Low-Rank Krylov Subspace Methods for the Solution of Non-Autonomous ODEs via the ⋆-Algebra

This thesis explores efficient yet unconventional numerical methods for solving linear non-autonomous ordinary differential equations, with particular attention to large-scale problems, arising, for example, in nuclear magnetic resonance (NMR) spin dynamics. The work adopts a recent strategy, the ⋆-algebra framework, to turn the original differential problem into an algebraic one, reformulating the time-ordered exponential associated with time-dependent systems as a ⋆-resolvent. After discretization by means of orthonormal Legendre polynomials, the ⋆-product is approximated through standard matrix multiplication, leading first to a linear system and then to a multiterm linear matrix equation, through vectorization and expoliting the Kronecker structure. The thesis then focuses on the numerical solution of this equation in matrix form, avoiding the explicit construction of large system matrices. In particular, matrix-form Krylov subspace methods for nonsymmetric problems, namely BiCG and BiCGSTAB, are developed together with their low-rank variants. Low-rank compression techniques based on factored representations and truncation strategies are employed to control memory growth and reduce computational cost while preserving accuracy. The proposed approach is tested on an NMR spin dynamics model built from real molecular data. Numerical experiments compare standard and low-rank Krylov methods in terms of accuracy, residuals, convergence behavior, runtime, and scalability. The results show that the ⋆-algebra reformulation combined with low-rank Krylov solvers provides an effective and promising strategy: it significantly reduces storage requirements and computational time with limited loss of accuracy, and enables the treatment of larger discretizations and higher-dimensional problems than standard full methods.