Related papers: Geometry, mixing properties and hypocoercivity of …
Several recent works have explored stochastic gradient methods for variational inference that exploit the geometry of the variational-parameter space. However, the theoretical properties of these methods are not well-understood and these…
We develop a geometric formulation of stochastic dynamics in which noise, diffusion, path probabilities, fluctuation theorems, and entropy production arise from the intrinsic geometry of an evolving manifold rather than from externally…
We introduce a notion of geometric tempering using exponentially-dampened Mittag-Leffler tempering functions and closely investigate the univariate case. Characteristic exponents and cumulants are calculated, as well as spectral densities.…
This work is concerned with the development of a family of Galerkin finite element methods for the classical Kolmogorov's equation. Kolmogorov's equation serves as a sufficiently rich, for our purposes, model problem for kinetic-type…
Adaptive Langevin dynamics is a method for sampling the Boltzmann-Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the…
In this paper, we study stochastic homogenization of a coupled diffusion-reaction system. The diffusion-reaction system is coupled to stochastic differential equations, which govern the changes in the media properties. Though homogenization…
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline…
A simple model of an irreversible process is introduced. The equation of iterations in the model includes a noise generation term. We study the properties of the system when the noise generation term is a stochastic process (e.g. a random…
We study computational and statistical consequences of problem geometry in stochastic and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which stochastic- and…
Probabilistic generative models based on measure transport, such as diffusion and flow-based models, are often formulated in the language of Markovian stochastic dynamics, where the choice of the underlying process impacts both algorithmic…
In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric…
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and…
Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing…
We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…
This note is devoted to a simple method for proving hypocoercivity of the solutions of a kinetic equation involving a linear time relaxation operator, i.e. the construction of an adapted Lyapunov functional satisfying a Gronwall-type…
Knitted fabrics exemplify a broad class of architected materials capable of large deformations, enabling shape morphing, mechanical biocompatibility, and embedded multifunctionality without material damage. Although geometric nonlinearity…
Our work is part of the close link between continuous-time dissipative dynamical systems and optimization algorithms, and more precisely here, in the stochastic setting. We aim to study stochastic convex minimization problems through the…
Chaotic micromixers such as the staggered herringbone mixer developed by Stroock et al. allow efficient mixing of fluids even at low Reynolds number by repeated stretching and folding of the fluid interfaces. The ability of the fluid to mix…