Alkalmazott Analízis Szeminárium

A BME Matematika Intézet Analízis és Differenciálegyenletek Tanszékének közös Alkalmazott Analízis Szemináriuma 2016. őszén indult Faragó István (Differenciálegyenletek Tanszék) kezdeményezésére az MTA-ELTE Numerikus Analízis és Nagy Hálózatok Kutatócsoporttal együttműködésben. A szeminárium célja, hogy elősegítse egy alkalmazott analízissel (funkcionálanalízis, differenciálegyenletek, numerikus módszerek) foglalkozó kutatói kör kialakítását az intézeten belül. A szemináriummal fórumot szeretnénk biztosítani az alkalmazott analízissel foglalkozó matematikusok és az analízist alkalmazó kutatók számára az együttgondolkodásra. További cél az érdeklődő hallgatók (MSc, PhD) bevonása a kutatói munkába.

Szemináriumunk 2017-től felvette a Farkas Miklós Alkalmazott Analízis Szeminárium nevet. Ezzel szeretnénk emléket állítani egyetemünk egykori tanszékvezető matematikaprofesszorának, aki elindította egyetemünkön a matematikus-mérnök képzést, és a stabilitáselmélet valamint a biomatematika terén elért jelentős tudományos eredményeivel ill. könyveivel nagyban hozzájárult az alkalmazott matematika erősödéséhez. (English version of the introduction.)

 From the autumn semester of 2017 the talks will be in English on a regular basis. / 2017. őszi félévétől az előadásokat angol nyelven tartjuk. 

Organizers / Szervezők: Faragó István1,2,3, Karátson János1,2,3 ,Horváth Róbert1,3 ,Mincsovics Miklós1,3 (1BME, 2ELTE, 3MTA-ELTE NUMNET)

Request for e-mail notifications and remarks to the organizers / Feliratkozás az e-mail listára ill. egyéb megjegyzések a szeminárium szervezőihez

Prospective speakers: 

  • 27/09/18: Ilona Nagy (BME)
  • 04/10/18: Miklós Rontó (University of Miskolc) 
  • 11/10/18: Róbert Kovács (BME)
  • 18/10/18: Stefan Filipov (Department of Computer Science, Faculty of Chemical System Engineering, University of Chemical Technology and Metallurgy, Sofia, Bulgaria)
  • 25/10/18: Mihály Pituk (University of Pannonia)
  • ...

Next seminar: 

27 September 2018, 10:15, H306

Ilona Nagy (BME, Department of Analysis)

Two Limit Cycles in a Two-Species Reaction

Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. About this reaction we show the following facts (using symbolic methods): the kinetic differential equation of the reaction has two limit cycles surrounding the stationary point of focus type in the positive quadrant. The outer limit cycle is always stable and the inner one is always unstable. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and Singular) because the symbolic calculations to carry out are too complicated to do by hand. Even to make characteristic drawings is very far from trivial: the inner limit cycle is very small. Some of the methods we use are relatively new, and all of them have never been used in reaction kinetics, although they can be used to have similar exact results on other models.


Previous seminars: 

20 September 2018

Professors Gustaf Söderlind and Carmen Arévalo will present three talks during their visit to Budapest. All three talks deal with adaptive methods of ODEs. Professor Söderlind’s first talk will be held at the Seminar on Applied Analysis of the Department of Applied Analysis at ELTE university (details below), while his second talk together with Professor Arévalo’s talk will be presented at the Miklós Farkas Seminar. Although professor Söderlind’s talks form a complete presentation together, yet they can be followed separately.

Gustaf Söderlind (Lund University, Sweden)

Adaptive Time-Stepping. Part II. Time transformations applied to reversible Hamiltonian dynamics and weakly dissipative systems

In the second talk on time step adaptivity, we focus on the special needs of conservative dynamical systems. This includes Hamiltonian problems, and weakly dissipative systems. In integrable Hamiltonian problems, the mathematical solution is time reversible, which precludes the use of classical controllers, which adapt the step size to manage the error observed in previous steps. Instead, a time reversible tracking algorithm is developed, which allows full reversibility of the adaptive computational process. This is shown to preserve first integrals over long times, and even improves the accuracy over constant step size symplectic integrators. We demonstrate the procedure in two examples from celestial mechanics, and then proceed to demonstrate how a similar approach can be combined with splitting methods in weakly dissipative systems. The latter approach has been put to effective use in rolling bearing dynamic simulation.

The first part of the talk: 

Adaptive Time-Stepping. Part I. How control theory and digital filters enhance the performance of IVP solvers

Seminar on Applied Analysis, 17 September 2018, 14:00, Department of Applied Analysis and Computational Mathematics, ELTE, Pázmány P.  1/C, room 3.719

In two talks we will describe a systematic approach to time-stepping adaptivity in the numerical integration of initial value problems. The first part will deal with the proper construction of discrete controllers that keep the local error equidistributed along the solution trajectory. The controllers will be based both on classical constructions such as proportional-integral controllers, and on modern digital low-pass filters, which are designed to maintain a smooth step size sequence. We will demonstrate that a regular step size sequence is of benefit as it increases the computational stability — small changes in the error tolerance will only produce a small change in the accuracy of the computed solution. While computational speedup is often only modest, the quality of the computed solution as well as the reliability of the software can sometimes be remarkably improved. Presentation

Carmen Arévalo (Lund University, Sweden)

An intrinsically adaptive formulation of multistep methods

Multistep methods are important tools for solving ordinary differential equations with initial conditions. In order for these methods to be efficient they must be adaptive, that is, they must allow the choice of an appropriate step-size for each integration step. We present a comprehensive way of formulating multistep methods that is adaptive by construction and show how this methodology can be applied to particular situations. We also show the application to strong stability preserving methods, used to solve ODEs arising from the semi-discretization of time-dependent partial differential equations (PDEs), especially hyperbolic PDEs with shocks. We finally demonstrate how we apply the formulation to differential algebraic equations (DAEs), an ODE system coupled with algebraic constraints.

Presentations in 2017/182016/17