Talk – May 26: Multilevel Monte Carlo Methods
Multilevel Monte Carlo Methods
Speaker: Rob Scheichl
Time: Wednesday, May 26, 2021, 14:00
Online: Please write a short e-mail to email@example.com to receive the registration link.
In this talk, I will introduce multilevel Monte Carlo methods for high dimensional quadrature and applications in Uncertainty Quantification (UQ). Modelling and simulation of physical, biological and engineering processes via differential equations plays a central role in many applications. Simulation tools have reached a high level of sophistication, making it possible to study ever more complex problems. However, model parameters and geometries are often uncertain or unknown, leading to uncertainty quantification (UQ) and parameter identification problems. Especially when the uncertainties are spatial distributed, this can lead to very high dimensional parametric problems. Sampling-based methods, such as Monte Carlo (for UQ) or Markov chain Monte Carlo (for parameter identification), are in principle dimension-independent approaches to solve the resulting high-dimensional quadrature problems, but they can be extremely inefficient in their most basic form. Multilevel Monte Carlo methods, which exploit a hierarchy of model approximations (naturally available in the context of differential equations) are able to significantly reduce the variance of standard Monte Carlo estimators. In the best case, they are able to provide the uncertainties in a PDE solution at a computational cost that is only a small multiple of the cost of solving the PDE for one fixed parameter value. The talk will introduce the basic idea for the methodology on two standard model problems. It will give insight into its theoretical justification, as well as presenting extensions, in particular in the direction of Bayesian inference for parameter estimation via Markov chain Monte Carlo methods.