# Farkas Miklós Alkalmazott Analízis Szeminárium előadásai 2016/17

2017. július 31.

Randall J. LeVeque, Boeing Professor of Applied Mathematics, University of Washington, a SIAM Board of Trustees tagja valamint a SIAM Journals Committee elnöke

Abstract: Time-dependent hyperbolic partial differential equations can be efficiently solved using adaptive mesh refinement, with a hierarchy of finer grid patches in regions where the solution is discontinuous or rapidly varying.  These patches can be adjusted every few time steps to follow propagating waves.  For many problems the primary interest is in tracking waves that reach one target location, perhaps after multiple reflections.  The solution to an adjoint equation solved backward in time from the target location can be used to identify the regions that require refinement.  These adjoint methods are incorporated in the Clawpack software for general hyperbolic problems and have been used in the GeoClaw software to track tsunami waves in the ocean that will reach a particular community of interest.

Marsha BergerCourant Institute, NYU,

Modeling and Simulation of Asteroid-Generated Tsunamis

Abstract: In 2013, an uncharted asteroid exploded in the atmosphere over Chelyabinsk, causing damage for a radius of 20 kilometers. We examine the question of what would happen if an asteroid burst over water instead of land. Could it generate a tsunami that would cause widespread damage far away? We present numerical simulations using the GeoClaw software and the shallow water equations in a variety of settings. We have a model problem with an explicit solution that explains the phenomena found in the computations. Finally, we discuss whether compressibility and dispersion are important effects that should be included, and show results using the linearized Euler equations that begin to address this.

2017. május 4.

Owe Axelsson (Institute of Geonics AS CR, IT4 Innovations, Ostrava, Csehország; Uppsala University, Svédország )

A survey of applications of a preconditioned iterative solution method in optimal control problems, constrained by PDEs

Abstract:  Optimal control problems,constrained by a state equation in the form of a partial differential equation (PDE),arise in many important applications, where one wants to steer the modelled process in order to have a solution close to some given target function.The discretized problems lead to a particular two-by-two block matrix form for which a very efficient preconditioner,leading to very tight eigenvalue bounds, will be presented. Various applications, such as in time-harmonic parabolic and Stokes equations and eddy current electromagnetic problems, will also be discussed.

2017. április 25.

Tim Healey (Dept. Mathematics, Cornell University)

Global symmetry-breaking bifurcation in a model for 2-phase lipid-bilayer vesicles - analysis and computation

Abstract: We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.

2017. április 20.

Csomós Petra (ELTE, Alkalmazott Analízis és Számításmatematika Tanszék)

Innovative Integrators

Abstract: Innovative integrators (operator splitting procedures, exponential integrators, and Magnus-type integrators) provide an efficient way to approximate the solution of nonlinear evolution equations. The lecture gives a first insight into the topic. We will introduce the various innovative integrators, show what kind of equations they are designed for, and sketch how to prove their convergence. We also consider an error estimation of the total numerical method, i.e., when innovative integrators are applied together with space and time discretization schemes. Since the study of evolution equations requires a functional analytic framework, we briefly recall the corresponding results in operator semigroup theory. As an application, we present our results on computing the optimal state in linear quadratic regulator problems, especially in the case of shallow water equations.

Related papers:

2017. április 6.

Liepa Bikulciene (Kaunasi Műszaki Egyetem, Litvánia)

Operator method in the theory of differential equations

Abstract: The Operator method for differential equations solving can be applied in nonlinear dynamics for exact solutions finding. Starting from basics of this method and Hankel matrices ranks possibilities of evaluation of DE and special PDE solutions using MAPLE mathematical software will be introduced. Examples of solved ODE and PDE: Huxley, Liouville, KdV equations and their soliton solutions will be presented. It will be shown that special solitary solutions exist only on a line in the parameter plane of initial and boundary conditions. This result may lead to important findings in a variety of practical applications as nonlinear evolution equations in mathematical physics. Joint work with Research Group for Mathematical and Numerical Analysis of Dynamical Systems https://nonlinear.fmf.ktu.lt/index.htm.

2017. március 30.

Kiss Márton (BME Differenciálegyenletek Tanszék)

A Chaotic Linear Operator

Abstract: Not just nonlinear systems, but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequences $l_{2}$ displays chaotic dynamics. We give an outline of the proof starting from Devaney's definition of chaos. Then we construct the corresponding operator on the space of periodic functions and provide its representation involving a principal value integral. We explicitly calculate its eigenfunctions, as well as its periodic points; and also provide examples of chaotic and unbounded trajectories. Joint work with Tamás Kalmár-Nagy.

2017. március 16.

Zsuppán Sándor (Berzsenyi Dániel Evangélikus Gimnázium (Líceum), Sopron)

Stokes problem, related inequalities, constants and representations

Abstract: Stable solvability of the Stokes problem describing fluid motion with small Reynolds number in a domain depends on the inf-sup condition, the discrete version of which is involved in the analysis of many numerical methods. The inf-sup constant figuring in the condition is connected to other domain specific constants figuring in other inequalities such as the Babuska-Aziz inequality for the divergence equation, the Friedrichs-Velte inequality for conjugate harmonic functions and the Korn inequality in linear elasticity. These constants are also connected to the Schur complement operator of the Stokes problem and in the planar case to the Friedrichs operator of the domain. Despite of their theoretical and practical importance exact values of these constants are known only for a few domains. On the other hand there exist useful estimations for planar and spatial domains as well. The values of these constants depend only on the geometry of the domain which is for simply connected planar domains encoded in analytical properties of the conformal mapping of the domain onto the unit disc. Utilization of conformal mapping provides useful results for the constants (exact values or estimations for special domains, and continuous domain dependence) and also for the operators. These results are partly generalizable for other types of domains. We also review some representations of solutions of the Stokes equation by harmonic potentials and formulte results about their equivalence. We also review recent results and outline possible ways for further research of this topic.

2017. március 9.

Gaussian curvature of piecewise flat manifolds

Abstract: Smooth surfaces are often approximated by triangulations. When the triangles are flat the curvature of the original surface becomes reflected in angle defects at vertices: the angles of the triangles meeting at a vertex do not sum to $2\pi$, and the difference is called the deficit angle. The deficit angle has been proposed as a measure of the gaussian curvature at vertices and I will give a rigorous interpretation and proof of this intuition.

2017. március 2.

Uniqueness of steady state, smooth shapes in a nonlocal geometric PDE and a model for the shape evolution of ooids

Abstract: We investigate steady state solutions of a nonlocal geometric PDE that serves as a simple model of simultaneous contraction and growth of grains called ooids in geosciences. As a main result of the talk I demonstrate that the parameters associated with the physical environment determine a unique, time-invariant (equilibrium) solution of the equation among smooth, convex curves embedded in $\mathbb{R}^2$. The model produces nontrivial shapes that are consistent with recorded shapes of mature ooids found in nature.

2017. február 23.

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.

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.

• Related papers or presentations: paper

2016. november 24.

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.

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.

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.

• Related papers or presentations: paper

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.

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.

• Related papers or presentations: paper