# Szemináriumok

## Matematikai játékok és egy 51-éves Pólya György-történet

Időpont:
2019. 12. 03. 10:30
Hely:
H306
Molnár Emil

## Multidimensional Sensitivity Analysis of Large-scale Mathematical Models & High-accuracy numerical methods for parabolic systems in air pollution modeling

Időpont:
2019. 11. 28. 10:15
Hely:
H306
Ivan Dimov & Venilin Todorov

Ivan Dimov: Multidimensional Sensitivity Analysis of Large-scale Mathematical Models

The aim of the talk is to present various approximation Sensitivity Analysis techniques of Large-scale Mathematical Models. A concept of Sensitivity Analysis and complexity in classes of algorithms will be presented. More precisely, randomized Quasi-Monte Carlo and modified Sobol sequences will be analyzed. As a case study the Unified Danish Eulerian model of air pollution transport (UNI-DEM) will be considered. UNI-DEM is carried out to compute more precisely interaction effects of inputs and sensitivity measures. Various Monte Carlo algorithms have been applied for numerical integration of multidimensional integrals to estimate the sensitivity measures. A comparison of their efficiency will be discussed.

Venelin Todorov: High-accuracy numerical methods for parabolic systems in air pollution modeling

We present different approach es for enhancing the accuracy of the second-order finite difference approximations of two dimensional semilinear parabolic systems. These are the fourth-order compact difference scheme and the fourth-order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential equations in air pollution modeling. We analyze numerical experiments to compare the two approaches with respect to accuracy, computational complexity, nonnegativity preserving, etc. The sixth-order approximation based on the fourth order compact difference scheme combined with Richardson extrapolation is also discussed numerically.
(Joint work with Juri Kandilarov, Ivan Dimov and Luben Vulkov.)

## Chemical Resonance and Beats in Periodically Forced Chemical Oscillatory Systems

Időpont:
2019. 11. 21. 10:15
Hely:
H306
Lagzi István (BME, MTA-BME)

We present the existence of resonance and beats in open and forced chemical oscillatory systems using a superimposed sinusoidal modulation on the inflow rates of the reagents. We demonstrate control over the periodicity of the forced oscillations and show that the time period of beats follows the relation known for forced physical oscillators. Based on experimental results and numerical model simulations, we could show that resonance and beats are internal properties of chemical oscillatory systems. A forced open chemical oscillatory system is a counterpart of the forced oscillators known form the classical mechanics (e.g., driven pendulum), in which instead of applying a periodic external driving force, the periodically changing chemical potential drives the open oscillatory systems.

(joint work with Hugh Shearer LawsonGábor HollóRóbert Horváth)

## A többváltozós Denton-probléma megoldása többcélfüggvényű optimalizálási megközelítéssel

Időpont:
2019. 11. 14. 14:15
Hely:
BME H. épület 405/a terem
Lovics Gábor

## Bifurcation and Discretization in Integrodifference Equations

Időpont:
2019. 11. 14. 10:15
Hely:
H306

Integrodifference equations are infinite-dimensional dynamical systems in discrete time. They are motivated by theoretical ecology in order to describe the spatial dispersal and temporal evolution of species having non-overlapping generations. In this talk, we review some recent work addressing two aspects concerning their long-term behavior:
(1) Bifurcation theory of periodic equations, which requires a combination of analytical and numerical techniques (joint work with Christian Aarset)
(2) Numerical Dynamics (persistence of dynamical properties under numerical discretization)

## Large-amplitude periodic orbits for delay equations

Időpont:
2019. 11. 07. 10:15
Hely:
H306
Vas Gabriella

Let us consider scalar delay differential equations of the form x'(t)=-ax(t)+f(x(t-1)), where a>0 and f is a nondecreasing C1-function. This talks gives an overview of the periodic orbits and the global attractor.

After showing some well-known results of Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, I introduce the notion of large-amplitude periodic (LAP) orbits. First we discuss the bifurcation and the existence of a pair of LAP orbits. Then we describe the geometric properties of the unstable set of a specific LAP orbit in detail. Complicated configurations of LAP orbits appear when the dynamical system has several unstable equilibria – we also consider this case. These are joint works with Tibor Krisztin and Szandra Beretka.

No preliminary knowledge of delay equations is presumed.

## Anchored expansion in supercritical percolation on nonamenable graphs

Időpont:
2019. 10. 24. 16:15
Hely:
H306
Jonathan Hermon (Cambridge)

## Numerical Solution of Fractional Diffusion Problems

Időpont:
2019. 10. 24. 10:15
Hely:
H306
Svetozar Margenov

This study is motivated by the recent achievements in fractional calculus and its numerous applications related to anomalous (super) diffusion. Let us consider a fractional power of a self-adjoint elliptic operator introduced through its spectral decomposition. It is self-adjoint but nonlocal. The nonlocal problems are computationally expensive. Several different techniques were recently proposed to localize the nonlocal elliptic operator, thus increasing the space dimension of the original computational domain.

An alternative approach [1,2,3] is discussed in this talk. Let $\cal A$ be a properly scaled symmetric and positive definite (SPD) sparse  matrix, arising from finite element or finite difference discretization of the initial (standard, local) diffusion problem. A method for solving algebraic systems of linear equations involving $\cal A^\alpha$, $0 < \alpha < 1$, is presented. The solution methods are based on best uniform rational approximations (BURA) of the scalar function $t^{\alpha}$, $0\le t\le 1$. The method has exponential convergence rate with respect to the degree of rational approximation $k$. The error estimates of the last variant of BURA methods are robust with respect to the spectral condition number $\kappa (\cal A)$. A stabilized modification of the Remez algorithm is developed to compute the BURA of $t^{\alpha}$. Although the fractional power of $\cal A$ is a dense matrix, the algorithm has complexity of order $O(N)$, where $N$ is the number of unknowns. At this point we assume that some solver of optimal complexity (say multigrid or multilevel) is used for the involved systems with matrices ${\cal A} + d_j \cal I$, $d_j \ge 0$, $j=1, \dots, k$. The comparative numerical tests confirm the advantages of the BURA method.

Acknowledgement: This research has been partially supported by the Bulgarian NSF Grant DN12/1.

## Körlemez felbontása egybevágó részekre

Időpont:
2019. 10. 22. 10:30
Hely:
H306