In this work, we explore the use of factorization techniques as a means to identify low- rank structures in nuclear interactions and many-body methods, which we aim to leverage in order to reduce computational cost and memory requirements of modern nuclear many-body calculations.Using the Singular Value Decomposition (SVD), we show that we can construct accurate low-rank models of modern nucleon-nucleon interactions, and we develop a new variant of the Similarity Renormalization Group (SRG), that allows us to evolve nuclear interactions directly in the SVD-factorized form. Next, We extend these developments to three-nucleon interactions, using randomized SVD (R-SVD) algorithms to mitigate the impact of the much greater basis dimensions. While we find evidence of low-rank structures in the three-nucleon interaction, we also identify obstacles to a persistent rank reduction that we trace back to SRG-induced interactions. Finally, we explore applications of the SVD for factorization and rank reduction within the context of the In-Medium SRG (IMSRG). We develop rank-reduced IMSRG flow equations, which, while promising, still pose computational challenges stemming from the treatment of the so-called particle-hole terms. As a first step towards tackling this problem, we perform a detailed breakdown of the scaling and importance of all contributions in the IMSRG flow in applications for a schematic model as well as infinite neutron matter based on realistic interactions from chiral Effective Field Theory.
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