Distinguished Lecture——Regularized dynamical parametric approximation
报告人：Christian Lubich (University of Tuebingen, Germany)
时间：2024-09-06 16:00-17:00
地点：智华楼四元厅
Abstract:
This talk is about the numerical approximation of evolution equations by nonlinear parametrizations $u(t)=\Phi(q(t))$ with time-dependent parameters $q(t)$, which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and 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. We derive approximation results for a regularized approach in the time-continuous case as well as in time-discretized cases. With a suitable choice of the regularization parameter and the time stepsize, the approach can be successfully applied in irregular situations, even though it runs counter to the basic principle in numerical analysis to avoid solving ill-posed subproblems when aiming for a stable algorithm. Numerical experiments with sums of Gaussians for approximating quantum dynamics and with neural networks for approximating the flow map of a system of ordinary differential equations illustrate and complement the theoretical results. The talk is based on joint work with Caroline Lasser, Jörg Nick and Michael Feischl.
Bio:
Christian Lubich is a professor and head of Numerical Analysis Groups at the University of Tuebingen, Germany. His research concerns with mathematical analysis, numerical analysis, differential equation, exponential integrator and numerical stability. He is an expert on numerical analysis of time-dependent problems, including ordinary/partial differential equations and evolutionary integral equations. His recent research interests include dynamical low-rank approximation of high-dimensional matrix and tensor differential equations with applications to quantum dynamics and plasma physics, highly oscillatory problems, stable numerical interior-exterior coupling of wave equations, and the numerical analysis of geometric evolution equations. He is a famous applied mathematician with many influential works. He won a SIAM Dahlquist Prize, and gave a plenary lecture at the International Congress of Mathematicians in Rio de Janeiro in 2018.
