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Xavier Fernández-Real

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math.AP Machine Learning math.OC

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preprint2026arXiv

Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$ -dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$ -norm, we obtain explicit polynomial convergence rates in $L^2$ -norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.

preprint2023arXiv

Regularity Theory for Elliptic PDE

This manuscript aims to provide a self-contained introduction to the regularity theory for elliptic PDE, focusing on the main ideas rather than proving all results in their greatest generality. It can be seen as a bridge between an elementary PDE course and more advanced textbooks. This is a draft of the book "Regularity Theory for Elliptic PDE". The final version has been published in Zurich Lectures in Advanced Mathematics, EMS Press, 2022.

preprint2022arXiv

Stable cones in the thin one-phase problem

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $n\geq3$ is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions $n\le 5$ is one-dimensional, independently of the parameter $s\in (0,1)$ .

preprint2021arXiv

Free boundary regularity for almost every solution to the Signorini problem

We investigate the regularity of the free boundary for the Signorini problem in $\mathbb{R}^{n+1}$ . It is known that regular points are $(n-1)$ -dimensional and $C^\infty$ . However, even for $C^\infty$ obstacles $φ$ , the set of non-regular (or degenerate) points could be very large, e.g. with infinite $\mathcal{H}^{n-1}$ measure. The only two assumptions under which a nice structure result for degenerate points has been established are: when $φ$ is analytic, and when $Δφ< 0$ . However, even in these cases, the set of degenerate points is in general $(n-1)$ -dimensional (as large as the set of regular points). In this work, we show for the first time that, "usually", the set of degenerate points is small. Namely, we prove that, given any $C^\infty$ obstacle, for "almost every" solution the non-regular part of the free boundary is at most $(n-2)$ -dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $(-Δ)^s$ , and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $(n-1-α_\circ)$ -dimensional for almost all times $t$ , for some $α_\circ > 0$ . Finally, we construct some new examples of free boundaries with degenerate points.

preprint2016arXiv

$C^{1,α}$ estimates for the fully nonlinear Signorini problem

We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local $C^{1,α}$ estimates on each side of the smooth obstacle, for some small $α> 0$ . Our results extend those of Milakis-Silvestre in two ways: first, we do not assume solutions nor operators to be symmetric, and second, our estimates are local, in the sense that do not rely on the boundary data. As a consequence, we prove $C^{1,α}$ regularity even when the problem is posed in general Lipschitz domains.

preprint2014arXiv

Boundary regularity for the fractional heat equation

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1,1}$ domains. More precisely, we consider solutions to $\partial_t u + (-Δ)^s u=0 \textrm{ in }Ω,\ t > 0$ , with zero Dirichlet conditions in $\mathbb{R}^n\setminus Ω$ and with initial data $u_0\in L^2(Ω)$ . Using the results of the second author and Serra for the elliptic problem, we show that for all $t>0$ we have $u(\cdot, t)\in C^s(\mathbb{R}^n)$ and $u(\cdot, t)/δ^s \in C^{s-ε}(\overlineΩ)$ for any $ε> 0$ and $δ(x) = \textrm{dist}(x,\partialΩ)$ . Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable Lévy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.

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Xavier Fernández-Real

Research overview

Is this your work?

6 publications

Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

Regularity Theory for Elliptic PDE

Stable cones in the thin one-phase problem

Free boundary regularity for almost every solution to the Signorini problem

$C^{1,α}$ estimates for the fully nonlinear Signorini problem

Boundary regularity for the fractional heat equation

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Xavier Fernández-Real

Research overview

Is this your work?

6 publications

Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow

Regularity Theory for Elliptic PDE

Stable cones in the thin one-phase problem

Free boundary regularity for almost every solution to the Signorini problem

C1,αC^{1,α}C1,α estimates for the fully nonlinear Signorini problem

Boundary regularity for the fractional heat equation

See the researcher in context

Building this map preview

$C^{1,α}$ estimates for the fully nonlinear Signorini problem