MaPHS Seminar Archive

19-02-2025:

Speaker: Natalia Jeszka

Title: Walking on the looptree

Abstract: Let us transform a random tree into a structure which might have been called its perfect opposite: a graph consisting only of cycles. It can be achieved with a simple operation of replacing every node by a cycle of length equal to the degree of the node and “gluing” these cycles in such a way that the layout resembles original tree.

This transformation results in a discrete structure known in the literature as a looptree (N. Curien, I. Kortchemski, Random stable looptrees, Electron. J. Probab.19 (2014), pp. 1-35). In the talk we will explore the global geometry of this graph as the number of nodes grows to infinity. We will also introduce a second level of randomness by considering random weights on the edges of the graph, which will influence both the discrete and limiting structures.

05 February 2025

Spreaker: Maik Overmars. 

Title: Convergence of off-policy temporal-difference learning for reversible Markov chains

Abstract:

Temporal-difference learning is one of the most fundamental algorithms within the field of reinforcement learning, and it is used to estimate the expected reward in a Markov chain. The algorithm works by iteratively updating its estimates by following a sequence of states sampled from the Markov chain. While standard temporal-difference learning has been shown to converge, the algorithm becomes instable when both off-policy learning and function approximation are introduced. However, both of these properties are common in many practical settings. To circumvent this, the usual approach in the literature is to modify the algorithm, but this can lead to worse performance or slower convergence.  Instead, we consider the standard algorithm when applied to a specific class of Markov chains, namely reversible chains.

We show convergence of off-policy temporal-difference learning with linear function approximation for this setting. By exploiting the reversibility property and the inherent structure of the update rule of the algorithm, we derive required conditions for convergence to hold. In particular, these conditions restrict off-policy behaviour of the algorithm. Our results show that temporal-difference learning can still be guaranteed in the off-policy case information is known about the structure of the Markov chain. We finally show how to apply our results on different classes of reversible Markov chains, such as random walks.

11 December 2024

Speaker: Natalia Jeszka

Title: Does the market jump? – on the Lévy-driven Ornstein-Uhlenbeck process

Abstract:

Stochastic differential equations are an important mathematical tool to describe diverse interacting systems. As such, they are often applied to model e.g. particles movements or relations between financial institutions.

In the talk, we will discover the significance of the dynamics’ continuity character: while particles are often assumed not to make jumps, some financial quantities may be allowed to do so. Our considerations will be based on one exemplary stochastic process which describes interactions between the insurance company and the market. 

27 November 2024

Speaker: Patryk Rygiel
Title: Geometric deep learning for blood flow modelling in aortic abdominal aneurysms (AAA)

Abstract:
The study of blood flow dynamics (hemodynamics) is an important step in assessing severity and progression of cardiovascular diseases. The usual way of in silico modelling of hemodynamics is to employ Computational Fluid Dynamics (CFD) which is a numerical approach of solving the Navier-Stokes equations that govern fluid dynamics. However accurate, the CFD simulations are known to be very computationally demanding in both time and resources required. To remedy that, in the recent years geometric deep learning methods, operating directly on 3D shapes, have been proposed as compelling surrogates of CFD by providing accurate estimates in just a few seconds.

In my work I focus on hemodynamics within abdominal aortic aneurysms (AAAs) which are pathologic dilations of the abdominal aorta. The development and progression of AAAs is very complex as they can stay asymptomatic for years causing no harm to the patient, as well as rupture with fatal consequences in ~80% of the cases. I will showcase my current research on developing the models for rapid hemodynamics estimation within AAA patients. I will go through the process of building a large-scale data population including 3D shapes and corresponding CFD simulations. Delving into the geometric deep learning methods, I will explain how to model the 3D shape in an informative way through projective geometric algebra and how to use it for training accurate hemodynamics.

• 13 November 2024
  MaPhS and Abacus will be teaming up to bring you a special edition of the Mathematics PhD Seminar, showcasing some of the interesting research conducted in the Applied Mathematics department to both students as well as fellow employees.
  You are all cordially invited to join us at 16:00 at Abscint, where two of our PhD students, Oki Amalia (SOR) and Filippo Testa (MAST) will be presenting their research (see the abstracts below). And of course there will also be plenty of time for discussions and getting to know each other over a nice drink.
  
  
  • Oki Amalia:  
    Scheduling surgeries with different priorities in dedicated operating room capacities
    In hospitals, patients who need treatment or surgery are often categorized based on their urgency levels, each with its own due date. In our work, we consider surgeries of two urgency levels: urgent surgery, which must start within 6 hours, and semi-urgent surgery, which must start within 24 hours. Scheduling urgent and semi-urgent patients is non-trivial, as patients have multiple priority levels and thus different due date targets. The scheduler continuously balances between reserving capacity for urgent patients that is potentially left idle and meeting the due dates for all patients. We present two policies to schedule urgent and semi-urgent patients in the dedicated operating room (OR) capacity: last-minute scheduling and near-online scheduling. Under last-minute scheduling, the scheduler assigns capacity to the waiting patients just before the start of the time slot in which the patient will be treated. Under near-online scheduling, the scheduler assigns a future time slot to the patient shortly after the patient's arrival. We developed a Markov decision process (MDP) for both scheduling policies. Further, we also propose two heuristics based on the policy that is easily applied by hospitals. In the first heuristic, we use all available capacity to schedule patients from the highest urgency level. In the second heuristic, we allow reserving capacity to anticipate urgent patient arrivals in the subsequent time slot. For the time-varying patient arrivals, we use nonstationary MDP, where we evaluate the performance of the optimal policies using discrete event simulation and compare it to the policies from the heuristics. We demonstrate the usefulness of our method by applying it to the case of our partnering hospital.
  • Filippo Testa: 
    Contact points between analysis and geometry: the Poincaré-Hopf theorem
    Given an object, it is often possible to establish a connection between its topological and geometrical properties (its shape) and the analytical properties of functions defined on it.
    Many results in this regard have been achieved, deeply intertwining the fields of geometry and analysis. This talk will introduce the Poincaré-Hopf theorem, which connects the shape of a manifold with the zeroes of vector fields defined on it, while also containing a basic introduction to the language of differential geometry. We will also go over some of its implications, like the hairy ball theorem.

30 October 2024:

Speaker: Hugo Hof

Speaker: Matthew Maat
Title: Running in circles: Cycle patterns and how algorithms use them for games
Abstract:
Parity games, mean payoff games and energy games are examples of games that are played on the vertices of a directed graph. The problem of finding optimal strategies or values for these games is a well-studied topic, with countless algorithms being proposed. It is interesting from a complexity-theoretic viewpoint, as it is one of the few problems in both NP and coNP for which no polynomial-time algorithm is known.
We introduce the notion of 'cycle pattern' to shed some light on the underlying structure of these games. We characterize which cycle patterns can be realized in a weighted graph. We show some hardness results related to cycle patterns and to computing the winner of a game using only cycle patterns. We also show some bounds on the maximum required size of weights in the graph, and what this implies for algorithms that solve mean payoff games

10 October 2024

Speaker: Lars Schroeder

Title: Stationary distribution of node2vec random walks on household models
Abstract: 
node2vec random walks are tuneable random walks that come from the popular computer science algorithm node2vec which is used for feature learning on networks. The transition probabilities of the random walks depend on the previous visited node and on the triangles that contain the current and the previous node. We will present results on the stationary distribution of these random walks on household models and compare it with the simple random walk.

26 June 2024:

Speaker: Maike de Jongh
Title: Controlling the Ising model using spatiotemporal Markov decision theory

Abstract: Dynamics that are governed by both spatial and temporal interaction structures occur in a wide range of disciplines, such as economics, ecology, logistics and healthcare. In many situations, it is of interest to control such spatiotemporal processes and to steer them towards desirable behaviours. In this talk, we introduce the spatiotemporal Markov decision process (STMDP), an extension of the classic Markov decision process that provides a framework for sequential decision making in spatiotemporal stochastic settings. We illustrate the framework by applying it to a dynamic version of the Ising model on a two-dimensional lattice. At low temperatures and under a small positive magnetic field, this model is known to transition from the all-minus configuration to the all-plus configuration through the formation of a droplet of +-spins that will eventually nucleate the entire lattice. Our aim is to speed up this nucleation process by means of external influences and find the most efficient strategy to drive the process towards the all-plus configuration. After casting this problem as an STMDP, we provide insights in the structure and performance of the optimal policy.

12 June 2024

Speaker:  Aditya Pappu
Title: "Watt a Network! A primer on Energy Networks”
Abstract: The goals of decarbonisation and energy security have led to an increasing deployment of renewables and increased electrification in the Netherlands. The increase in renewables leads to a mismatch in demand and supply which our failing grid infrastructure cannot keep up with. In many parts of the Netherlands the grid is running out of capacity and connection requests are being denied.

But fear not, all is not lost! 

In this presentation, we take a look at the emergence and importance of local energy networks and how distributed energy management can turn the tide in our favour in the energy transition. 

This presentation is for those:

• 1 - Who want a ‘gezellig’ introduction to energy networks
• 2 - Who want to know what the weird energy people on the 5th floor are up to
• 3 - Who want to see some funny (energy) memes
  
  Disclaimer: This presentation contains no math. It is given by an engineer. You have been warned. 

15 May 2024:
Speaker: Hongliang Mu
Title: Piece-wise Symplectic Model Reduction on Quadratically Embedded Manifolds 

Abstract:
In this work, we present a piece-wise symplectic model order reduction (MOR) method for Hamiltonian systems on quadratically embedded manifolds. For Hamiltonian systems, which suffer from slowly decaying Kolmogorov N-widths, linear-subspace reduced order models (ROMs) of low dimension can have insufficient accuracy. The recently proposed symplectic manifold Galerkin projection combined with the quadratic manifold cotangent lift approximation (SMG-QMCL) is a symplectic MOR method that achieves higher accuracy than linear-subspace symplectic MOR methods. In this paper, we improve the efficiency of the SMG-QMCL by proposing a piece-wise symplectic MOR approach. 

First, the QMCL map is approximated piece-wisely by a linear symplectic map on each discrete time-interval. Then the symplectic Galerkin projection is applied to obtain a series of reduced-order Hamiltonian systems. In case that the Hamiltonian of the full-order model is a polynomial, the series of the Hamiltonians of the ROMs can be preserved up to a multiple of a pre-given tolerance used in the Newton iteration. In the numerical example, we demonstrate the approximation quality and the energy-preservation of the proposed algorithm. 

01 May 2024:
Speaker: Alexander Wierzba
Title: Of port-Hamiltonian systems, a Neumann series and a very special matrix or: BIBO stability, vol. 2

Abstract:
Port-Hamiltonian systems (pHS) provide a useful tool for modelling physical systems such as e.g. flexible beams within mechanical systems or flow phenomena in fluid dynamics and chemistry. This is in particular due to the close connection of their mathematical structure with the concepts of energy-flows between and energy-conservation within systems.

In this talk we want to tackle the question of when a distributed port-Hamiltonian system is bounded-input bounded-output (BIBO) stable - that is whether bounded inputs will always result in bounded outputs - continuing recent research on subtleties of this notion for infinite-dimensional systems. Analysing and utilizing the special structure of the transfer function of distributed pHS, allows us to derive several sufficient conditions for BIBO stability in this case.

10 April 2024. -   !15:00 - 16:00 Hours!

Clément Lezane
Title: "From functional analysis to complexity analysis in optimisation"

Abstract:
Modern industries provide a large spectrum of complex optimisation problems which, most of the time, cannot be solved exactly. In these cases, we are often called to use an approximated model to balance between the accuracy and the complexity. Choosing the right model requires a great knowledge about different algorithms complexity.  This talk will focus on minimizing functions in compact sets. We will start by reviewing some fundamental concepts in functional analysis and make the connection with current research directions in optimisation.

03 April 2024: Petr Zamolodtchikov

Title: "Transfer Learning under Covariate Shift: Local k-Nearest Neighbours Regression with Heavy-Tailed Design"

20 March 2024: Nicolò Botteghi
Title: Parametric PDE Control with Deep Reinforcement Learning and Differentiable L0 Polynomial Policies.
Abstract:
Optimal control of parametric partial differential equations (PDEs) is crucial in many applications in engineering and science. In recent years, the progress in scientific machine learning has opened up new frontiers for the control of parametric PDEs. In particular, deep reinforcement learning (DRL) has the potential to solve high-dimensional and complex control problems in a large variety of applications. Most DRL methods rely on deep neural network (DNN) control policies. However, for many dynamical systems, DNN-based control policies tend to be over-parametrized, which means they need large amounts of training data, show limited robustness, and lack interpretability. In this work, we leverage dictionary learning and differentiable L0 regularization to learn sparse, robust, and interpretable control policies for parametric PDEs. Our sparse policy architecture is agnostic to the DRL algorithm and can be used in different policy-gradient and actor-critic DRL algorithms without changing their policy-optimization procedure. We test our approach on the challenging tasks of controlling parametric Kuramoto-Sivashinsky and convection-diffusion-reaction PDEs. We show that our method (1) outperforms baseline DNN-based DRL policies, (2) allows for the derivation of interpretable equations of the learned optimal control laws, and (3) generalizes to unseen parameters of the PDE without retraining the policies.

06 March 2024: Riccardo Michielan

Abstract:

In this talk we discuss a realistic social network model, where individuals connect whenever they share a group. Formally, the individual and group sets are chosen as realization of two independent homogeneous Poisson processes on R^d, with parameter lambda and mu. Then, any individual v will be part of group u with some probability g(v − u): in particular, where g is a fixed non-increasing radial function, so that individuals are more likely to join close-by groups. The social network obtained out of this construction is called Geometric Random Intersection Graph (GRIG). Firstly, we analyze basic properties of GRIGs, such as edge probability and degree distribution. Later, we discuss percolation on GRIGs, that is, the emergence of a connected component of infinite size. In particular, we see that the interplay between lambda and mu gives rise to a percolation phase transition.

21 February 2024: Leonardo Del Grande

Title: Exact Sparse Representation Recovery in Convex Optimization

Abstract: We show the recovery of the sparse representation of data in general infinite-dimensional optimization problems regularized by convex functionals. It is possible to define a suitable non-degeneracy condition on the minimal-norm dual certificate, extending the well-known non-degeneracy source condition (NDSC). In our general setting, we study how the dual certificate is acting, through the duality product, on the set of extreme points of the ball of the regularizer, seen as a metric space. This justifies the name Metric Non-Degenerate Source Condition (MNDSC). By assuming the validity of the MNDSC, together with the linear independence of the measurements on these extreme points, we establish that, for a suitable choice of regularization parameters and noise levels, the minimizer is unique and is uniquely represented as a linear combination of n extreme points (exact recovery). Finally, we obtain explicit formulations of the MNDSC for three problems of interest:

• Total variation regularized deconvolution problems, where we show that the classical NDSC implies our MNDSC;
• 1-dimensional BV functions regularized with their BV-seminorm;
• - Pairs of measures regularized with their mutual 1-Wasserstein distance.

07 February 2024: Julian Suk
Title: LaB-GATr: geometric transformer for large biomedical surface and volume meshes
Abstract: The transformer architecture is the de-facto gold standard in natural language processing and many computer vision problems but suffers from the computational load of computing an (n x n) attention matrix to model attention between all tokens. We construct an SE(3)-equivariant token reduction scheme to streamline the use of geometric algebra transformers for large-scale biomedical surface and volume meshes.

• 24 January 2024: Giacomo Cristinelli
  Title: Graph cuts in discretized perimeter-minimizing problems
  Abstract: Perimeter-minimizing problems are concerned with finding shapes or configurations that minimize the total length or area of their boundary subject to certain constraints. A classical example is Plateau problem. Discretizing the configuration space allows us to reformulate some of these problems as optimal segmentations on graphs, enabling the use of efficient discrete optimization methods for finding solutions. In this talk, I will present two examples of applications of this technique: a conditional gradient approach to optimal control with TV-regularization, and a discretized mean curvature flow for capillary surfaces.

20 December 2023: Robin Markwitz

Titel:  Modelling interval-censored data using marked point processes

Abstract: Interval-censored data are made up of a time span in which an event certainly occurred, but the exact time of occurrence is unknown. For example, the police are often able to determine a time interval within which a crime occurred, but not the exact time. Our approach for modelling interval-censored data is by using a marked point process to model occurrence times, with the intervals represented as marks. Previous approaches when modelling aoristic data usually assume that censoring occurs homogeneously in time. We have developed a non-homogeneous, semi-Markov-inspired censoring mechanism that is able to capture seasonal variations and external factors. A Bayesian approach is used to perform the missing data estimation, and MCMC methods are used to estimate the parameters of the complete model.

• 22 November 2023: Phillip Preußler
• 06 December 2023: Sander Dijkstra

• 08 November 2023: Yanna Kraakman & Anna Dankers
  Title Yanna: How to Randomize a Directed Hypergraph
  Title Anna: Analyzing radiology on-call hours: insights from queueing models

• 25 October 2023: Erwin Luesink
  Title: Numerical explorations of 2D turbulence
  Abstract:
  In this talk I will explain the role that geometric structure plays in fluid dynamics and use it to obtain structure-preserving numerical algorithms. These algorithms are shown to have several favourable properties and are used to generate simulations of several models that one encounters in geophysical fluid dynamics.

Title: A unified approach to von Neumann’s inequality and Crouzeix’s conjecture

Abstract:
We discuss bounds for a class of homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix–Palencia, Okubo–Ando, and von Neumann in a unified way and to refine a recent result by Crouzeix–Greenbaum. This talk is based on joint work with Felix Schwenninger, arXiv:2302.05389.

Title: Local convergence rates of the nonparametric least squares estimator with applications to transfer learning

Abstract: Convergence properties of empirical risk minimizers can be conveniently expressed in terms of the associated population risk. To derive bounds for the performance of the estimator under covariate shift, however, pointwise convergence rates are required. Under weak assumptions on the design distribution, it is shown that least squares estimators (LSE) over 1-Lipschitz functions are also minimax rate optimal with respect to a weighted uniform norm, where the weighting accounts in a natural way for the non-uniformity of the design distribution. This moreover implies that although least squares is a global criterion, the LSE turns out to be locally adaptive. We develop a new indirect proof technique that establishes the local convergence behavior based on a carefully chosen local perturbation of the LSE. These local rates are then used to construct a rate-optimal estimator for transfer learning under covariate shift.

Title: Two-Stage Facility Location Games with Strategic Clients and Facilities

Abstract: We consider non-cooperative facility location games where both facilities and clients act strategically and heavily influence each other. This contrasts established game-theoretic facility location models with non-strategic clients that simply select the closest opened facility. In our model, every facility location has a set of attracted clients and each client has a set of shopping locations and a weight that corresponds to her spending capacity. Facility agents selfishly select a location for opening their facility to maximize the attracted total spending capacity, whereas clients strategically decide how to distribute their spending capacity among the opened facilities in their shopping range. We focus on natural client behaviors with clients minimizing their waiting times for getting serviced, where a facility's waiting time corresponds to its total attracted client weight.
For multiple different versions of this model, we give results on the existence/approximation of subgame perfect equilibria and also provide almost tight constant bounds on the Price of Anarchy and the Price of Stability, which even hold for a broader class of games with arbitrary client behavior.

Title: Physics-informed Deep kernel learning solving linear partial differential equations

Abstract: This work proposes the Physics-informed Deep Kernel Learning (PI-DKL) model, a probabilistic approach that integrates physical laws with Gaussian processes (GPs) to solve linear partial differential equations (PDEs) and handle inverse problems by treating unknown coefficients as hyper-parameters. The work integrates Gaussian processes, a non-parametric Bayesian machine learning technique, to provide a flexible prior distribution over functions and a fully probabilistic workflow for estimating uncertainty. The approach also uses deep kernel learning as feature extractors and dimensionality reduction methods for more robust GP regression, offering non-parametric flexibility in kernel learning to capture non-stationary structures within the data.

Title: "Capacity planning in the court of law"

Abstract: The court of law faces many challenges regarding timely and predictable jurisdiction. It is a complex logistical puzzle to optimally plan cases and schedule judges, legal assistants, and rooms. After a general introduction on the court of law and its current planning strategies, I will elaborate more on the analysis of the number of incoming law cases.

Title: “Riemannian Geometry, shape/image analysis, and Deep Learning"

Title: “Data-driven fluid models via a filtering approach”
Abstract:
In large-eddy simulation (LES) large-scale flow structures are solved, while the effect of small-scale motions is modelled. With the increase of computational power, high-fidelity numerical results become increasingly accessible and may serve to construct models for coarse-grid fluid simulations. By decomposing space- and time-dependent data into fixed spatial profiles and corresponding time series, already existing theory from data assimilation and Kalman filtering can be used to derive (stochastic) fluid models. In this talk, we provide a low-level introduction to fluid modelling and demonstrate how filtering approaches improve simulations of the two-dimensional Euler equations on the sphere.

Title: “Introduction to Asymptotic Homogenization for Phase Change Materials in Thermal Energy Storage System.”

Abstract:

Thermal Energy Storage (TES) systems are increasingly being used as a means of energy storage and management, especially in the field of renewable energy. TES refers to a system that stores heat energy for later usage and is based on the working principle of sensible or latent energy, where the latter is also known as phase change material (PCM). On small scales (microstructure), the properties of composite materials vary greatly due to their heterogeneous nature, making it challenging to predict their behaviour.

• 
  Abstract: Osteoarthritis is a common disease that causes the breakdown of cartilage, mostly affecting people aged 60 and older. Currently there is no effective treatment for osteoarthritis. Recently, much work has been put into the use of stem cells to develop personalised treatments for OA patients. In such treatments, stem cells would differentiate to new healthy cartilage, which could be injected in an affected joint. However, the mechanism underlying stem cell differentiation towards cartilage cells is not yet well understood, hindering the development of these treatments. In this talk, we discuss the use of ODE models based on gene regulatory networks to better understand this mechanism. We first discuss the inference of gene regulatory networks from experimental data. Subsequently, we discuss the ODE model posed on this network, and how such models can be used to better understand stem cell differentiation

Title: Of BIBO-stability and infinite-dimensional systems
Abstract:  BIBO-stability – that is the property that the output of a system will remain bounded if the input is bounded  – has been an important concept employed in the study of systems and control theory of ODEs in the past decades. Most recently it has in particular been employed in the context novel control techniques.
In this talk, we consider BIBO-stability of a class of systems described by infinite-dimensional linear state-space models as they are used in the study of PDEs. In doing so we highlight the crucial differences to the finite-dimensional case that require a more careful consideration of concepts such as that of a solution and lead to the emergence of a priori counterintuitive results.
In addition, to aid investigations of concrete systems in practical and theoretical applications, we study the preservation of the BIBO-property under multiplicative and additive perturbations of the system.

Speaker: Tjeerd Jan Heeringa

Title:“Different activation functions for machine learning, and why higher order matters.”

Abstract:

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Title: "From Christmas trees to cancer treatment: dominating the dynamics of spatio-temporal processes"

Abstract:
Processes that are governed by both spatial and temporal dynamics occur in a wide range of different disciplines, such as economics, ecology, logistics and healthcare. In various situations, it is of interest to interfere with the dynamics of such spatio-temporal processes in order to steer them towards a certain desirable behaviour. In this talk, we introduce the spatio-temporal Markov decision process, an extension of the classic Markov decision process that can serve as a framework for decision making in spatio-temporal stochastic settings. We illustrate the framework by means of a dynamic version of the renowned Ising model in Christmas spirit. Also, we touch upon the potential application of spatio-temporal Markov decision theory in solving current challenges in developing effective treatment for cancer, which is a strong motivation for this research project

Title: "Understanding radiative transfer with polarization" 

Abstract: In the high frequency limit unpolarized light is commonly described by the radiative transfer equation (RTE), a linear scalar partial differential equation. However, for some applications knowledge of the polarization of the light is required and the RTE should be replaced by a system of four coupled PDE’s. In this talk we will explain the physics behind these equations and demonstrate that they can be cast into a metriplectic formulation, which show that there is an energy associated to these equations that is conserved, and an entropy that is dissipated.

Title: "Improved Smoothed Analysis of 2-Opt for the Euclidean TSP"

Abstract:
The 2-opt heuristic is a simple local search heuristic for the Travelling Salesperson Problem (TSP). Although it usually performs well in practice, its worst-case runtime is poor. Attempts to reconcile this difference have used smoothed analysis, in which adversarial instances are perturbed probabilistically.

We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the points in an adversarial instance are perturbed by Gaussian random variables. This model was previously used by Manthey \& Veenstra, who obtained smoothed complexity bounds polynomial in $n$, the dimension $d$, and the perturbation strength $\sigma^{-1}$. However, their analysis only works for $d \geq 4$. The only previous analysis for $d \leq 3$ was performed by Englert, R\"oglin \& V\"ocking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in $n$ and $\sigma^{-d}$, and super-exponential in $d$.

As it is somewhat unsatisfactory that no direct analysis existed for Gaussian perturbations that yields polynomial bounds for all $d$, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds.

Title:  "Duality for Neural Networks through Reproducing Kernel Banach Spaces"

Abstract:
The universal approximation theorem guarantees that neural networks (NN) with infinite width can approximate any continuous functions. However, the question is how which functions are easy or hard to represent by a NN. Reproducing Kernel Hilbert spaces (RKHS) have been a very successful tool in various areas of machine learning. However, for neural networks the Hilbert setting is not sufficient due to strong nonlinear coupling of the weights. To study how neural networks approximate arbitrary functions, we need the more general Reproducing Kernel Banach spaces (RKBS). This class of integral RKBS can be understood as an infinite union of RKHS spaces. As the RKBS is not a Hilbert space, it is not its own dual space. However, we show that its dual space is again an RKBS where the roles of the data and parameters are interchanged, forming an adjoint pair of RKBSs including a reproducing property in the dual space.  

Title: Optimal Algorithms for Stochastic Complementary Composite Minimization

Abstract:
Inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. This problem corresponds to the minimization of the sum of a (weakly) smooth function endowed with a stochastic first-order oracle, and a structured uniformly convex (possibly nonsmooth and non-Lipschitz) regularization term. Despite intensive work on closely related settings, prior to our work no complexity bounds for this problem were known. We close this gap by providing novel excess risk bounds, both in expectation and with high probability. Our algorithms are nearly optimal, which we prove via novel lower complexity bounds for this class of problems. We conclude by providing numerical results comparing our methods to the state of the art.

Title: "Going off-grid: implicit neural representations for 3D vascular models"

Abstract:
Personalised 3D vascular models are valuable for monitoring and treatment planning of patients with cardiovascular diseases. Traditionally, these models have been constructed using explicit representations, e.g. meshes or 3D voxelmasks. Obtaining these models from 3D image data is, however, a cumbersome task. Moreover, these representations require costly preprocessing for downstream tasks, such as CFD. We propose to represent these structures implicitly by the zero levelset of their signed distance function, represented by a neural network. This allows for continuous modeling of these surfaces, hence the term: off-grid. These representations can be obtained from a sparse point cloud. We will demonstrate the potential of this approach for vascular modeling.

Title: “Realizing the energy transition by using local energy markets”

Abstract:
To stop global warming, an energy transition must take place. However, doing so is causing large problems in the electricity distribution grid. As traditional methods of solving grid problems are no longer feasible, new solutions must be created. The most promising of these is ‘demand side management’ for which several algorithms have been created. Local energy markets must now be used to implement such a method in practice. In this talk, we give more details behind the problems that currently arise and the possible solutions. Furthermore, we discuss the current research directions of the PhD track

Title: "Resilience of the internet"

Abstract:
Cellular networks have become one of the critical infrastructures as many services depend increasingly on wireless connectivity. Therefore, it is important to quantify the resilience of existing cellular network infrastructures against potential risks, ranging from natural disasters to security attacks, that might occur with a low probability but might lead to severe disruption of the services. To investigate this, we did a data-driven analysis using existing data of the Dutch wireless network. Moreover, in this talk, we give some answers to the question how we can improve this resilience with new techniques that are introduced in 5G wireless networks.