Approximation of moments of Banach-valued random variables using Monte Carlo methods (Prof. Tommaso Vanzan)

10.06.2025 14:00 – 15:00

This talk addresses the numerical approximation of k-th moments of Banach-valued random variables using Monte Carlo methods.

One major difficulty is that classical textbook convergence and complexity analysis of Monte Carlo methods does not apply
since it heavily relies on the scalar product structure of an underlying Hilbert space.
Consequently, we will begin reviewing known results on the approximation of the mean of random variables valued in infinite-dimensional Banach spaces,
introducing the concepts of the Rademacher type and constant of a Banach space.

However, a simple numerical experiment shows that these analyses are not sharp when considering finite dimensional approximations, as typical in numerical analysis.
This motivates us to refine the available theory by either taking into account the dependency of Rademacher constants on the dimension of the approximation space, or by using a particular argument valid exclusively in $L^p$ spaces.

Numerically experiments will show that our results precisely describe the asymptotic complexity of a Monte Carlo estimator.
If time permits, we will finally introduce a novel sparse estimator for general k-th moments, along with corresponding convergence and complexity results.

This is based on an ongoing work with Kristin Kirchner (Delft TU & KTH), Fabio Nobile (EPFL) and Christoph Schwab (ETH).

Lieu

Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

entrée libre