As part of the research seminars at the Institute of Applied Analysis, Dr Kaori Nagatu and Mr Louis Carillo will speak on 12 January 2026 in room E.60 (Helmholtzstrasse 18) on the following topics:

4 p.m.
Dr Kaori Nagatu (Karlsruhe Institute of Technology) will talk about "Computer Assisted Proofs for Nonlinear Partial Differential Equations", and

5 p.m.
Mr Louis Carillo (CERMICS) on "Entropic Metastability in the Narrow Escape Problem".

Abstracts:

Dr Nagatou 4pm:

Computer Assisted Proofs for Nonlinear Partial Differential Equations
Numerical methods that ensure the reliability of numerical results obtained by a computer, sometimes referred to as "numerical verification methods", have attracted a lot of attention in recent years. These methods not only guarantee a bound for the error between an approximate solution and an exact solution, but also prove the existence of an exact solution within the computed error bounds. That is why such methods are also referred to as "Computer Assisted Proofs". In particular, they can cover cases where purely analytical methods have failed. This lecture will introduce the fundamental concepts of computer assisted proofs for partial differential equations (in particular elliptic boundary value problems) and show some examples, including the current project on non-selfadjoint spectral problems related to self-similar blowup in nonlinear wave equations.

Mr Carillo 5 pm:

Entropic metastability in the narrow escape problem
The narrow escape problem is a prototypical example to study entropic metastability, inspired by concrete applications in biology and chemistry. The problem is the determination of the exit time and position of a Brownian particle trapped in a domain with narrow holes. Unlike the usual double well potential, this form of metastability is not of energetic form as there is no potential involved, hence no Eyring-Kramers formula. Our goal is to study this problem in any dimension for arbitrary shapes of hole, and thanks to the quasi-stationary distribution approach to metastability, to rigorously derive the counterpart Eyring-Kramers type formulas in this context. Our numerical predictions are illustrated by numerical simulations, both Monte Carlo like and deterministic ones based on solving appropriate partial differential equations.