[AN] Kristin Kirchner: Monte Carlo methods: From the law of large numbers to estimating moments in Banach spaces

28 November 2023 16:00 till 17:00 - Location: Chip (36.HB.01.600) | Add to my calendar

The law of large numbers states that the sample average of the results obtained from a large number of independent identical trials will be close to the expected value. As more trials are performed, this average tends to become closer to the mean value or, in other words, the standard Monte Carlo estimator converges to the first moment of the corresponding probability distribution.

In my talk I will take you on a journey of the history of Monte Carlo methods and their relevance for applications, with emphasis on numerical integration and computational uncertainty quantification. After discussing existing results on standard and multilevel Monte Carlo methods for real-valued and Hilbert space valued random variables, I will present our recent work on the estimation of expected values of random variables taking values in Banach spaces. Besides the mean value, our results provide explicit convergence rates for the estimation of higher-order moments. There are numerous applications conceivable, and I will detail one of them: The approximation of cross-correlations for solutions to stochastic differential equations by means of (multilevel) Monte Carlo Euler-Maruyama methods.

This talk is based on joint work with Christoph Schwab (ETH Zürich).