Sylwester Arabas (Jagiellonian University, Kraków, Poland)

Title:

Overview of PySDM and PyMPDATA: two new packages for numerically solving coagulation and transport problems in cloud physics and beyond

Abstract:

PySDM is a Pythonic high-performance (CPU & GPU) implementation of the Super-Droplet Method (SDM) Monte-Carlo algorithm for representing collisional growth of particles (such as cloud droplets) - a probabilistic alternative to the Smoluchowski coagulation equation.

PyMPDATA offers Numba-accelerated implementation of the MPDATA algorithm for solving generalised transport equations modelling conservation/balance laws, scalar-transport problems, convection-diffusion phenomena.

The talk will provide an overview of the features of the two packages developed recently at UJ (https://github.com/atmos-cloud-sim-uj).

Nikodem Szyszka (Jagiellonian University, Poland)

Title:

Globalne bifurkacje równowagi problemu (n+1) - ciał.

Abstract: 

Na podstawie pracy García-Azpeitia, C., and J. Ize. "Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators." Journal of Differential Equations 251.11 (2011): 3202-3227. (https://www.sciencedirect.com/science/article/pii/S0022039611002981) przedstawię globalne bifurkacje równowagi układu składającego się z n jednakowych obiektów obracających się wokół obiektu centralnego. Następnie przyjmując stosowne założenia, przedstawię rozwiązanie dla zagadnienia (n+1) - ciał.

Szymon Drenda (Jagiellonian University, Poland)

Title:

Niejawny algorytm C^1 oparty na wielomianach Czebyszewa dla autonomicznych równań różniczkowych z opóźnieniem

Abstract:

Na podstawie pracy "A Rigorous Implicit C^1 Chebyshev Integrator for Delay Equation" (https://link.springer.com/article/10.1007/s10884-020-09880-1) chciałbym zaprezentować proponowany przez autorów algorytm C^1 do ścisłego całkowania numerycznego równań różniczkowych zwyczajnych z opóźnieniem, oparty na wielomianach Czebyszewa. Postaram się również przedstawić pewne oszacowania błędów, oraz część kroków użytyw do stworzenia skończenie wymiarowej aproksymacji rozwiązania.

Anna Gierzkiewicz-Pieniążek (University of Agriculture in Kraków) 

Title:

Generalization of Sharkovskii's Theorem to multi-dimensional maps with an attracting periodic orbit 

Abstract:

In a joint work with Piotr Zgliczyński, we show that the methods used by Burns and Hasselblatt to prove Sharkovskii's Theorem can be generalized to the case of higher-dimensional maps with an attracting periodic orbit.

As an application, we prove the existence of $n$-periodic orbits for almost all $n\in\mathbb{N}$ in the R\"ossler system with an attracting periodic orbit, for two sets of parameters. The proof is computer-assisted.  

Jakub Zelek (Jagiellonian University, Poland)

Title:

Homologie persystentne odwzorowań próbkowanych w oparciu o reprezentacje kołczanowe

Abstract:

Podczas seminarium będziemy omawiać nowe podejście do analizy filtracyjnej odwzorowań próbkowanych, która bazuje na homologiach persystentnych. Całość będzie realizowana w oparciu o reprezentacje kołczanowe, które dostarczą nam odpowiednich narzędzi do rekonstruowania pewnych odwzorowań.

We wstępie zostaną krótko omówione problemy, które mogą zostać rozwiązane przy pomocy nowego podejścia.Następnie zostaną rozważone pewne obiekty matematyczne oraz twierdzenia, które pozwolą nam przejść ze świata kołczanów do świata homologii. Całość prezentacji zostanie podsumowana przykładami eksperymentów numerycznych, które pokażą siłę jak i pewne słabości podejścia.

Całość zostanie zrealizowana w oparciu o publikację:

Takeuchi, H. The persistent homology of a sampled map: from a viewpoint of quiver representations. J Appl. and Comput. Topology (2021). https://doi.org/10.1007/s41468-021-00065-3

Daniel Wilczak (Jagiellonian University, Poland)

Title:

Recent advances in rigorous computation of Poincaré maps

Abstract:

I will present an algorithm for rigorous computation of Poincar\'e maps. It addresses the following questions:

• What is the optimal choice (in the sense of obtained enclosures) of a Poincar\'e section near a periodic orbit?

• If the sections are fixed (like guards in hybrid systems), can we reduce overestimation by appropriate choice of coordinates on these sections?

• Can we take advantage from the knowledge on internal representation of the solution to IVP in a rigorous ODE solver to reduce overestimation in computation of Poincar\'e maps?

The algorithm [3] is already implemented and freely available as a part of the CAPD library [1,2]. Theoretical considerations will be supported by numerical tests [3].

References:

[1] CAPD::DynSys Computer Assisted Proofs in Dynamics, a C++ package for rigorous numerics. http://capd.ii.uj.edu.pl/

[2] T. Kapela, M. Mrozek, D. Wilczak, P Zgliczyński, {CAPD::DynSys:} a flexible {C}++ toolbox for rigorous numerical analysis of dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 10.1016/j.cnsns.2020.105578

[3] T. Kapela, D. Wilczak, P Zgliczyński, Recent advances in rigorous computation of Poincar\'e maps, Communications in Nonlinear Science and Numerical Simulation, in review 

Filip Oskar Przybycień (Jagiellonian University, Poland)

Title: 

Topological Computation Analysis of Meteorological Time-Series Data

Abstract: 

A topological computation method is applied to noisy time-series data obtained from meteorological measurement. This method is based on the idea of the Morse decomposition, which is a decomposition of the dynamics into invariant sets, called the Morse sets, and their gradient-like connections. However, time-series data obtained from real measurements are often highly stochastic due to the presence of noise. A decomposition is, therefore, determined statistically. We will consider data taken from the pressure patterns in the troposphere and the stratosphere in the northern hemisphere. The motions detected by the analysis are consistent with changes between characteristic pressure patterns that have been previously recognised in meteorological studies. Based on a paper by Hidetoshi Morita, Masaru Inatsu, and Hiroshi Kokubu.

Andreas Rauh (ENSTA Bretagne, France)

Title:

Verified Parameter Identification of Quasi-Linear Cooperative System Models: A Combination of Branch-and-Bound as well as Contractor Techniques

Abstract:

Many dynamic system models in (control) engineering, especially in the frame of thermo-fluidic applications, are described after a first-principle modeling by sets of either discrete-time difference equations or by sets of ordinary differential equations that have certain monotonicity properties. The most important property that allows for a simplification of parameter identification tasks is the cooperativity of the state equations that often results from properties such as conservation of mass and energy. In this case, (quasi-)linear system models are characterized by dynamics matrices that can be bounded by interval-valued Metzler matrices. Those matrices are characterized by non-negative off-diagonal elements. In the case that all system states are additionally ensured to be non-negative, the property of cooperativity leads to a decoupling of lower and upper bounding systems that enclose all possible state trajectories with certainty. This presentation gives an overview of branch-and-bound techniques as well as contractor-based approaches for the parameter identification of cooperative systems on the basis of uncertain measurements. An overview of current research towards the GPU-based identification of nonlinear models of the thermal behavior of high-temperature fuel cells concludes this talk.

Jim Wiseman (Agnes Scott College, Decatur, USA)

Title:

Persistence of Morse decompositions for finite-resolution dynamics

Abstract:

We are interested in the behavior of a dynamical system given by a self-map of a compact metric space. We approximate the system with a finite discretization, and study the topological persistence of the dynamics as the resolution of the discretization changes. In particular, we look at the Morse decomposition of the dynamics into recurrent and gradient-like pieces, which gives both local and global information.

Uli Bauer (Technical University of Munich, Germany)

Title:

Persistent matchmaking

Abstract: 

In this talk, I will give an overview of some recent results motivated by the computation and applications of persistent homology – a theory that creates a bridge between the continuous world of topology and the discrete world of data, and assigns stable and structurally simple invariants to geometric point sets, parametrized over geometric scale.

I will explore why cohomology is so much more efficient in such computations, and how it can also be employed to efficiently compute the induced matchings that exhibit the stability of persistence diagrams. The main task in this construction is to compute the image of a morphism of persistence modules, which will be reviewed in light of the most recent advances in computing persistence. I will also explain how the goal of finding the perfect partner helps with cohomology computation at a large scale, involving hundreds of trillions (!) of simplices.

Boumediene Hamzi (Imperial College, London, UK)

Title:  

Machine Learning and Dynamical Systems meet in Reproducing Kernel Hilbert Spaces

Abstract: 

Since its inception in the 19th century through the efforts of Poincaré and Lyapunov, the theory of dynamical systems addresses the qualitative behaviour of dynamical systems as understood from models. From this perspective, the modeling of dynamical processes in applications requires a detailed understanding of the processes to be analyzed. This deep understanding leads to a model, which is an approximation of the observed reality and is often expressed by a system of Ordinary/Partial, Underdetermined (Control), Deterministic/Stochastic differential or difference equations. While models are very precise for many processes, for some of the most challenging applications of dynamical systems (such as climate dynamics, brain dynamics, biological systems or the financial markets), the development of such models is notably difficult. On the other hand, the field of machine learning is concerned with algorithms designed to accomplish a certain task, whose performance improves with the input of more data. Applications for machine learning methods include computer vision, stock market analysis, speech recognition, recommender systems and sentiment analysis in social media. The machine learning approach is invaluable in settings where no explicit model is formulated, but measurement data is available. This is frequently the case in many systems of interest, and the development of data-driven technologies is becoming increasingly important in many applications. The intersection of the fields of dynamical systems and machine learning is largely unexplored and the objective of this talk is to show that working in reproducing kernel Hilbert spaces offers tools for a data-based theory of nonlinear dynamical systems. In this talk, we introduce a data-based approach to estimating key quantities which arise in the study of nonlinear autonomous, control and random dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success- once the nonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energies for nonlinear systems. We apply this approach to the problem of model reduction of nonlinear control systems. It is also shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system. We also show how kernel methods can be used to detect critical transitions for some multi scale dynamical systems. We also use the method of kernel flows to predict some chaotic dynamical systems. Finally, we show how kernel methods can be used to approximate center manifolds, propose a data-based version of the centre manifold theorem and construct Lyapunov functions for nonlinear ODEs. This is joint work with Jake Bouvrie (MIT, USA), Peter Giesl (University of Sussex, UK), Christian Kuehn (TUM, Munich/Germany), Romit Malik (ANNL), Sameh Mohamed (SUTD, Singapore), Houman Owhadi (Caltech), Martin Rasmussen (Imperial College London), Kevin Webster (Imperial College London), Bernard Hasasdonk, Gabriele Santin and Dominik Wittwar (University of Stuttgart).

Tomasz Kaczyński (Université de Sherbrooke, Canada)

Title: 

Towards an Application-Driven Extension of Forman's Discrete Morse Theory to Vector Functions

Abstract: 

Our primary motivation for persistent homology is in its applications to shape similarity measures. Multidimensional or multiparameter persistence comes into play in that context when two objects are to be simultaneously compared according to several features. The ideas go back to early 1900s when Pareto’s optimal points of multiple functions were studied with applications to economy on mind.

In our previous work, we developed an algorithm that produces an acyclic partial matching (A, B, C) on the cells of a given simplicial complex, in the way that it is compatible with a vector-valued function given on its vertices. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. Until now, any simplex added to C by our algorithm has been defined as critical. It was legitimate to do so, because an application-driven extension of Forman’s discrete Morse theory to multi-parameter functions has not been carried out yet. In particular, no definition of a general combinatorial critical cell has been given in this context. We now propose new definitions of a discrete Morse vector function (for short, dmv function), of its gradient field, its regular and critical cells. We next show that the function fused as input for our algorithm gives rise to a dmv function g with the same order of sublevel sets and the same partial matching as the one produced by our algorithm. This is a joint work with Madjid Allili, Claudia Landi, and Filippo Masoni.

Barbara Mahler (Oxford University, UK)

Title:

Contagion Dynamics for Manifold Learning

Abstract: 

Contagion maps exploit activation times in threshold contagions to assign vectors in high-dimensional Euclidean space to the nodes of a network.  A point cloud that is the image of a contagion map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. In this talk, I will present how contagion maps perform as a manifold-learning tool on a number of different real-world and synthetic data sets, and compare their performance to that of Isomap, one of the most well-known manifold-learning algorithms. I will show that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, while Isomap fails due to noise-induced error. 

Dorota Nowak (Jagiellonian University, Poland)

Title: 

Algorithm for rigorous integration of systems with conservation laws

Abstract:

In general case for solving systems of differential equations we often use rigorous numerical techniques, where all operations are conducted on intervals instead of numbers. Those algorithms (if given problem has a solution) compute bounds that contain this solution. On the other hand, there is also a large group of systems with specific properties, for example Hamiltonian systems, which preserve energy (as long as they are autonomous) and their flow is a symplectic transformation. These additional information are not used in rigorous algorithms.  In order to obtain more accurate results in case of rigorous methods for systems with specific properties, we could modify the existing algorithms by adding conditions, for example condition on symplecticity of the flow in these systems or condition on energy preservation.

Donald Woukeng (Jagiellonian University, Poland)

Title:

Topological data analysis of time series data

Abstract:

I will present the sliding windows technique - a method for analyzing the periodicity in time-series data. The first part of the seminar will be about the persistence diagram's stability, a tool that we will use for the analysis. After that, we will have Taken's delay embedding for time series to construct point cloud data, and from the point cloud, we will give the method to compute the persistence diagram using the matrix reduction algorithm. The last part of the talk will be the implementation of the method to study time-series data on the monthly average temperature in two countries. 

Roberto Barrio, (University of Zaragoza, Spain)

Title:

Homoclinic phenomena in mathematical neuron models

Abstract:

Bursting phenomena are found in a wide variety of fast–slow systems. In this talk, I  present a global analysis of the main elements, focusing on the homoclinic bifurcations, involved in the bursting dynamics, exemplified in the Hindmarsh–Rose neuron model. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast–slow perspective no longer applies. Based on this information, we propose a global theoretical description that includes surfaces of codimension-one homoclinic bifurcations that are exponentially close to each other, isolas of homoclinic bifurcations, etc. These curves organize the bifurcations associated with the spike-adding process common in most mathematical neuron models. This talk is based on the recent publications: "Homoclinic organization in the Hindmarsh–Rose model: A three parameter study", R Barrio, S Ibañez, L Perez, Chaos 30 (5), 053132, 2020 and "Spike-adding structure in fold/hom bursters", R Barrio, S Ibañez, L Perez, S Serrano, Communications in Nonlinear Science and Numerical Simulation 83, 105100, 2020. 

Damian Sadowski (Jagiellonian University, Poland)

Title:

Przybliżanie potoku wielościanami

Abstract:

Na podstawie artykułu:   E. Boczko, W. D. Kalies i K. Mischaikow, Polygonal approximation of flows, przedstawię metodę ścisłego przybliżania zbiorów niezmienniczych izolowanych w rozkładzie Morse'a poprzez konstrukcję podziału przestrzeni na wielościany o transwersalnym przepływie potoku przez ściany kowymiaru jeden. 

Vitaliy Kurlin (University of Liverpool, UK)

Title:

Introduction to Periodic Geometry for applications in Crystallography and Materials Science.

Abstract:

 A periodic crystal is modeled as a periodic set of zero-sized points in 3-space. Such a periodic set is usually given by a unit cell (a parallelepiped defined by 3 edge-lengths and 3 angles) and a motif of points with fractional coordinates in this cell. Representing a crystal as a unit cell plus a motif is highly ambiguous. Hence a reliable comparison of rigid crystals should be based on isometry invariants that are preserved under any rigid motions, hence are independent of a unit cell and a motif. The talk will introduce a stable and complete isometry classification of periodic crystals that will accelerate materials discovery. Most past invariants of crystals such as symmetry groups are unstable or discontinuous under atomic vibrations. The new classification establishes foundations of periodic geometry and continuous crystallography.

Jason James (Florida Atlantic University, US)

Title:

Computer assisted existence proofs for collision orbits in the planar circular restricted three body problem

Abstract:

I'll discuss some recent work with Maciej Capinski and Shane Kepley where we study collisions in the restricted three body problem using computer assisted methods of proof.  The idea is to combine rigorous numerical integrators with classical Levi-Civita regularization methods so that questions about the existence of collisions are reformulated as equivalent questions about solutions of some two point boundary value problems.  We study the resulting boundary value problems using interval Newton methods.  We focus on collision-to-collision orbits and equilibrium-to-collision orbits.

David Mosquera Lois (Universidade de Santiago de Compostela, Spain)

Differential Topology on finite spaces

Abstract:

The study of finite topological spaces, or equivalently, partially ordered sets, has regain some popularity during the last decade. Mainly, due to its connections with some problems on low dimensional topology, combinatorial topology, combinatorics and group theory. The purpose of this talk is to present the combinatorial or discrete counterparts of several notions of classical differential topology in the context of finite topological spaces. In particular, we will pay extra attention to a Morse Theory for this setting and its relationship with a notion of Lusternik-Schnirelmann category.

Jacek Brodzki (University of Southampton, UK)

Title:

Geometry, Topology, and the structure of data

Abstract:

Modern data is astonishing in its variety, and is far removed from anything that could be held in a spreadsheet or analyzed using traditional methods. Through intense research effort, topology has emerged as a source of novel methodology to provide insight into the structure of very complex, high dimensional data. It provided us with tools like persistent homology, which are used to compute numerical topological characteristics of the data. More recently, these methods have been augmented by geometric insights, which are valuable in capturing the structure of and relationships between complex shapes. 

In this talk, I will provide an overview of new techniques from topology and geometry and illustrate them on particular examples. One set of data was created through the study of CT scans of human lungs, and another addresses the problem of classification of three-dimensional shapes.

 Auguste Bourgois (ENSTA Bretagne, France)

Title:

Set-membership methods for stability analysis of dynamical systems

Abstract:

In this talk, I will present some of the results obtained in the course of my thesis, entitled "Safe & collaborative autonomous underwater docking".

This thesis consisted in developing new methods to prove the feasibility of an underwater docking mission. Our first approach was based on stability analysis of the dynamical system composed of the robot and its target. In that aim, we developed a new tool, the stability contractor, allowing to prove asymptotic stability of discrete, continuous and hybrid system.

Our second approach was based on reachability analysis of the robot, and uses in particular constraint programming methods. We developed a new constraint programming tool, the Lohner contractor, allowing to contract intervals of trajectories according to differential constraints.

The first part of this talk will focus on stability analysis of hybrid system : we will illustrate the stability contractor and new applications for the CAPD library through a practical example. The second part of this talk will introduce the Lohner contractor, and apply it to find an outer approximation of the basin of attraction of a system.

Kelly Spendlove (University of Oxford, UK)

Title:

Morse, Conley, and Computation

Abstract:

Algebraic topology and dynamical systems are intimately related: the algebra may constrain or force the existence of certain dynamics. Morse homology is the prototypical theory grounded in this observation. Conley theory is a far-reaching topological generalization of Morse theory. Within the Conley theory the connection matrix is the mathematical object which transforms the approach into a truly homological theory: it is the Conley-theoretic generalization of the Morse boundary operator.

We’ll discuss a new formulation of the connection matrix theory, which casts the connection matrix in categorical, homotopy-theoretic language.  This enables the efficient computation of connection matrices via the technique of reductions in combination with algebraic-discrete Morse theory. We will also discuss a software package for such computations and demonstrate the theory and computations with some applications to classical examples as well as a Morse theory on braids.

Stefano De Marchi (University of Padova, Italy)

Title:

A short tour on Radial Basis Functions approximation and VSDK (in about 50 slides)

Abstract:

In this talk I wish to present the most rilevant results I have obtained in the last years using Radial Basis Functions (RBF) in approximation of data and functions. After a short overview of the RBF approximation problem with emphasis to some stability issues, I’ll focus on Variably Scaled Kernels (VSK) for approximating discontinuous functions, called Variably Scaled Discontinous Kernels (VSDK) [1]. Numerical experiments and applications in imaging will conclude our journey [2].

[1] S. De Marchi, F. Marchetti and E. Perracchione: “Jumping with Variably Scaled Discontinuos Kernels (VSDK) “, BIT Numerical Mathematics 60 (2020), pp. 441-463.

[2] Stefano De Marchi, Wolfgang Erb, Francesco Marchetti, Emma Perracchione, Milvia Rossini: ”Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI”. SIAM J. Sci. Comput. 42(2), (2020) pp. B472–B491.

Piotr Kopacz (Maritime University, Gdynia, Poland)

Title:

Towards improving search patterns by time-minimal paths

Abstract:

In this talk we shall consider improving the standard search patterns under the effect of the drift caused by a perturbation of a vector field, basing on the time-minimal paths as the local solutionsto Zermelo's navigation problem. The real-world applications of the study refer in particular to the aeronautical search and rescue operations.

Pedro Chocano (Universidad Complutense, Madrit, Spain)

Title:

Lefschetz fixed point theorems for finite spaces and applications.

Abstract:

We adapt the definition of the Vietoris map to the framework of finite topological spaces and we prove a coincidence theorem. From here, we deduce Lefschetz fixed point theorems for single valued and multivalued maps. Finally, it is given an application of the theory developed to the approximation of discrete dynamical systems in polyhedra.

Ryan Slechta (Purdue University, West Lafayette, IN, US)

Title:

Persistence of the Conley Index in Combinatorial Dynamical Systems

Abstract:

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to track features in a changing multivector field.

Marian Mrozek (Jagiellonian University, Kraków, Poland)

Title:

Combinatorial Topological Dynamics

Abstract: 

The ease of collecting enormous amounts of data in the present world together with problems in gaining useful knowledge out of it stimulate the development of mathematical tools to deal with the situation. Since, in the case of data collected from dynamic processes, there are no sufficient theoretical grounds for building mathematical models (differential or difference equations), one concerns a general polynomial equation and chooses coefficient to match the available data. However, there is an alternative: instead of building artificial equations one can use the data to construct a combinatorial counterpart of a dynamical system and adapt to is available topological tools. In the lecture I'll present an overview of some recent results in this direction.