# 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.

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

Feliratkozás az e-mail listára ill. egyéb megjegyzések a szeminárium szervezőihez.

2017. február 23. (csütörtök), 10:15, H306

Elliptikus parciális differenciálegyenletek hálónélküli megoldása az
alapmegoldások módszerével

Kivonat: A parciális differenciálegyenletek hagyományosnak számító numerikus módszerei a véges differencia és a véges elem módszerek (FDM, FEM). Előbbi módszerben a tartományok diszkretizálása egy számítási ráccsal, az utóbbiban végeselemes háló segítségével történik. Mindkét módszer ún. tartomány típusú, azaz a teljes tartományt diszkretizálni kell, ami végeredményben egy sokismeretlenes lineáris algebrai egyenletrendszerre vezet. Továbbá, egy bonyolult tartományra jól illeszkedő végeselem-háló kialakítása maga is igen bonyolult probléma lehet. Ezen hátrányok kiküszöbölésére születtek az ún. hálónélküli módszerek, melyek intenzívebb kutatása nagyjából az ezredforduló környékén kezdődött. Itt a tartományon és annak peremén semmiféle rács- vagy hálóstruktúra kialakítása felesleges: a diszkretizálás struktúra nélküli ponthalmazzal történik. Így a hálógenerálás problémája automatikusan megoldódik. A bevezetett ismeretlenek száma jellemzően sokkal kevesebb, mint az FDM ill. a FEM esetén. Ezen előnyök ára, hogy a módszer olyan lineáris egyenletrendszerre vezet, melynek mátrixa teljesen kitöltött, nemszimmetrikus és általában rosszul kondícionált. Az előadáson egy speciális hálónélküli módszert mutatunk be, az alapmegoldások módszerét. Itt a közelítő megoldást a szóban forgó differenciálegyenlet alapmegoldása segítségével konstruáljuk, melyet bizonyos külső pontokba (forráspontokba) tolunk el. A módszer rendkívül egyszerűen programozható, és ugyanakkor sok esetben nagyon pontos. Hátránya a már említett rosszul kondícionált mátrixok megjelenése, valamint a forráspontok optimális meghatározása. Az előadáson részletezzük e hátrányok csökkentésének lehetőségeit, és vázoljuk azt is, hogy a módszer hogyan terjeszthető ki inhomogén problémák megoldására: ez utóbbira egy ún. szórt alappontú interpolációs technikát alkalmazunk.

2016. december 1.

Garay Barnabás

Metastability of a periodic orbit

Abstract: A ring of $N=2M$ identical neuron cells with piecewise linear and saturated bidirectional coupling nonlinearities is considered. For certain values of the coupling parameters $\alpha$ and $\beta$, existence of a hyperbolic periodic solution with cyclic symmetry is established. With $M \rightarrow \infty$, the dominant Floquet multiplier converges to $1$ and the remaining $2M-2$ nontrivial Floquet multipliers  converge to $0$. In both cases - based on root asymptotics of certain families of lacunary polynomials - sharp exponential estimates are given. Waveform asymptotics as well as the asymptotics of the dominant eigenvector are also presented. The results deal with the two simplest types of periodic rotating waves with maximal symmetry. The theory is almost complete for what we term as Type One central waves. The paper ends with several remarks and conjectures on the more general picture. The entire work was motivated by electrical circuit experiments.

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2016. november 24.

Havasi Ágnes

Richardson extrapolation and its applications in environmental models

Abstract: Richardson extrapolation is a numerical procedure by which we can eficiently enhance the accuracy of a time integration method. During this procedure a weighted average of two numerical solutions obtained by the same numerical method, but two different values of the time-step sizes is calculated.  In this talk we present the theoretical background of Richardson extrapolation, and give some results about its convergence and stability. The applicability of the method will be illustrated on different environmental models (simlplified carbon-dioxide model, air chemistry model, advection equation).

2016. november 10.

Kovács Balázs

Numerical analysis of parabolic problems with dynamic boundary conditions

Abstract: Space and time discretisations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual inner product is replaced by an suitable inner product on a problem specific Hilbert space. The class of parabolic equations considered includes linear problems with time- and space-dependent coefficients and semi-linear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretisation by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration, which yields surprising numerical results.

2016. október 27.

Izsák Ferenc

Space-fractional diffusion problems: modeling and numerical solution

Abstract: Space-fractional diffusion processes were observed in the last decades in many real-life situations. To simulate such phenomena, we first need an appropriate model. It turned out that a classical topic, the so-called fractional-order calculus can be applied here. We also discuss some approaches for the numerical solution of the corresponding problems, both finite difference and finite element methods. The matrix transformation method is investigated in details. We also mention some open questions of the topic.

2016. október 20.

Error analysis of waveform relaxation method for reaction-diffusion equations

Abstract: The method of waveform relaxation (WR) is an iterative method which can be applied for a large class of problems. It was first used to solve a system of ordinary differential equations describing large scale circuits. Since then many works have been devoted to investigate the convergence of the method for different types of problems, for example reaction--diffusion equations, delay differential equations. All of them consider time dependent problems that are either a system of ODEs originally or obtained from partial differential equations by spatial discretization. The key to prove convergence in every case is the Lipschitz property of the function acting on the right hand side.

In PDEs describing diffusion or advection processes the spatial differentiation is not a Lipschitz-continuous operation. Consequently the usual formulation of WR fails when the method is applied directly on reaction-diffusion or reaction-advection problems. Furthermore the convergence rates of the spatially discretized problem depend on the discretization parameter, thus the results can not automatically be transferred to the original continuous model.

In this talk, I propose to apply the WR method directly on the continuous problem.

Using the concept of strongly continuous one-parameter semigroups a large class of continuous problems can be discussed, including systems of reaction-diffusion and reaction-advection equations in multiple spatial dimensions. Better error estimations can be given, that are also explicit.

After that the subproblems can be solved in each iteration numerically, with spatial discretization. This scenario allows to investigate the effect of the numerical treatment as well, an overall error estimation can be formulated which includes the iteration error and the cumulative numerical error.

Faster convergence can be achieved by dividing the time interval into subintervals and applying WR on these time windows one after another. This procedure is called windowing. Convergence of windowing is proven for a large class of PDEs.

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2016. október 13.

Equivalent operator preconditioning for elliptic problems

Abstract: For a vast number of models for real-life problems, including various partial differential equations, the numerical solution is ultimately reduced to the solution of linear algebraic systems. The efficiency of this last step often depends on the proper choice of a preconditioning matrix. A class of efficient preconditioners for discretized elliptic problems can be obtained via equivalent operator preconditioning. This means that the preconditioner is chosen as
the discretization of a suitable auxiliary operator that is equivalent to the original one, Under proper conditions one can thus achieve mesh independent convergence rates. Hence, if the discretized auxiliary problems possess efficient optimal order solvers (e.g. of multigrid type) regarding the number of arithmetic operations, then the overall iteration also yields an optimal order solution, i.e. the cost O(N) is proportional to the degrees of freedom.

In this talk first some theoretical background is summarized, including both linear and superlinear mesh independent convergence, then various applications are shown. The results can be applied, among other things, for parallel preconditioning of transport type systems,
for streamline diffusion preconditioning of convection-diffusion problems, and to achieve superlinear convergence under shifted Laplace preconditioners for Helmholtz equations.

2016. szeptember 29.

Horváth Róbert

Qualitative properties of numerical solutions of PDE models of disease propagation

Abstract: The large pandemics in the human history show that infectious diseases are able to cause widespread devastation. This is why we want to prevent their outbreak by all means. Mathematical models help us to understand the dynamics of epidemics. However, the popular compartmental models (such as the SIR model) do not take into account the spatial positions of the individuals. This is why we use partial differential equations (PDE) models.

In this talk we consider two types of PDE models. The first one comes from the localised nature of the disease transmission and the second one from adding diffusive terms to the SIR model. We formulate some typical qualitative properties (nonnegativity, monotonicity) of the above models and investigate the validity of these properties both for the continuous and the discrete (obtained by the finite difference method) problems. We give sufficient conditions for the mesh sizes that guarantee the qualitative properties a priori in the case of different problem settings. The results are demonstrated on several numerical test problems. (Joint work with István Faragó)

2016. szeptember 22.

The Mathematics of Stiffness. History and Evolution of a Concept.

Abstract: The notion of stiffness was introduced in 1952, by Curtiss and Hirschfelder. It was recognized that some well-posed initial value problems could not be solved numerically except by using dedicated implicit methods. For many years, attempts were made to characterize stiffness. These attempts were contrived, sometimes right, sometimes wrong, or otherwise flawed. A significant problem was that mathematical properties of the problem were mixed with operational criteria, such as the choice of discretization method and the accuracy requirement.

In the end, it was recognized that every numerical analyst learns what is a stiff problem by solving a few. However, it is highly unsatisfactory that a proper definition does not exist. At least, there should be a single, mathematical necessary condition for when to look out for stiffness.

In this talk we outline the history of the concept of stiffness, and end by introducing a new, unexpected criterion. This is simple in the sense that it relates a problem property (completely defined in terms of the differential equation) to a time scale. The latter is, in turn, related to the rage of integration, or to the desired time step. It turns out that the mathematically necessary condition for stiffness depends only on the divergence of the vector field of the ODE, and on the range of integration. A new theory will be introduced and explained, with numerous examples of how stiffness can be identified also in strongly nonlinear systems.

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