Computational and Applied Mathematics seminar

Abstract: This talk is about the numerical approximation of solutions to initial value problems of high-dimensional ordinary differential equations or evolutionary partial differential equations such as the Schrödinger equation by nonlinear parametrizations $u(t)=\Phi(q(t))$ with time-dependent parameters $q(t)$, which are to be determined in the computation. Our motivation comes from approximations by multiple Gaussians in quantum dynamics, by tensor networks, and by neural networks. In all these cases, the parametrization is typically irregular: the derivative $\Phi'(q)$ can have arbitrarily small singular values and may have varying rank. The talk is about approximation results for a regularized approach, which can still be successfully applied in such irregular situations, even if it runs counter to the basic principle in numerical analysis to avoid solving ill-posed subproblems when aiming for a stable algorithm.

The talk is based on joint work with Jörg Nick, Caroline Lasser and Michael Feischl.