@phdthesis{20563,
  abstract     = {The theory of optimal transport provides an elegant and powerful description of many evolution
equations as gradient flows. The primary objective of this thesis is to adapt and extend the
theory to deal with important equations that are not covered by the classical framework,
specifically boundary value problems and kinetic equations. Additionally, we establish new
results in periodic homogenization for discrete dynamical optimal transport and in quantization
of measures.
Section 1.1 serves as an invitation to the classical theory of optimal transport, including the
main definitions and a selection of well-established theorems. Sections 1.2-1.5 introduce the
main results of this thesis, outline the motivations, and review the current state of the art.
In Chapter 2, we consider the Fokker–Planck equation on a bounded set with positive Dirichlet
boundary conditions. We construct a time-discrete scheme involving a modification of the
Wasserstein distance and, under weak assumptions, prove its convergence to a solution of this
boundary value problem. In dimension 1, we show that this solution is a gradient flow in a
suitable space of measures.
Chapter 3 presents joint work with Giovanni Brigati and Jan Maas. We introduce a new theory
of optimal transport to describe and study particle systems at the mesoscopic scale. We prove
adapted versions of some fundamental theorems, including the Benamou–Brenier formula and
the identification of absolutely continuous curves of measures.
Chapter 4 presents joint work with Lorenzo Portinale. We prove convergence of dynamical
transportation functionals on periodic graphs in the large-scale limit when the cost functional
is asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances converge
to 1-Wasserstein distances constructed from crystalline norms on R
d
.
Chapter 5 concerns optimal empirical quantization: the problem of approximating a measure
by the sum of n equally weighted Dirac deltas, so as to minimize the error in the p-Wasserstein
distance. Our main result is an analog of Zador’s theorem, providing asymptotic bounds for
the minimal error as n tends to infinity.
},
  author       = {Quattrocchi, Filippo},
  issn         = {2663-337X},
  keywords     = {optimal transport, kinetic equations, boundary value problems, quantization, gradient flows, homogenization},
  pages        = {240},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures}},
  doi          = {10.15479/AT-ISTA-20563},
  year         = {2025},
}

@unpublished{20570,
  abstract     = {We investigate the minimal error in approximating a general probability
measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with
prescribed cardinality $n$. The error is measured in the $p$-Wasserstein
distance. In particular, when $1\le p<d$, we establish asymptotic upper and
lower bounds as $n \to \infty$ on the rescaled minimal error that have the
same, explicit dependency on $\mu$.
  In some instances, we prove that the rescaled minimal error has a limit.
These include general measures in dimension $d = 2$ with $1 \le p < 2$, and
uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform
measures, we prove the limit existence for $p \ge d$ as well.
  For a class of compactly supported measures with H\"older densities, we
determine the convergence speed of the minimal error for every $p \ge 1$.
  Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper
estimate of the minimal error when $1 \le p < d$.
  In the initial sections, we survey the state of the art and draw connections
with similar problems, such as classical and random quantization.},
  author       = {Quattrocchi, Filippo},
  booktitle    = {arXiv},
  keywords     = {optimal empirical quantization, vector quantization, Wasserstein distance, semidiscrete optimal transport, Zador’s Theorem, Pierce’s Lemma},
  title        = {{Asymptotics for optimal empirical quantization of measures}},
  doi          = {10.48550/arXiv.2408.12924},
  year         = {2024},
}