Title: Solving Differential-algebraic Equations by Taylor Series (I): Computing Taylor Coefficients

Abstract: Presented paper is a part of a series on the DAETS code for solving DAE initial value problems using Taylor series expansion, explains the method for computing Taylor coefficients (TCs) with automatic differentiation. It covers fully implicit, nonlinear DAEs with higher-order derivatives. The paper provides algorithmic details and proves that the method either successfully computes the TCs of the local solution or identifies specific error conditions.

References: "Solving Differential-algebraic Equations by Taylor Series (I): Computing Taylor Coefficients" Nedialko S. Nedialkov and John D. Pryce BIT, Submitted March 2005

Title: Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

Abstract: We will review computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, this will provide an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation.

This talk will be based on the work of the same name by van den Berg, Jan Bouwe; Queirolo, Elena.

References: van den Berg, J. B., & Queirolo, E. (2022). Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE. Communications in Nonlinear Science and Numerical Simulation, 108, 1-22. [106133]. doi.org/10.1016/j.cnsns.2021.106133

Title: Zbieżność wyższego rzędu dla inkluzji różniczkowych (Higher Order Method for Differential Inclusions)

Abstract: Inkluzje różniczkowe są uogólnieniem równań różniczkowych zwyczajnych i można je postrzegać jako jako systemy (niestacjonarne) zależne od czasu. Wykorzystywane mogą być m.in. w analizie złożonych systemów lub w teorii sterowania. Podczas seminarium zreferuję pracę, której autorzy proponują algorytm znajdowania rozwiązań inkluzji różniczkowych, zdolny do osiągnięcia kwadratowej i sześciennej zbieżności. Zaprezentuję wyniki otrzymane i umieszczone w artykule dla wybranych nieliniowych systemów.

References: "Higher Order Method for Differential Inclusions" (2020) arxiv.org/abs/2001.11330

Title: Neural ODEs and its applications

Abstract: During our meetings, we will focus on findings from the article Neural Ordinary Differential Equations by Chen et. al. (2019). We will talk about the most important issues related to ODENets - a family of neural networks aimed at learning representations based on the latent state dynamics of the network. We will introduce the most important concepts related to flow-based generative models. We will also present selected applications of the presented methods.

References: Neural Ordinary Differential Equations by Chen et. al. (2019)

https://doi.org/10.48550/arXiv.1806.07366 

Title: Neural ODEs and its applications

Abstract: During our meetings, we will focus on findings from the article Neural Ordinary Differential Equations by Chen et. al. (2019). We will talk about the most important issues related to ODENets - a family of neural networks aimed at learning representations based on the latent state dynamics of the network. We will introduce the most important concepts related to flow-based generative models. We will also present selected applications of the presented methods.

References: Neural Ordinary Differential Equations by Chen et. al. (2019)

https://doi.org/10.48550/arXiv.1806.07366

Title: Abstract Domains for Constraint Programming with Differential Equations 

Abstract: Cyber-physical systems and their physical parts are frequently described by ordinary differential equations (ODEs). In the presented article, framework for validating such systems is proposed. Validation begins with the correct over-approximation of the reachability states of the given system. Then, the proposed solution employs abstract domains to solve constraint satisfaction problems involving ODEs, introducing two kinds of abstract domains that aim to leverage the continuity of ODE solutions.  

Based on the article: Abstract domains for constraint programming with differential equations by Ziat et al. (2020)

https://doi.org/10.1145/3427762.3429453 

Title: Equivalence of strength of the Szymczak and Leray functors 

Abstract: A necessary step in the construction of a Conley Index is choosing the right normal functor. The functor described by Szymczak is the most powerful choice, as it preserves the most information about the system. However, it is usually difficult to compute. On the contrary, the Leray functor is very straightforward to compute, at least on categories of modules. We will present the generalised construction of the Leray functor on any abelian category and show that on certain subcategories it has the same classifying power as the Szymczak functor. 

Title: Komputerowo wspierany dowód dyfuzji w problemie trzech ciał 

Abstract: W referacie przedstawię główne tezy przygotowanej przeze mnie rozprawy doktorskiej pod tytułem "Komputerowo wspierany dowód dyfuzji w problemie trzech ciał". W rozprawie tej przedstawiony został komputerowo wspierany dowód dyfuzji w eliptycznym ograniczonym problemie trzech ciał na płaszczyźnie. Rozważany jest kołowy ograniczony problem trzech ciał na płaszczyźnie. Wprowadzając parametr zaburzenia, otrzymujemy problem eliptyczny. Niezaburzony układ zachowuje energię, lecz można pokazać, że dla dostatecznie małej perturbacji istnieją orbity ze zmianą energii, która nie zależy od rozmiaru tej perturbacji. W dowodzie badamy przecięcia rozmaitości stabilnej i niestabilnej normalnie hiperbolicznego cylindra, wykorzystując odwzorowanie rozpraszające oraz teorię śledzenia. Udowadniamy istnienie pseudo-orbit, które śledzą właściwe orbity układu o interesujących nas zmianach energii. 

Rozprawa: http://wms.mat.agh.edu.pl/~nwodka/Rozprawa%20doktorska_Natalia_Wodka-Cholewa.pdf

Kod: https://github.com/NataliaWodka/Drift_Orbits_3bp

Title: A Dynamic Look at Persistent Homology 

Abstract: In the present-day methodology of persistent homology dominates the approach grounded on an algebraic decomposition theorem of persistence modules. However, the original approach by Edelsbrunner, Letscher, Zomorodian (2002), based on a simplicial complex and the concept of a filter has more geometric flavor with birth-death pairing of cells. A filter may be interpreted as a Morse function on the simplicial complex.

This Morse function induces a combinatorial dynamical system via a class of birth-death pairs, called shallow pairs. Other birth-death pairs may

be indicated by heteroclinic connections in this dynamical system. They require cancellation of shallow pairs to become visible as new shallow

pairs in the reduced complex. This leads to depth poset, a poset structure in the collection of birth-death pairs. The induced dynamical

system together with the depth poset may be used to get more insight into the shape of data.

Based on research in progress with Herbert Edelsbrunner. 

Title: Vines and Vineyards by Updating Persistence in Linear Time

Abstract: The original algorithm that computes persistence diagrams takes worst-case time cubic in the number of simplices. The main result is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm’s analysis is an elementary proof of the stability of persistence diagrams in the special case of piecewise-linear functions. The algorithm can be used to compute 1-parameter families of diagrams. 

Based on the article: Vines and vineyards by updating persistence in linear time by

D. Cohen-Steiner, H. Edelsbrunner and D. Morozov.

Title:  Kompleks Viertorisa-Ripsa 

Abstract: Kompleks Viertorisa-Ripsa jest abstrakcyjnym kompleksem symplicjalnym zdefiniowanym dla zbioru punktów w przestrzeni Euklidesowej który istotnie korzysta z odległości między punktami. Stanowi on łatwo obliczalną aproksymację dla typu homotopii zbioru z którego pochodzą punkty. Można pokazać dobre zachowanie tej konstrukcji w przypadku planarnym. 

Title: Optimal approximation of solutions to SDEs driven by countably dimensional Wiener process 

Abstract: In this talk we focus on numerical approximation of stochastic differential equations (SDEs) with countably dimensional noise

structure. First, we cover pointwise approximation problem for jump-diffusion SDEs with jumps driven by Poisson random measure [1].

Second, we investigate global approximation for models with additive noise [2]. Our analysis is performed in the spirit of Information-based

Complexity (IBC). In both cases, we deliver asymptotic lower error bounds holding for any algorithm in the predefined classes of methods.

We also construct implementable (randomised or with adaptive step-size control) Euler-Maruyama schemes which asymptotically attain these

estimates. Finally, we report results of numerical experiments conducted via CUDA API on Nvidia graphic cards and multiprocessing library on

CPUs.

References:

[1] Przybyłowicz, P., Sobieraj, M., Stępień, Ł.: Efficient approximation

of SDEs driven by countably dimensional Wiener process and Poisson

random measure, SIAM J. Numer. Anal. 60 (2022), 824–855.

[2] Stępień, Ł.: Adaptive step-size control for SDEs driven by countably

dimensional Wiener process, Numer. Algor. (2023).

Title: Global dynamics for steep sigmoidal nonlinearities in two dimensions (III)

Abstract: I will present the article „Global dynamics for steep sigmoidal nonlinearities in two dimensions” written by T. Gedeon, S. Harker, H. Kokubu, K. Mischaikow and H. Oka. Its focus is the construction of mathematical, combinatorial model of dynamics given by biological structures known as regulatory networks. In the first seminar I will present motivation and goals. It will be followed by remarks on some basic concepts (lattice, Morse sets, multivalued maps) and lastly I will be focusing on building two concepts: switching system and associated continuous switching system. 

Title: Vines and Vineyards

by Updating Persistence in Linear Time

Abstract: In this talk we will first introduce persistent homology and the stability theorem for persistence diagram, then we will talk about vine and vineyards of pairwise distance function. 

Title: Global dynamics for steep sigmoidal nonlinearities in two dimensions (II)

Abstract: I will present the article „Global dynamics for steep sigmoidal nonlinearities in two dimensions” written by T. Gedeon, S. Harker, H. Kokubu, K. Mischaikow and H. Oka. Its focus is the construction of mathematical, combinatorial model of dynamics given by biological structures known as regulatory networks. In the first seminar I will present motivation and goals. It will be followed by remarks on some basic concepts (lattice, Morse sets, multivalued maps) and lastly I will be focusing on building two concepts: switching system and associated continuous switching system. 

Title: Global dynamics for steep sigmoidal nonlinearities in two dimensions (I)

Abstract: I will present the article „Global dynamics for steep sigmoidal nonlinearities in two dimensions” written by T. Gedeon, S. Harker, H. Kokubu, K. Mischaikow and H. Oka. Its focus is the construction of mathematical, combinatorial model of dynamics given by biological structures known as regulatory networks. In the first seminar I will present motivation and goals. It will be followed by remarks on some basic concepts (lattice, Morse sets, multivalued maps) and lastly I will be focusing on building two concepts: switching system and associated continuous switching system. 

Title: Application of Radii Polynomial Approach to the Swift-Hohenberg differential equation (II)

Abstract: I will present the extension of Newton’s method to work with infinite dimensional Banach spaces. The main goal of this is to develop an abstract framework, called Radii Polynomial Approach, for rigorous numerical solving initial condition of differential equations. The main idea is that solving differential equation is equivalent to finding a zero of differential operator which is a map between functional spaces. Later I will show the application of Radii Polynomial Approach to find periodic orbit of Swift-Hohenberg equation which forces chaotic behavior of the system. The presentation is based on the article “Introduction to rigorous numerics in dynamics: general functional analytic setup and an example that forces chaos” by Jan Bouwe van den Berg.  

Title: Application of Radii Polynomial Approach to the Swift-Hohenberg differential equation 

Abstract: I will present the extension of Newton’s method to work with infinite dimensional Banach spaces. The main goal of this is to develop an abstract framework, called Radii Polynomial Approach, for rigorous numerical solving initial condition of differential equations. The main idea is that solving differential equation is equivalent to finding a zero of differential operator which is a map between functional spaces. Later I will show the application of Radii Polynomial Approach to find periodic orbit of Swift-Hohenberg equation which forces chaotic behavior of the system. The presentation is based on the article “Introduction to rigorous numerics in dynamics: general functional analytic setup and an example that forces chaos” by Jan Bouwe van den Berg.  

Title: An Introduction to an Introduction to Homotopy Type Theory (II)

Abstract: Homotopy Type Theory is an alternative foundational theory of mathematics. One of its major strengths is its ability to be mechanised — that is, explained to a computer. It naturally leads to functional programming and, at the same time, proof theory, allowing automated proof checking and maybe even proof searching. We are going to start with Intentional Type Theory and try to get a broad overview of the subject and its connections to constructive and classical mathematics, programming and logic.

Title: An Introduction to an Introduction to Homotopy Type Theory (I)

Abstract: Homotopy Type Theory is an alternative foundational theory of mathematics. One of its major strengths is its ability to be mechanised — that is, explained to a computer. It naturally leads to functional programming and, at the same time, proof theory, allowing automated proof checking and maybe even proof searching. We are going to start with Intentional Type Theory and try to get a broad overview of the subject and its connections to constructive and classical mathematics, programming and logic.

Title: Combinatorial approach to sampled dynamics based on surrogate Gaussian Process modeling (III)

Abstract: We propose a novel method combining combinatorial topological dynamics and Gaussian Process (GP) modeling, whereby we can characterize the global dynamics from a finite amount of data, with high confidence. More specifically, the data is used to construct a surrogate GP model. We then describe the dynamics using algebraic topological invariants inferred from a combinatorial multivector field built on the basis of the GP model.  Our experiments show that relatively sparse data is sufficient to obtain a qualitative description of the underlying dynamics with high confidence.

In the talk I will also present the basics of both concepts: combinatorial multivector fields introduced by M. Mrozek [M. Mrozek, Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes, Foundations of Computational Mathematics 17(2017), 1585--1633 ] and GP modeling. 

Title: Combinatorial approach to sampled dynamics based on surrogate Gaussian Process modeling (II)

Abstract: We propose a novel method combining combinatorial topological dynamics and Gaussian Process (GP) modeling, whereby we can characterize the global dynamics from a finite amount of data, with high confidence. More specifically, the data is used to construct a surrogate GP model. We then describe the dynamics using algebraic topological invariants inferred from a combinatorial multivector field built on the basis of the GP model.  Our experiments show that relatively sparse data is sufficient to obtain a qualitative description of the underlying dynamics with high confidence.

In the talk I will also present the basics of both concepts: combinatorial multivector fields introduced by M. Mrozek [M. Mrozek, Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes, Foundations of Computational Mathematics 17(2017), 1585--1633 ] and GP modeling. 

Title: Combinatorial approach to sampled dynamics based on surrogate Gaussian Process modeling (I)

Abstract: We propose a novel method combining combinatorial topological dynamics and Gaussian Process (GP) modeling, whereby we can characterize the global dynamics from a finite amount of data, with high confidence. More specifically, the data is used to construct a surrogate GP model. We then describe the dynamics using algebraic topological invariants inferred from a combinatorial multivector field built on the basis of the GP model.  Our experiments show that relatively sparse data is sufficient to obtain a qualitative description of the underlying dynamics with high confidence.

In the talk I will also present the basics of both concepts: combinatorial multivector fields introduced by M. Mrozek [M. Mrozek, Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes, Foundations of Computational Mathematics 17(2017), 1585--1633 ] and GP modeling. 

Title: Wprowadzenie do metod ścisłej analizy numerycznej. (3)

Abstract: Opowiem o podstawowych narzędziach ścisłej analizy numerycznej i ich przykładowych zastosowaniach. 

room 0094

Title: Wprowadzenie do metod ścisłej analizy numerycznej. (2)

Abstract: Opowiem o podstawowych narzędziach ścisłej analizy numerycznej i ich przykładowych zastosowaniach. 

Title: Wprowadzenie do metod ścisłej analizy numerycznej. 

Abstract: Opowiem o podstawowych narzędziach ścisłej analizy numerycznej i ich przykładowych zastosowaniach. 

room 0094

Title: Informal introduction to persistent homology. 

Abstract: Homology theory makes topology a computational subject. The discovery of persistent homology about 25 years ago facilitated applications of topology in data analysis and today Topological Data Analysis (TDA) is a very hot subject. In the talk I'll present basics of persistent homology together with the algorithm for its computation.