Dear all,

We are starting this week a new series of NOMADS seminars on Numerical methods, Matrix analysis and Data Science.

This week's seminar will take place on Wednesday December 1 at 17:00 (CET) in the GSSI Main Lecture Hall (the "red" conference hall by the entrance of the lecture rooms building).

The speaker is Matthew J Colbrook from University of Cambridge and ENS Paris, with a talk on the computation of semigroups of fractional PDEs.

Title and abstract are below.

In person participation is encouraged, but remote attendance will also be possible via the link:

Further info about past and future meetings are available at the webpage:
https://num-gssi.github.io/seminar/

Hope to see you all on Wednesday! And, please feel free to distribute this announcement as you see fit.

Francesco and Nicola

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Title:
Computing semigroups and time-fractional PDEs with error control

Abstract:
We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t > 0$, an arbitrary initial vector $u_0$ and an error tolerance $\epsilon > 0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$. The (parallelisable) algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules and the adaptive computation of resolvents in infinite dimensions. As a particular case, we deal with semigroups on $L^2(R^d)$ that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups, we provide a quadrature rule whose error decreases like $\exp(−cN/ log(N))$ for $N$ quadrature points, that remains stable as $N \to \infty$, and which is also suitable for infinite-dimensional operators. Finally, we extend the method to time-fractional PDEs (where it avoids singularities as $t \to  0$ and large memory consumption). Numerical examples are given, including: Schrödinger and wave equations on the aperiodic Ammann–Beenker tiling and fractional beam equations arising in the modelling of small-amplitude vibration of viscoelastic materials. The spectral analysis (which is always needed for contour methods) is considerably simplified due to an infinite-dimensional “solve-then-discretise” approach.