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.
The seminar will be continued in the next semester
7 December 2017
Research reports of PhD students
9:30-9:45: Rahele Mosleh - Mathematical models for malaria disease
9:45-10:00: Császár Szilvia - Approximation of homoclinic orbits
10:00-10:15: Takács Bálint - Epidemic models with spatial dependence
10:15-10:30: Neogrády-Kiss Márton - Two simple models with inhibitory and excitatory neurons.
10:30-10:45: Maros Gábor - Analysis of fractional diffusion problems
30 November 2017
Yiannis Hadjimichael (BME Institute of Mathematics & MTA-ELTE NUMNET Research Group)
Optimal strong stability preserving time-stepping methods with upwind- and downwind-biased operators
A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases, it is difficult to satisfy certain physical properties while maintaining high order of accuracy. Strong stability preserving (SSP) time discretizations were developed to ensure that nonlinear stability properties of the solution are maintained when coupled with suitable spatial discretizations.
In the first part of this talk, we review the development of optimal SSP Runge-Kutta and multistep methods for nonlinear problems. We emphasize the usage of an alternative representation of Runge-Kutta methods that reveals the SSP properties of such methods. Numerical examples illustrate the effectiveness and usefulness of SSP methods.
In the second part, we present some recent results related to perturbed methods that use both upwind- and downwind-biased spatial discretizations. We introduce a novel family of third-order implicit Runge–Kutta methods with arbitrarily large SSP coefficient and investigate the stability and accuracy of these methods. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems.
23 November 2017
Lubin G. Vulkov (Angel Kanchev University of Ruse, Bulgaria)
Adequate numerical methods for nonlinear parabolic problems in mathematical finance
The prices and hedging strategies in the real financial market models are often described by fully nonlinear versions of the standard Black-Scholes equation. We concentrate on two classes of models: first, nonlinear Black-Scholes equations in which the volatility depends on second space derivatives of the price(=solution) and then on regime-switching models described by systems of semilinear parabolic equations with exponential nonlinearities. The following characteristic properties of these parabolic problems are typical: unbounded domain, boundary degeneration, maximum-minimum principle and nonnegativity preservation. We develop effective discretizations that reproduce these properties.
9 November 2017
Miklós Mincsovics (BME, Department of Differential Equations)
What is the difference between weakly and strongly stable linear multistep methods?
There are two main approaches to explain the differences between them. The first one relies on the role of the parasitic roots (this is what we usually teach). The second one is more indirect and based on the general definition of stability. Spijker was the first who presented a norm pair in which the midpoint method is not stable. This example can be extended to the general weakly stable case. Finally, we upgrade this latter approach keeping its advantages and eliminating its weak point.
26 October 2017
Mónika Polner (Bolyai Institute, University of Szeged)
A space-time finite element method for neural field equations with transmission delays
Neural field equations are models that describe the spatio-temporal evolution of (spatially) coarse grained variables such as synaptic or firing rate activity in populations of neurons. We consider a single population of neurons, distributed over some bounded, connected, open region, whose state is described by their membrane potential. These potentials are assumed to evolve according to an integro-differential equation with space dependent delay.
Neural field models with transmission delay may be cast as abstract delay differential equations, which is the starting formulation for our numerical discretization. The numerical treatment of these systems is rare in the literature and has several restrictions on the space domain and the functions involved. The aim of this work is the development of an accurate numerical method without introducing limitations to its applicability. We present and analyze a novel time-discontinuous Galerkin finite element method. We give a theoretical analysis of the stability and order of accuracy of the numerical discretization and demonstrate the method on a number of neural field computations in one and two space dimensions.
19 October 2017
Miklós Horváth (BME, Institute of Mathematics)
Inverse scattering: Mathematical properties of the phase shifts
The inverse scattering is one of the most useful devices to obtain information about quantum mechanical forces. The solutions of the radial Schrödinger equation, describing mathematically the situation, behave asymptotically like sine functions with shifted phase. The sequence of these phase shifts is a central notion in scattering theory. In this talk some classical and recent results about phase shifts will be presented and some open questions will be discussed.
12 October 2017
Imre Fekete (Eötvös Loránd University & MTA-ELTE NUMNET)
On the zero-stability of multistep methods on smooth nonuniform grids
In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. The grid points are constructed as the image of an equidistant grid under a smooth deformation map. We show that for all strongly stable linear multistep methods, there is an $N^*$ such that a condition of zero stability is always fulfilled for $N > N^*$ under a smoothness condition. Examples are given for Adams and BDF type methods.
5 October 2017
An explicit analytic solution of a coupled first order partial and ordinary differential equation system for a discontinuous initial-boundary value problem
Non-polynomial series solution of a coupled first order partial and ordinary differential equation (PDE-ODE) system for a discontinuous initial and boundary condition has been developed. Linear equation systems are constructed to calculate the constant coefficients of the series solution. Explicit expressions have been found to the solution of these linear equation systems. Different forms of the solution have been compared to the numerical solution of the PDE-ODE system and the rate of the convergence is also investigated. The studied first order PDE-ODE system describes an unsteady convection dominated heat transfer process induced by a buoyant plume entrainment.
28 September 2017
Devilish eigenvalues: hysteresis and mechanistic turbulence
We consider the adjacency matrix associated with a graph that describes transitions between 2^N states of the discrete Preisach memory model. This matrix can also be associated with the "last-in-first-out" inventory management rule. We present an explicit solution for the spectrum by showing that the characteristic polynomial is the product of Chebyshev polynomials. The eigenvalue distribution (density of states) is explicitly calculated and is shown to approach a scaled Devil's staircase. The eigenvectors of the adjacency matrix are also expressed analytically. This is joint work with Andreas Amann, Daniel Kim, and Dmitrii Rachinski. We also examine a mechanistic model of turbulence, a binary tree of masses connected by springs. We analyze the behavior of this linear model: a formula is presented for the analytical calculation of the eigenvalues and the optimal damping - at which the decay of the total mechanical energy is maximized. The discrete energy spectrum of the mechanistic model (defined as the total mechanical energy stored in each level) can be tuned to display the features of the Kolmogorov-spectrum. This is joint work with Bendegúz Dezső Bak.
21 September 2017
Qualitatively reliable numerical models of time-dependent problems
In the modeling process we construct mathematical and numerical models. Both models should preserve the basic (physically, biologically, etc. motivated) qualitative properties of the original phenomena. In this talk this problem will be discussed. We examine the different qualitative properties (maximum principles, non-negativity preservation, maximum norm contractivity) for both models and we show the relation between them for the linear problems. For the numerical models we give the condition for the construction of the mesh under which the above qualitative properties are valid. The results will be demonstrated in different real-life problems. The main attention will be focused to the heat conduction problem. Briefly we discuss the compartmental epidemic models which take into the account the space dependence, and also some simple discrete Lotka-Volterra models.
- Presentation or related papers: presentation
Presentations in 2016/17