Presenter: Karsten Urban (Ulm University, Germany)
The Reduced Basis Method in Space and Time: Challenges, Limits and Perspectives
In many engineering applications, a partial differential equation (PDE) has to be solved very often (“multi-query”) and/or extremely fast (“realtime”) and/or using restricted memory/CPU (“cold computing”). Moreover, the mathematical modeling yields complex systems in the sense that
(i) each simulation is extremely costly, its CPU time may be in the order of several weeks;
(ii) we are confronted with evolutionary, time-dependent processes with long time horizons or time-periodic behaviors (which often requires long-time horizons in order to
find the time-periodic solution). All problems rely on time-dependent parameterized partial differential equations (PPDEs);
(iii) the processes often involve transport and wave-type phenomena as well as complex coupling and nonlinearities.
Without significant model reduction, one will not be able to able to tackle such problems. Moreover, there is a requirement in each of the above problems to ensure that the reduced simulations are certified in the sense that a reduced output comes along with a computable indicator which is a sharp upper bound of the error.
The Reduced Basis Method (RBM) is a well-established method for Model Order Reduction of PPDEs. We recall the classical framework for well-posed linear problems and extend this setting towards time-dependent problems of heat, transport, wave and Schrödinger type. The question of optimal approximation rates is discussed and possible benefits of ultraweak variational space-time methods are described.