Time-fractional operators: From theory to simulation

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Barbara Wohlmuth
TUM, Germany
Thursday 23rd November 2023
Aula Consiglio VII piano - Dipartimento di Matematica
In this talk, we discuss several aspects of time-fractional differential operators. The non-local nature of the operator is challenging from a theoretical and algorithmic point of view but allows to account for memory effects in the mathematical model. We consider a wide class of time-fractional PDE models, including gradient flow, viscoelasticity, random field generators and polymeric fluids. Of special interest is the equivalence of the time-fractional gradient flow model with an integer order system in a higher dimension. Exploiting this equivalence allows us to construct efficient algorithms for time fractional PDEs. Replacing the kernel by a finite sum of exponentials transforms the non-local PDEinto a integer-order system for the modes. To balance the discretization error in time and space with the kernel approximation error, the number of modes has to grow logarithmically. However, we observe numerically that the number of modes is very moderate if the finite sum of exponentials is based on a rational approximation of the kernel function. In this talk we present theoretical existence results, algorithmic aspects and show several applications in which fractional operators play an important role. Numerical examples illustrate the large flexibility of the proposed techniques. Joint work with Jonas Beddrich (TU München, Germany), Marvin Fritz (RICAM, Austria), Brendan Keith (Brown University, USA), Ustim Khristenko (Safran, France), and Endre Süli (University of Oxford, UK). Contatto: luca.formaggia@polimi.it
The research of Professor Wohlmuth examines the numerical simulation of partial differential equations. Special areas of interest here are discretization techniques, multi-scale solvers, coupled multi-field problems and predictive modelling. Interdisciplinary cooperation with engineering experts is an important part of her work. Professor Wohlmuth studied mathematics at TUM and at Grenoble. She also received her doctorate from TUM (1995) and completed her postdoctoral teaching qualification at Augsburg. Various positions as visiting scientist or professor led her to the US, to France, Hong Kong and Norway. In 2001, she was appointed to her first chair position at the University of Stuttgart, followed by professorship offers at Darmstadt and Berlin and later on at UT Austin and EPFL. In 2010, Professor Wohlmuth joined the faculty of TUM. She serves on academic boards of highly renowned institutions in the US and Europe. Since 2016 she serves as Director of the International Graduate School of Science and Engineering at TUM and forges new paths for training young scientists. She is an elected member of the Bavarian Academy of Sciences, the Leopoldina and the European Academy of Sciences.