The derivation of kinetic equations as mean field limits from microscopic dynamics is a longstanding problem in kinetic theory which goes back to the work of Dobrushin in the late 70s. Recently, this derivation has been resolved for first order systems with Riesz type singularities in the work of Serfaty-Duerinckx and Bresch-Jabin-Wang via the renormalized modulated energy method. The aim of this talk would be to review these developments and explain how this approach can be incorporated within the framework of systems of trajectories which include weights evolving in time according to some appropriate ODE. At the macroscopic PDE level this corresponds to a nonlocal transport equation with a nonlocal source terms. Joint work with Jose Carrillo and Pierre-Emmanuel Jabin.

We revisit the classic simple symmetric exclusion process, which has the same hydrodynamic limit as a system of independent random walkers. Our aim is to provide a deeper understanding why the exclusion mechanism does not influence the hydrodynamic limit. We do this by interpreting each time the exclusion mechanism is invoked as a collision between particles, then keep track of the number of collisions in the system and pass to the hydrodynamic limit. In fact we study four of such variables under different scaling regimes and obtain a zoo of hydrodynamic limits - some deterministic and some stochastic.

Solving Integer Programs (IPs) is generally NP-hard. But this does not imply that all instances are inherently hard. In fact, a substantial body of research has focused on identifying tractable subclasses and developing efficient (fixed-parameter tractable) time algorithms for those. This talk will give a little overview of some of the key results and techniques.

According to Sklar’s theorem, any multidimensional absolutely continuous distribution function can be uniquely represented as a copula, which captures the dependence structure among the vector components. In real data applications, the interest of the analyses often lies on specific functionals of the dependence, which quantify aspects of it in a few numerical values. A broad literature exists on such functionals, however extensions to include covariates are still limited. This is mainly due to the lack of unbiased estimators of the conditional copula, especially when one does not have enough information to select the copula model. Several Bayesian methods to approximate the posterior distribution of functionals of the dependence varying according covariates are presented and compared; the main advantage of the investigated methods is that they use nonparametric models, avoiding the selection of the copula, which is usually a delicate aspect of copula modelling. These methods are compared in simulation studies and in two realistic applications, from civil engineering and astrophysics.

Given that all models are wrong, it is important to understand the performance of methods when the settings for which they have been designed are not met, and to modify them where possible so they are robust to these sorts of departures from the ideal. We present two examples with this broad goal in mind. \[ \] We first look at a classical case of model misspecification in (linear) mixed-effect models for grouped data. Existing approaches estimate linear model parameters through weighted least squares, with optimal weights (given by the inverse covariance of the response, conditional on the covariates) typically estimated by maximizing a (restricted) likelihood from random effects modelling or by using generalized estimating equations. We introduce a new ‘sandwich loss’ whose population minimizer coincides with the weights of these approaches when the parametric forms for the conditional covariance are well-specified, but can yield arbitrarily large improvements when they are not. \[ \] The starting point of our second vignette is the recognition that semiparametric efficient estimation can be hard to achieve in practice: estimators that are in theory efficient may require unattainable levels of accuracy for the estimation of complex nuisance functions. As a consequence, estimators deployed on real datasets are often chosen in a somewhat ad hoc fashion and may suffer high variance. We study this gap between theory and practice in the context of a broad collection of semiparametric regression models that includes the generalized partially linear model. We advocate using estimators that are robust in the sense that they enjoy root n consistent uniformly over a sufficiently rich class of distributions characterized by certain conditional expectations being estimable by user-chosen machine learning methods. We show that even asking for locally uniform estimation within such a class narrows down possible estimators to those parametrized by certain weight functions and develop a new random forest-based estimation scheme to estimate the optimal weights. We demonstrate the effectiveness of the resulting estimator in a variety of semiparametric settings on simulated and real-world data.

We apply computational techniques of convex stochastic optimization to optimal operation and valuation of electricity storages in the face of uncertain electricity prices. Our approach is based on quadrature approximations of Markov processes and on the Stochastic Dual Dynamic Programming (SDDP) algorithm which is widely applied across the energy industry. The approach is applicable to various specifications of storages, and it allows for e.g. hard constraints on storage capacity and charging speed. Our valuations are based on the indifference pricing principle, which builds on optimal trading strategies and calibrates to the user's initial position, market views and risk preferences. We illustrate the effects of storage capacity and charging speed by numerically computing the valuations using stochastic dual dynamic programming. If time permits, we provide theoretical justification of the employed computational techniques.

In this talk, I will discuss recent results which characterize the long-time behavior of a conditional McKean-vlasov equation related to interacting diffusions on regular trees. This is joint work with Kavita Ramanan.

We study a problem arising in the management of staff positions in institutions with a cameralistic budgeting system, for example in public universities in Germany. When a new employee is hired, she needs to be assigned to one or (partially) to several of the available positions. These position may already be (partially) assigned to other staff members during certain time periods. Some positions are better suited for the new hire due to, e.g., their associated pay grades or other administrative reasons. One seeks a solution with assignments to suitable open positions and wants few changes of those assignments over time. This yields a multi-objective optimization problem with a lexicographic objective function which can be seen as a scheduling problem with non-availability periods for the machines.

We derive structural insights into this problem and present several MIP-formulations for it. Their solutions are optimal w.r.t. the three most important objectives and optimal or near-optimal w.r.t. the least important objective, respectively. In particular, we are able to solve our problem faster than with a straight-forward approach if one is willing to potentially sacrifice a bit of accuracy in the least important objective. In addition, we present very fast combinatorial algorithms for important special cases of the problem. Overall, we can solve most practically relevant instances in less than a few seconds. Our optimization tool was developed in collaboration with the administration of the School of Computation, Information, and Technology (CIT) at the Technical University of Munich where it is now used on a regular basis.

Providing guarantees on the reproducibility of discoveries is essential when drawing inferences from high-dimensional data. Such data is common in numerous scientific domains, for example, in biomedicine, it is imperative to reliably detect the genes that are truly associated with the survival time of patients diagnosed with a certain type of cancer, or in finance, one aims at determining a sparse portfolio to reliably perform index tracking. This talk introduces the Terminating-Random Experiments (T-Rex) selector, a fast multivariate variable selection framework for high-dimensional data. The T-Rex selector provably controls a user-defined target false discovery rate (FDR) while maximizing the number of selected variables. It scales to settings with millions of variables. Its computational complexity is linear in the number of variables, making it more than two orders of magnitude faster than, e.g., the existing model-X knockoff methods. An easy-to-use open-source R package that implements the TRexSelector is available on CRAN. The focus of this talk lies on high-dimensional linear regression models, but we also describe extensions to principal component analysis (PCA) and Gaussian graphical models (GGMs).

We introduce a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic PDE system describing the evolution of the population density and the chemical signal. The key novelty lies in the transport term for the population, which depends on the second-order derivatives of the chemical field. This term is derived as a limiting anticipation-reaction mechanism for an infinitesimally small ant. We establish global existence and uniqueness of solutions, as well as the propagation of regularity of the initial data. We then analyze the long-time behavior of the system: we prove the existence of a compact global attractor and show that the homogeneous steady state becomes nonlinearly unstable under an inviscid instability criterion. Additionally, we provide a lower bound on the dimension of the attractor. Conversely, we prove that for sufficiently small interaction strength, the homogeneous steady state is globally asymptotically stable. Finally, we present several numerical simulations illustrating the model's dynamics.

References: Curvature in chemotaxis: A model for ant trail pattern formation, Charles Bertucci, Matthias Rakotomalala, Milica Tomasevic Existence and dimensional lower bound for the global attractor of a PDE model for ant trail formation, Matthias Rakotomalala, Oscar de Wit

A rather natural “Lagrangian particle” discretisation of a 1d scalar conservation law leads to the so-called “Follow the Leader” approximation of it. While this has been formally known for many decades (see e.g. the 1974 book by Whitham), the first rigorous results came only in the past ten years. It is by know quite well known that a suitable “upwind” version of the FtL approximation converges (as the number of particles goes to infinity and through a suitable reconstruction of the Eulerian density) to the unique entropy solution to the corresponding conservation law in Kruzkov’s sense. We will review the main results in the literature, including a series of paper in collaboration with M. Rosini, S. Fagioli, and G. Russo, a set of results by Holden and Risebro, and a very recent result by Storbugt. Recurring strategies to prove compactness of the scheme include BV estimates and (more recently) compensated compactness. An alternative approach tries to catch the L1 to L infinity “smoothing effect” of the PDE through a discrete one-sided Lipschitz condition. We will describe the latter approach in more detail, based on a paper in collaboration with G. Stivaletta (2022). We will also address some possible extension of the whole theory and list some open problems.

We study a class of learning models known as inverse optimization (IO), where the goal is to replicate the behaviors of a decision-maker (i.e., optimizer) with an unknown objective function. We discuss recent developments in IO concerning convex training losses and optimization algorithms. The main message of this talk is that IO is a rich learning model that can capture complex, potentially discontinuous behaviors, while the training phase is still a tractable convex program. We motivate the discussion with applications from control (learning the MPC control law), transportation (2021 Amazon Routing Problem Challenge), and robotics (comparing with state-of-the-art neural networks in MuJoCo environments).

We study travelling fronts in a two-component singularly perturbed (FitzHugh-Nagumo-like) reaction-diffusion system with spatially heterogeneous coefficients. A front solution has a sharp interface and connects two equilibrium states (background states) of the PDE, but due to the spatial inhomogeneity these background states are not constant in space. This renders a comoving frame approach in order to construct travelling fronts practically useless. The difficulty is worsened by the fact that both the speed and the shape of the profile a front are varying in time, as clearly seen in numeric simulations.

To deal with the challenges, we develop new techniques to deal with the spatiotemporal nature of the problem. We manage to prove existence of travelling fronts for a large range of parameters. In doing so, we find numerous insights on the circumstances for an interface to persist in time. In addition, we find a satisfying description of how travelling fronts move. Specifically, we discover that the position of a travelling front can be described in the singular limit by means of a delay differential equation. Numerically this delay equation seems to provide a very precise approximation for the position of the front for small values of the singularly perturbed parameter.

The Vertex Reinforced Jump Process (VRJP) is a self-reinforcing stochastic process on a graph. There are open questions about its behaviour on Z^d, for example, whether there exists a unique phase transition for d > 2. In order to help build an intuition for these questions, we develop an efficient algorithm for the simulation of the VRJP's random environment on large grid graphs, which can serve as approximations to its dynamics on Z^d. By exploiting the properties of the VRJP's beta-field, we propose an iterative algorithm for its efficient simulation. Its complexity is linear in the number of vertices and cubic in the maximal vertex degree. The random environment can be derived from this beta field by solving a linear system, whose size equals the number of vertices in the graph. To solve this linear system, we use Conjugate Gradients and give a brief discussion of some preconditioning and optimization strategies based on the adjacency matrices of the grid graphs

Matrix Concentration inequalities such as Matrix Bernstein inequality (Oliveira and Tropp) have played an important role in many areas of pure and applied mathematics. These inequalities are intimately related to the celebrated noncommutative Khintchine inequality of Lust-Piquard and Pisier. While these tend to be optimal when the underlying matrices are commutative, they are known to be sub-optimal in several other instances. Recently, we have leveraged ideas from Free Probability to fully remove the sub-optimal dimensional dependencies in these inequalities in a range of instances, yielding sharp bounds in many settings of interest. In this talk I will describe these results, some of the recent and ongoing work that it has sparked, and several open problems. Includes joint work with: March Boedihardjo (MSU); Ramon van Handel and Giorgio Cipolloni (Princeton); Petar Nizic-Nikolac, Anastasia Kireeva, Kevin Lucca, and Dominik Schroder (ETH); Xinmeng Zeng (Stanford); Dustin Mixon (OSU); Dmitriy Kunisky (Johns Hopkins); Thomas Rothvoss (U Washington); Haotian Jiang (U Chicago); Sivakanth Gopi (MSR). ______________________

Invited by Prof. Holger Rauhut