Ober­sem­in­ar Nu­merik parti­eller Dif­fer­en­tialgleichun­gen: Jo­nas Kusch (Nor­we­gi­an Uni­ver­sity of Life Sci­ences): Dy­nam­ic­al low-rank ap­prox­im­a­tion: From ra­di­ation ther­apy to quantum mech­an­ics

Many problems in physics, engineering, and computer science require substantial computational and memory resources. Three prominent examples are radiation therapy, quantum mechanics, and neural networks. This is because evolving the wave function in quantum mechanics in time, calculating the radiation therapy dose, or training neural network weights involves solving large,
often prohibitively complex, matrix or tensor differential equations.

However, it has been observed that solutions in all three fields often exhibit low-rank structures. To efficiently evolve solutions as low-rank representations, we employ dynamical low-rank approximation (DLRA), as introduced in [Koch and Lubich (2007)]. DLRA is closely related to the Dirac–Frenkel time-dependent variational principle (see, e.g., [Dirac (1930), Beck (2000)]), representing solutions through low-rank matrix or tensor factorizations and deriving time evolution equations for the low-rank factors. Since these equations are highly stiff, novel numerical methods tailored to the geometry of low-rank tensor manifolds are necessary. I will discuss the development and analysis of new time integration methods that can efficiently simulate problems in radiation therapy, quantum mechanics, and neural network training while preserving essential structures of the original full-rank dynamics.