Abstract: Standard numerical integrators, such as Lie splitting, Strang splitting, and exponential integrators, experience order reduction when applied to semilinear dispersive problems with non-smooth initial data. To address this issue, a recent development introduces a new class of integrators known as low-regularity integrators. These integrators use the variation-of-constants formula and employ resonance-based approximations in Fourier space, demonstrating improved convergence rates at low regularity. However, the estimation of nonlinear terms in the global error still relies on classical bilinear estimates derived from Sobolev embeddings. At very low regularity, traditional error analysis in Sobolev spaces is hampered by the lack of suitable embeddings. A novel framework, inspired by Bourgain's techniques, has been developed that allows the analysis of methods applicable to very low regularity initial data. This approach has been applied to various problems, including the nonlinear Schrödinger equation and the `good' Boussinesq equation.

Introduction of the speaker: Alexander Ostermann is a professor of numerical analysis and scientific computing at the University of Innsbruck in Austria. He earned his Ph.D. from the same institution and was a postdoctoral fellow at the University of Geneva in Switzerland. His research focuses on the numerical solution of partial differential equations. Specifically, he has collaborated with Christian Lubich on implicit and linearly implicit Runge-Kutta methods, with Marlis Hochbruck on the development of exponential integrators and their stiff order conditions, with Lukas Einkemmer on the correct treatment of nontrivial boundary conditions in splitting methods, and with Katharina Schratz on numerical integrators for dispersive equations with extremely rough initial data.

Ostermann was the dean of the School of Mathematics, Computer Science, and Physics at the University of Innsbruck for eight years. He is also a member of several scientific societies and boards. He has led the university's Scientific Computing research area for over a decade. In this position, he oversees the computing infrastructure at the university and serves as a board member of Austria's largest computing cluster, organized in Austrian Scientific Computing. He has also recently taken on responsibility for the installation of a quantum computer at the university.