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01/14
4:00pm |
Bloc 117 |
Ye He |
Beyond Log-Concavity: Sampling Challenges and Advances in Multimodal and Heavy-Tailed Distributions
Sampling from non-log-concave distributions poses significant challenges in a variety of fields, from Bayesian inference to computational physics and machine learning. Unlike log-concave distributions, which offer theoretical guarantees for efficient sampling, non-log-concave distributions often feature complex landscapes, including multimodality and heavy tails, that hinder standard algorithms from exploring the state space effectively. In this talk, I will discuss key obstacles and recent advances in sampling algorithms for non-log-concave distributions. First, I will explore the behavior of classical methods, such as Langevin Monte Carlo (LMC) and Proximal Sampler in the presence of multiple modes and heavy-tailed behaviors, highlighting issues like metastability and slow mixing. I will then introduce techniques designed to overcome these challenges, including using denoising diffusion and novel modifications to the Gaussian noise. This presentation aims to shed light on how these innovations bridge the gap between theory and practice, offering a more nuanced understanding of sampling in complex, high-dimensional spaces. By addressing these fundamental challenges, we can deepen our insight into the behavior of advanced sampling algorithms in non-log-concave regimes. |
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01/16
4:00pm |
Bloc 117 |
Adrian van Kan |
From numerical simulations of rotating Rayleigh-Bénard convection at very low Ekman numbers to stochastic dynamics in quasi-two-dimensional turbulence
Rapidly rotating Rayleigh-Bénard convection (RRRBC) provides a paradigm for direct numerical simulations (DNS) of geo- and astrophysical fluid flows, but the accessible parameter space, despite great computational efforts, has remained restricted to moderately fast rotation (Ekman numbers $Ek \gtrsim 10^{-8}$), while realistic values of $Ek$ for applications are orders of magnitude smaller. Reduced equations of motion, the non-hydrostatic quasi-geostrophic equations (NHQGE) describing the leading-order behavior in the limit of rapid rotation ($Ek\to 0$) cannot capture finite rotation effects. This leaves the physically most relevant part of parameter space with small but finite $Ek$ currently inaccessible. I will describe the rescaled incompressible Navier-Stokes equations (RiNSE) [1,2] – a reformulation of the Navier-Stokes-Boussinesq equations informed by the scaling laws valid for $Ek\to 0$. I present the first fully nonlinear DNS of RRRBC at unprecedented rotation strengths down to $Ek=10^{-15}$ and below, showing numerically that the RiNSE predicts statistics which agree favorably with the NHQGE at very low $Ek$. This work opens the door to the exploration of a large region in the parameter space of rotating convection.
Beyond the stiffness of the Navier-Stokes equations in the presence of a small parameter such as the Ekman number, the chaotic nature of turbulence also presents a significant challenge. The Navier-Stokes equations in two dimensions (2D) differ significantly from three dimensions (3D) due to additional conservation laws. Solving the 3D Navier-Stokes equations in a thin-layer geometry, there is a critical layer height $H$ below which rigorous bounding arguments show that 3D modes decay due to viscosity, leading to 2D flow. Close to this critical threshold, the energy contained in 3D modes exhibits highly intermittent dynamics. In the second part of this talk, motivated by numerical simulations of this phenomenon, I will describe stochastic dynamics of a single mode in the vicinity of a b |
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