NCSM) approach is employed throughout this study, though much of the discussion relate to other ab initio approaches as well.

We demonstrate the effects of basis truncation on the approximation of two-nucleon (2N) wavefunctions and observables in a relative Harmonic Oscillator (HO) basis, and investigate two approaches for matrix renormalization. We illustrate the effects of Okubo-Lee Suzuki renormalization on 2N observables for the root-mean-square point-proton radius, electric quadrupole moment, magnetic dipole moment, Gamow-Teller transition and neutrinoless double-beta decay operator using nucleon-nucleon interactions from Chiral Effective Field Theory (CEFT), both with and without a confining HO trap. We examine the 2N observables at each chiral order of the potential, verify agreement with experiment to at least a few percent (where applicable), and set the stage for comparing

the results to a theoretically consistent CEFT-treatment in a future study. Renormalization effects tend to be largest in the weaker traps and smaller basis spaces, suggesting applications to heavier nuclei with transitions dominated by weakly-bound nucleons would be subject to more significant renormalization effects.

In another study, we use the NCSM approach to benchmark the results of the Multi-Reference In-Medium Similarity Renormalization Group (MR-IMSRG) approach when calculating the matrix elements of neutrinoless double-beta decay mediated by light Majorana neutrino exchange in a light nucleus. The goal is to identify the MR-IMSRG approach as a possible candidate for calculating the decay's matrix elements in heavy nuclei. We compare results of the two approaches as a function of the basis cutoffs, and determine their predictions through extrapolation. Differences between the two approaches are observed for the square nuclear radii, and signal the effects of correlations that are omitted by the MR-IMSRG(2) truncation at the two-body level. The ground-state

energies and neutrinoless double-beta decay matrix elements show good agreement, and the implications for calculations in heavier nuclei are discussed.