相关论文: Mixing Estimates for Passive Scalar Transport by $…

We consider mixing problems in the form of transient convection--diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial…

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case…

In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation…

We develop an iterative framework for Bayesian inference problems where the posterior distribution may involve computationally intensive models, intractable gradients, significant posterior concentration, and pronounced non-Gaussianity. Our…

Given a bounded autonomous vector field $b \colon \mathbb R^d \to \mathbb R^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u=…

We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Amp{\`e}re equation, which is based on the maximum…

We study mixing for a divergence-free passive vector field $u$ transported by another divergence-free vector field $U$, where $u$ evolves according to $ \partial_t u + (U \cdot \nabla) u + \nabla p = 0.$ In recent years, a lot of attention…

We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded initial data for which the Cauchy problem for the…

This paper establishes new applications of the Christ-Journ\'e singular integral estimate to the transport equation for divergence-free vector fields in the Sobolev class $W^{1,p}$ with $p>1$. Our main result is a stability estimate for the…

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline…

We prove the persistence of boundary smoothness of vortex patches for a non-linear transport equation in $\mathbb{R}^n$ with velocity field given by convolution of the density with an odd kernel, homogeneous of degree $-(n-1)$ and of class…

In Bayesian applications, there is a huge interest in rapid and accurate estimation of the posterior distribution, particularly for high dimensional or hierarchical models. In this article, we propose to use optimization to solve for a…

We study stochastic perturbations of linear systems of the form $$ dv(t)+Av(t)dt = \epsilon P(v(t))dt+\sqrt{\epsilon}B(v(t)) dW (t), v\in\mathbb{R}^{D}, (*) $$ where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed…

We present an analysis of multilevel Monte Carlo techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients ({\em cross-sections}) are heterogenous random fields. To do this,…

We provide a survey of recent results on model calibration by Optimal Transport. We present the general framework and then discuss the calibration of local, and local-stochastic, volatility models to European options, the joint VIX/SPX…

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is…

This paper presents an iteration method for solving linear particle transport problems in binary stochastic mixtures. It is based on nonlinear projection approach. The method is defined by a hierarchy of equations consisting of the…

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire's formula [6], which requires the…

Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal…

Vector-valued Gaussian mixtures form an important special subset of vector-valued distributions. In general, vector-valued distributions constitute natural representations for physical entities, which can mutate or transit among alternative…