# Szemináriumok

## 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
Lángi Zsolt

## Axiomatic Cake Cutting

Időpont:
2019. 10. 17. 14:15
Hely:
BME H. épület 405/a terem
Sziklai Balázs MTA KRTK

## Renormalized and non-renormalized solutions of the transport equation

Időpont:
2019. 10. 17. 10:15
Hely:
H607 !!!
Székelyhidi László

The linear transport equation is the possibly simplest of all PDE, describing the evolution of a density under the flow of a given vectorfield. It is well-known that, as long as the vectorfield is Lipschitz continuous,  solutions of the linear transport equation are closely related, via the method of characteristics, to the Lagrangian flow map generated by the vectorfield. However, this link breaks down if the vectorfield is merely Sobolev, since in this case the ODE does not make sense classically. Of course, formulation of the PDE poses no problems even without differentiability assumptions. On the other hand there are numerous applications in fluid mechanics and kinetic theory, where the transport equation appears with a vectorfield which is in some Sobolev space, possibly even continuous, but not Lipschitz. For such cases DiPerna and Lions developed in the late 1980s a theory of renormalization, leading to a well-posed solution concept for both the PDE and the ODE. In the talk we discuss the limits of this theory and present examples showing that unless certain additional integrability conditions are imposed, the theory of renormalization does not lead to a unique solution.

## Szférikus és kombinatorikus megvilágítási probléma

Időpont:
2019. 10. 15. 10:30
Hely:
H306