The submitted Master's thesis by Jakub Osička investigates implicit numerical solutions of evolutionary partial differential equations in a one-dimensional setting. The approach is based on a discretization of the Jordan–Kinderlehrer–Otto (JKO) scheme.

The thesis provides a thorough review of the necessary theoretical background, including optimal transport theory, Wasserstein spaces, and gradient flows. As these topics are not part of the standard curriculum at our institution, the student had to study them independently, which he has done successfully.

The thesis interprets distributional solutions of evolution PDEs as gradient flows in the space of probability measures equipped with the Wasserstein metric and studies their time discretization via the JKO scheme. Exploiting an integral representation of the Wasserstein distance on R, the author develops a spatial discretization of the JKO scheme and reformulates the resulting variational problem as an optimization task.

A principal result of the thesis is the proof of 𝛤-convergence of the discretized scheme to the variational JKO formulation as the spatial discretization parameter tends to zero. Furthermore, the author derives the analytical solution of the p-Laplacian equation (the Barenblatt profile), identifies conditions under which it generates a gradient flow associated with a convex functional, and computes stationary solutions in the presence of a confinement potential. The thesis also includes numerical simulations implemented in Python and presents several additional observations, including the empirical convergence rate of the proposed numerical scheme compared with the Barenblatt profile.

Jakub regularly attended consultations and made steady progress on the thesis throughout the entire period.

In conclusion, we consider the thesis to be of a very good standard. We have no formal objections and recommend the thesis for defense with the grade A (excellent).

The review was written by the cosupervisors, Prof. dr hab. Błażej
Miasojedow, Ing. Matej Benko. I fully agree with the statement.

I was very pleased with the master's thesis by Jakub Osička. It deals with a topic of Wasserstein gradient flows, which is much advanced for a master's level.

Firstly, it is by no means an easy task to gather the literature on this topic and present it in a concise and consistent manner. It is my opinion that Jakub dealt with that challenge splendidly: the first half of the thesis does a great job of introducing the reader to the topic, from the general optimal transport theory, to Wasserstein metric spaces, and then to the gradient flow theory. The exposition is excellent, very readable and comprehensive. I am convinced that Jakub have well understood the (highly delicate) theoretical background that was required to engage the research part of his thesis.

The classical numerical approach to Wasserstein gradient flows leads through the JKO scheme, which discretizes the flow in time. At a given time step it still remains to discretize in space. The novel idea in Jakub's thesis is to approximate the unknown probability distributions by absolutely continuous probability measures of piece-wise constant density. The setting is limited to 1D where exact formulas for the Wasserstein distance are available.

I find Jakub's approach very natural comparing to the atomic approximations used in the literature: if the functional involves entropy, then the probability distributions are forced to be absolutely continuous. The main theoretical result of the thesis involves the Gamma-convergence for the approximation scheme. The proof is elegant and efficient. Finally, the scheme has been experimentally validated with the analytical benchmarks being known solutions of parabolic PDEs. It is worth mentioning that an original derivation of one of the solutions is included.

 

Of course, no thesis is perfect, and below I point to several aspects that could potentially be improved, for instance in the possible future publication.

My main remark concerns the lower semi-continuity for the entropy functional $\mathcal{U}$. My opinion is that compactness of the domain is immaterial to the l.s.c. The superlinearity of $U$ is essential either way. This is provided one defines $\mathcal{U}$ on the whole space of probabilities, by putting $+\infty$ for measure that are not absolutely continuous. To see this, it is enough to take $U(s) = s$ and a sequence of compactly supported mollifying kernels converging to a Dirac delta. Please also see Remark 9.3.8 in Ambrosio-Giglie-Savaré. Accordingly, the superlinearity must be also added to the assumptions (A1)-(A5) to achieve Gamma-convergence (Gamma-limit functionals are always lower semi-continuous).

The coverage of the theoretical background has been prepared based on several references. I found that (only on couple of occasions) this was a reason of slight inconsistencies. The primary example concerns the definition of the inner product in eq. (2.17), which is in the spirit of Ambrosio-Giglie-Savaré. Then, in Definition 3.6 the Wasserstein gradient is defined (formally) in the sense of Otto, which is well covered in Figalli-Glaudo. This gradient is a scalar function, and it does not fit to the inner product (2.17), as claimed prior to Def. 3.6.

The title of Section 3.3 reads "Gradient flows in $\mathscr{P}_2^{ac}(\Rd)$". I am slightly sceptical if we should treat $\mathscr{P}_2^{ac}(\Rd)$ as the metric space and consider gradient flows on it. After all, this space is not complete. It is probably safer to work with the space $\mathscr{P}_2(\Rd)$, and let the entropy functional to take care of the absolute continuity.

Finally, I would discourage from using contraction: can't, shouldn't, etc., in the text.

Several additional minor remarks are to be found below:

- As far as I understand, the energy $U$ in the Example 2.28 is a special case of the one in Example 2.27, i.e. for $m=3-p$, but with the constants conveniently chosen for the q-Laplacian case. Meanwhile, on the first read I had the impression that Example 2.28 covers something essentially new, e.g. because of the way the regime for $p$ is derived from scratch. I would suggest rewriting.

- I would define the functional $\mathscr{F}_h$ on the whole space $\mathscr{P}_p(\Rd)$ and consider Gamma-convergence in this space – it is clearer when the space is common for the sequence and the limit functional.

- The first sentence in Proposition 6.7 should be deleted.

- Prop. 6.9 should finish with $\mathcal{V}_h(\rho_h) \to \mathcal{V}(\rho)$. Similar remark applies to Prop. 6.11. Topics for thesis defence:

1. The numerical scheme that is put forth in the paper is based on the idea of approximating probability measures on the real line by a piece-wise constant absolutely continuous measure. Then, both the functional $\mathcal{U}$ and the Wasserstein distance is computed for such measures directly. Meanwhile, the potential and interaction energies $\mathcal{V}$ and $\mathcal{W}$ are additionally approximated by $\mathcal{V}_h$ and $\mathcal{W}_h$. The way I see it, $\mathcal{V}_h(\rho_h) = \mathcal{V}(\tilde{\rho}_h)$, where $\tilde{\rho}_h$ is the discrete measure with the atoms situated at the nodes $x_i$. Similar remark applies to $\mathcal{W}_h$. Has the author considered using the functionals $\mathcal{V}$ and $\mathcal{W}$ directly instead? It would probably be more fitting to the continuous approach proposed in the thesis, and, in some sense, the method would fall within the scope of the family of finite element methods.