This presentation focuses on the study of the regulartiy of random wavelet series. We first study their belonging to certain functional spaces and we compare these results with long-established results related to random Fourier series. Next, we show how the study of random wavelet series leads to precise pointwise regularity properties of processes like fractional Brownian motion. Additionally, we explore how these series helps create Gaussian processes  with random Hölder exponents.

One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and infinite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We discuss some dynamic games of this type, from static to two player to the associated mean field game, with some numerical and existence-uniqueness results.

Based on works with Andrew Ledvina and with Emre Parmaksiz.

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting  the temporal dynamics of the time series distribution. We  estimate the drift function from data samples by nonparametric, e.g. kernel regression methods,  and the simulation of the SB diffusion  yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments.  First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion,  and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. 

This talk explores recent advancements in stress and flux-based finite element methods. It focuses on addressing the limitations of traditional finite elements, in order to describe complex material behavior and engineer new metamaterials.

Stress and flux-based finite element methods are particularly useful in error estimation, laying the groundwork for adaptive refinement strategies. This concept builds upon the hypercircle theorem [1], which states that in a specific energy space, both the exact solution and any admissible stress field lie on a hypercircle. However, the construction of finite element spaces that satisfy admissible states for complex material behavior is not straightforward. It often requires a relaxation of specific properties, especially when dealing with non-symmetric stress tensors [2] or hyperelastic materials.

Alternatively, methods that directly approximate stresses can be employed, offering high accuracy of the stress fields and adherence to physical conservation laws. However, when approximating eigenvalues, this significant benefit for the solution's accuracy implies that the solution operator cannot be compact. To address this, the solution operator must be confined to a subset of the solution that excludes the stresses. Yet, due to compatibility conditions, the trial space for the other solution components typically does not yield the desired accuracy. The second part of this talk will therefore explore the Least-Squares method as a remedy to these challenges [3].

To conclude this talk, we will emphasize the integration of those methods within global solution strategies, with a particular focus on the challenges regarding model order reduction methods [4].

[1] W. Prager, J. Synge. Approximations in elasticity based on the concept of function space.

Quarterly of Applied Mathematics 5(3), 1947.

[2] FB, K. Bernhard, M. Moldenhauer, G. Starke. Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity, Numerical Methods for Partial Differential Equations 37(4), 2021.

[3] FB, D. Boffi. First order least-squares formulations for eigenvalue problems, IMA Journal of Numerical Analysis 42(2), 2023.

[4] FB, D. Boffi, A. Halim. A reduced order model for the finite element approximation of eigenvalue problems,Computer Methods in Applied Mechanics and Engineering 404, 2023.

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for the analysis, by means of deep-learning techniques, of data produced by computational science and engineering applications.  At the core of these methods is the supervised training algorithm to learn the neural network realization, a highly non-convex optimization problem that is usually solved using stochastic gradient methods.

However, distinct from deep-learning practice, scientific machine-learning training problems feature a much larger volume of smooth data and better characterizations of the empirical risk functions, which make them suited for conventional solvers for unconstrained optimization.

In this talk, we empirically demonstrate the superior efficacy of a trust region method based on the Gauss-Newton approximation of the Hessian in improving the generalization errors arising from regression tasks when learning surrogate models for a wide range of scientific machine-learning techniques and test cases. All the conventional solvers tested, including L-BFGS and inexact Newton with line-search, compare favorably, either in terms of cost or accuracy, with the adaptive first-order methods used to validate the surrogate models.

Euclidean Ramsey Theory is a natural multidimensional version of Ramsey Theory. A subset of Euclidean space is called Ramsey if, for any $k$, whenever we partition Euclidean space of sufficiently high dimension into $k$ classes, one class much contain a congruent copy of our subset. It is still unknown which sets are Ramsey. We will discuss background on this and then proceed to some recent results.