Tommaso Vanzan
Approximation of moments of Banach-valued random variables using Monte Carlo methods abstract
Abstract:
his 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 applysince 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).
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Henri Darmon (McGill University, Montreal)
Maximal tori in orthogonal groups and explicit class field theory abstract
Abstract:
Maximal tori in an orthogonal group of signature (n,2) give rise to special points on an orthogonal Shimura variety which are defined over class fields of an associated CM field. A largelyconjectural framework for associating class invariants to orthogonal groups and maximal tori of arbitrary realsignatures will be discussed.
14:15 • EPF Lausanne, MA A1 12
Abdul-Lateef Haji-Ali (Heriot-Watt University, Edinburgh, UK)
Multiindex Monte Carlo method for semilinear stochastic partial differential equations abstract
Abstract:
In this talk, I will present an exponential-integrator based mulitiindex Monte Carlo method (MIMC) for weak approximations of mild solutions of semilinear stochastic partial differential equations (SPDE). I will present the recent theoretical results on multiindex coupled solutions of the SPDE, namely that such couplings are stable and satisfy multiplicative error estimates, and describe how this theory can be utilized to obtain a tractable MIMC method. Numerical examples show that MIMC outperforms alternative methods, such as multilevel Monte Carlo, in settings with low regularity. I will also briefly discuss another recent work that extends the analysis of the truncated-Milstien scheme with an antithetic estimator to SPDEs.
16:15 • EPF Lausanne, MA A1 10