Partial differential equations, complex networks and applications  Page description

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Details of project

  
Identifier
81403
Type K
Principal investigator Simon, Péter
Title in Hungarian Parciális differenciálegyenletek, komplex hálózatok és alkalmazásaik
Title in English Partial differential equations, complex networks and applications
Keywords in Hungarian differenciálegyenlet, hálózat
Keywords in English differential equations, network
Discipline
Mathematics (Council of Physical Sciences)100 %
Ortelius classification: Differential equations
Panel Mathematics and Computing Science
Department or equivalent Applied Analysis and Computational Mathematics (Eötvös Loránd University)
Participants Bátkai, András
Besenyei, Ádám
Faragó, István
Farkas, Bálint
Garay, Barnabás
Horváth, Tamás
Sikolya, Eszter
Simon, László
Starting date 2010-04-01
Closing date 2015-06-30
Funding (in million HUF) 15.461
FTE (full time equivalent) 16.60
state closed project
Summary in Hungarian
A tudomány számos területén, például fizikában, műszaki tudományokban, kémiában, biológiában találkozhatunk bonyolult struktúrájú komplex rendszerekkel. Ha a komponensek szabályos struktúrába rendeződnek, pl egy kristályba, akkor a rendszer gyakran parciális differenciálegyenletekkel írható le. Azonban sokszor a struktúra bonyolultabb, mint pl az internet, vagy egy sejt metabolikus hálózata. Ekkor a rendszer komplex hálózatokkal modellezhető. Érdeklődésünk középpontjában a hálózatokon zajló folyamatok állnak, melyek differenciálegyenletekkel írhatók le. Ezek a folyamatok két csoportra bonthatók: egyrészt csúcspontokban vezérelt folyamatokra, mint pl. a celluláris neurális hálózatok, vagy járványterjedési modellek, másrészt az éleken vezérelt hálózatokra. Az első csoportba tartozók közönséges, míg a második csoportba tartozók parciális differenciálegyenletekkel írhatók le. A parciális differenciálegyenletek gyakran nehezen vizsgálhatók elméletileg, ezért numerikus vizsgálatokat is tervezünk, elsősorban az operátorszeletelés módszerének alkalmazásával.

Az Alkalmazott Analízis és Számításmatematikai Tanszéken évek óta folyik parciális differenciálegyenletek kutatása, és az utóbbi időben komplex hálózatok vizsgálatába is bekapcsolódtunk. Az eddigi kutatásainkat az OTKA több pályázatban is támogatta (T 4385, T 019460, K 49819, F 034840, F 022228, F 049624). Szeretnénk az eddig megkezdett kutatásainkat folytatni, illetve kiterjeszteni az alábbi területekre.

Nemlineáris parabolikus, valamint funkcionál differenciálegyenletek,
Komplex hálózatok,
Celluláris neurális hálózatok,
Parciális differenciálegyenletek numerikus vizsgálata operátorszeleteléssel.
Summary
Complex systems, consisting of many components connected in a possibly complicated structure arise in many areas of the science, e.g. in physics, engineering, chemistry or biology. If the structure has some regularity like a crystal then the system can usually be described by partial differential equations. However, in many cases the structure can be very complicated, like the Internet or the metabolic network of a cell. Then the system can be modelled by a so-called complex network. Our main field of interest is processes on networks that are described by differential equations. Network processes can be divided into the two classes: processes governed at the vertices of the graph, like cellular neural networks or epidemic spread models and processes governed at the edges of the graph subject to boundary conditions at the vertices. The first type of models are described by ODEs, while the second class leads to PDEs. PDE models ar often difficult to treat theoretically, hence we also plan their numerical study using the operator splitting method.

Our research group at the Department of Applied Analysis and Computational Mathematics has remarkable results in the field of partial differential equations, and recently has joined to the investigation of complex networks. Our previous research on the numerical and qualitative investigation of differential equations was supported by several OTKA grants (T 4385, T 019460, K 49819, F 034840, F 022228, F 049624). Our aim is to continue this research in the field of partial differential equations, to extend it to complex networks and to study different applications that are motivated by real life problems. We plan research in the following fields.

Nonlinear and functional parabolic differential equations,
Complex network processes (governed at vertices or edges),
Cellular neural networks,
PDE numerics using operator splitting.




 

Final report

  
Results in Hungarian
Kutatásainkat a kutatási terv szerint az alábbi területeken végeztük. Hálózati folyamatokat leíró differenciálegyenletekkel kapcsolatos munkánk fő célja olyan közelítő egyenletek levezetése és vizsgálata, melyek segítségével a hálózaton zajló folyamat vizsgálható. Kutatásaink során foglalkoztunk a közelítések pontosságának becslésével, a statikus gráfon ismert eredmények kiterjesztésével adaptív hálózatok esetére (ekkor a gráf is változik a folyamat során), valamint a hálózati folyamat vezérlésével. Parciális funkcionál differenciálegyenletek témakörében nem lokális tagokat, pl. az ismeretlen függvény integrálját tartalmazó időfüggő nemlineáris funkcionál differenciálegyenleteket és rendszereket vizsgáltunk, parabolikus és hiperbolikus esetben. Az egyes esetekben bebizonyítottuk globális megoldás létezését véges és végtelen időintervallumon, továbbá kiegészítő feltételek mellett igazoltuk a megoldások korlátosságát, illetve stabilizációját, ha az idő végtelenhez tart. Parciális differenciálegyenletek numerikus vizsgálata során a maximumelv diszkretizáció során való megőrződését vizsgáltuk. Ezenkívül az operátorszeletelés módszerét alkalmaztuk nagyméretű valós feladatokra. Celluláris neurális hálózatokban metastabil oszcillációk létrejöttét tanulmányoztuk a Floquet-elmélet alkalmazásával.
Results in English
The research was carried out according to the research plan in the following directions. The aim of our research concerning differential equations describing network processes was to derive and study approximating models, in order to investigate the properties of the process. We derived estimates for the accuracy of the approximations, extended the known results for static graphs to adaptive networks (when the network is also changing during the process) and studied the controllability of epidemic propagation on networks. The main topic of our research, related to partial functional differential equations, were non-linear time dependent functional differential equations and systems containing non-local terms, e.g. the integral of the unknown function, both in the parabolic and hyperbolic case. We proved the existence of the global solution in finite and infinite time intervals in several cases. Moreover, under stronger conditions, we verified the boundedness of the solutions and their stability when time tends to infinity. In the course of numerical investigation of partial differential equations we studied the preservation of the maximum principle under discretization. We applied operator splitting to large scale real reaction-diffusion problems. The appearance of metastable oscillations was studied in cellular neural networks by using Floquet theory.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=81403
Decision
Yes





 

List of publications

  
Sharkey,. K.J., Kiss, I.Z, Wilkinson, R.R., Simon, P.L.: Exact equations for SIR epidemics on tree graphs, Bull. Math. Biol. 77(4), 614.645, 2015
Bátkai, A., Havasi, Á., Horváth, R., Kunszenti-Kovács, D., Simon, P.L.: PDE approximation of large systems of differential equation, Operators and Matrices, 9 (1), 147-163, 2015
L. Simon: Semilinear hyperbolic functional equations, Banach Center Publications, 101, 207-224, 2014
L. Simon: On systems of semilinear hyperbolic functional equations, Stud. Univ. Babes-Bolyai Math. 59, 479-495, 2014
L. Simon: On semilinear hyperbolic functional equations with state-dependent delays, Recent Advances in Delay Differential and Difference Equations, Springer Proceedings in Mathematics and Statistics 94, Springer, 233-250, 2014
I. Faragó, Z. Zlatev: Application of Richardson Extrapolation for multi-dimensional advection equations, Computer and Mathematics with Application, 67, 2279-2293, 2014
Zlatev Z, Farago I, Havasi:A: Mathematical treatment of environmental models, Springer Series "Mathematics in Industry", Volume "Progress in Industrial Mathematics at ECMI 2012, 65-70., 2014
Simon P L, Taylor M, Kiss I Z: Exact epidemic models on graphs using graph-automorphism driven lumping, J MATH BIOL 62: (4) 479-508, 2011
Kiss I Z, Cassell J, Recker M, Simon P L: The impact of information transmission on epidemic outbreaks, MATH BIOSCI 225: (1) 1-10, 2010
Csörgő Gábor, Simon L. Péter: Numerical and analytical study of bifurcations in a model of electrochemical reactions in fuel cells, COMPUT MATH APPL 65: (3) 325-337, 2013
Nagy N, Simon PL: Monte Carlo simulation and analytic approximation of epidemic processes on large networks, CENT EUR J MATH 11: (4) 800-815, 2013
Hatzopoulos V, Taylor M, Simon P L, Kiss I Z: Multiple sources and routes of information transmission: Implications for epidemic dynamics, MATH BIOSCI 231: (2) 197-209, 2011
B.Indig, B.M.Garay: Chaos in Vallis' asymmetric model for El Nino, Chaos Solitons and Fractals, 75, 253--262, 2015
M.Forti, B.M.Garay, M.Koller, L.Pancioni: Long transient oscillations in a class of cooperative cellular neural networks, Int. J. Circuit Theory Applications 43, 635--655, 2015
Á. Besenyei: On a system consisting of three different types of differential equations, Acta Math. Hungar., 127(1-2), 178-194., 2010
Á. Besenyei: On some systems containing a parabolic PDE and a first order ODE, Math. Bohemica, 135(2), 133-141, 2010
L. Simon: On some singular systems of parabolic functional equations, Math. Bohemica 135, 123-132, 2010
L. Simon: On nonlinear functional parabolic equations with state-dependent delays of Volterra type, Internat J. Qualitative Theory Differential Equations Appl. 2, 88-103, 2010
K.-J. Engel, M. Kramar Fijavž, B. Klöss, R. Nagel, E. Sikolya: Maximal controllability for boundary control problems, Appl. Math. Optim. 62, 205-227, 2010
Kiss., I.Z., Cassell, J., Recker, M., Simon, P.L.: The impact of information transmission on epidemic outbreaks, Math. Biosci. 225, 1-10, 2010
Simon, P.L., Taylor, M., Kiss., I.Z.: Exact epidemic models on graphs using graph-, J. Math. Biol., 62, 479–508, 2011
András Bátkai, Petra Csomós, Bálint Farkas, and Gregor Nickel: Operator splitting for nonautonomous evolution equations, J. Funct. Anal. 260, 2163-2190, 2011
I. Faragó: Discrete maximum principle for finite element parabolic models in higher dimensions, Math. Comp. Sim., 80, 1601-1611, 2010
G.Colombo, M.Feckan, B.M.Garay: Multivalued perturbations of a saddle dynamics, Diff. Eq. Dyn. Syst., 18, 29--56, 2010
R.Csikja, B.M.Garay, J.Tóth: Chaos via two--valued interval maps in a piecewise affine model example for hysteresis, Proceedings of the 18th International Symposium on the Mathematical Theory of Networks and Systems (MTNS), Budapest, July 2010, pp. 187--194., 2010
B.M.Garay: The Euler--Poincaré formula for systems with hysteresis in two dimension, Ann. Univ. Budapest Sect. Math., 53, 59-68, 2010
L. Simon: On singular systems of parabolic functional equations, Operator Theory: Advances and Applications, 216, 317-330, 2011
L. Simon: Nonlinear second order evolution equations with state-dependent delays, EJQTDE, Proc. 9th Coll. QTDE, No. 14, 1-12., 2012
Hatzopoulos, V., Taylor, M., Simon, P.L., Kiss., I.Z.: Multiple sources and routes of information transmission: implications for epidemic dynamics, Math. Biosci., 231, 197-209, 2011
Bátkai, A., Kiss, I.Z., Sikolya.E., Simon, P.L.: Differential equation approximations of stochastic network processes: an operator semigroup approach, Netw. Heter. Media., 7, 43-58, 2012
Szabó, A., Simon, P.L., Kiss., I.Z.: Detailed study of bifurcations in an epidemic model on a dynamic network, Differ. Equ. Appl., 4, 277-296, 2012
I. Faragó, S. Korotov, T. Szabó: On modifications of continuous and discrete maximum principles for reaction-diffusion problems, Adv. Appl.Math. Mech., 3, 109-120, 2011
Bátkai A, Csomós P, Farkas B: Operator splitting for nonautonomous delay equations, COMPUTERS AND MATHEMATICS WITH APPLICATIONS 65, 315-324, 2013
Bátkai A, Csomós P, Farkas B, Nickel G.: Operator splitting with spatial-temporal discretization, OPERATOR THEORY : ADVANCES AND APPLICATIONS 221: pp. 161-171, 2012
András Bátkai, Petra Csomós, Klaus-Jochen Engel, Bálint Farkas: Stability and Convergence of Product Formulas for Operator Matrices, INTEGRAL EQUATIONS AND OPERATOR THEORY 74:(2) pp. 281-299, 2012
András Bátkai, Ullrich Groh, Dávid Kunszenti-Kovács, Marco Schreiber: Decomposition of operator semigroups on W*-algebras, SEMIGROUP FORUM 84: pp. 8-24, 2012
A. Besenyei, P. Simon: Asymptotic output controllability via Dynamic Matrix Control, Differ. Eq. Appl., 4, 495-519, 2012
G.Colombo, M.Feckan, B.M.Garay: Inflated deterministic chaos and Smale's horseshoe, J. Difference Eq. Appl. 18, 471--488, 2012
B.M.Garay, A.Simonovits, J.Tóth: Local interaction in tax evasion, Economics Letters 115, 412-415, 2012
M.Forti, B.M.Garay, M.Koller, L.Pancioni: An experimental study on long transient oscillations in cooperative CNN rings, Proceedings of the 13th International Workshop on Cellular Nanoscale Networks and their Applications, 2012
I. Faragó, J. Karátson, S. Korotov: Discrete maximum principles for the FEM solution of some nonlinear parabolic problems, IMA Numerical Analysis, 32, 1541–1573, 2012
Mincsovics M. E., Horváth L. T: On the differences of the discrete weak and strong maximum principles for elliptic operators, Lecture Notes in Computer Science, Springer 7116, 614—621, 2012
Horváth L. T., Izsák F: Implicit a posteriori error estimation using patch recovery techniques, Cent. Eur. J. Math. 10(1), 55-72, 2012
András Bátkai, Eszter Sikolya: The norm convergence of a Magnus expansion method, Cent. Eur. J. Math.10: pp. 150-158., 2012
Taylor, M., Simon, P.L., Green, D.M., House, T., Kiss., I.Z: From Markovian to pairwise epidemic models and the performance of moment closure approximations, J. Math. Biol., 64, 1021-1042, 2012
Simon, P.L., Kiss, I.Z.: New moment closures based on a priori distributions with applications to epidemic dynamics, Bull. Math. Biol., 74,1501-1515, 2012
Simon, P.L., Kiss, I.Z.: From exact stochastic to mean-field ODE models: a new approach to prove convergence results, IMA J. Appl. Math., 2012, doi: 10.1093/imamat/hxs001, 2012
Kiss., I.Z., Berthouze, L., Taylor, T.J., Simon, P.L.: Modelling approaches for simple dynamic networks and applications to disease transmission models, Proc.Roy.Soc.A, 468 (2141), 1332-1355, 2012
Nagy, N., Simon, P.L.: Monte-Carlo simulation and analytic approximation of epidemic processes on large networks, Central European Journal of Mathematics, 11(4), 800-815, 2013
Taylor, T.J., Hartley, C., Simon, P.L., Kiss., I.Z., Berthouze, L.: Identification of criticality in neuronal avalanches: I. A theoretical investigation of the non-driven case, J. Math. Neuroscience, doi:10.1186/2190-8567-3-5, 2013
Csörgő, G., Simon, P.L.: Numerical and analytical study of bifurcations in a model of electrochemical reactions in fuel cells, Computers and Mathematics with Applications, 65, 325-337, 2013
Ghazaryan, A., Schecter, S., Simon, P.L.: Gasless combustion fronts with heat loss, SIAM J. Appl. Math., 73(3), 1303-1326, 2013
Faragó István, Havasi Ágnes, Zlatev Zahari: Advanced Numerical Methods for Complex Environmental Models: Needs and Availability, BENTHAM SCIENCE PUBL. LTD, 2013
Farago I, Izsak F, Szabo T.: An IMEX scheme combined with Richardson extrapolation methods for some reaction-diffusion equations, IDŐJÁRÁS / QUARTERLY JOURNAL OF THE HUNGARIAN METEOROLOGICAL SERVICE 117:(2), 2013
András Bátkai, Petra Csomós, Bálint Farkas: Operator splitting for dissipative delay equations, IMA Journal of Numerical Analysis (accepted), 2013
M.Forti, B.M.Garay, M.Koller, L.Pancioni: Long transient oscillations in a class of cooperative cellular neural networks, Int. J. Circuit Theory Applications, DOI: 10.1002/cta.1965, 2013
M.DiMarco, M.Forti, B.M.Garay, M.Koller, L.Pancioni: Multiple metastable rotating waves and long transients in cooperative CNN rings, European Conference on Circuit Theory and Design (ECCTD), 2013
B.M.Garay, A.Simonovits, J.Tóth: Local interaction in tax evasion, Economics Letters 115, 412-415, 2012
Nagy N, Kiss IZ, Simon PL: Approximate master equations for dynamical processes on graphs, MATH MODEL NAT PHENO 9: (2) 43-57, 2014
Szabó-Solticzky A, Simon PL: The effect of graph structure on epidemic spread in a class of modified cycle graphs, MATH MODEL NAT PHENO 9: (2) 89-107, 2014
Kiss IZ, Morris CG, Selley F, Simon PL, Wilkinson RR: Exact deterministic representation of Markovian epidemics on networks with and without loops, J MATH BIOL 70: (3) 437-464, 2015
Sélley F, Besenyei Á, Kiss IZ, Simon PL: Dynamic control of modern, network-based epidemic models, SIAM J APPL DYN SYST 14: (1) 168-187, 2015
Simon PL, Kiss IZ: From exact stochastic to mean-field ODE models: a new approach to prove convergence results, IMA J APPL MATH 78: (5) 945-964, 2013




 

Events of the project

  
2013-03-13 14:30:00
Résztvevők változása
2012-03-23 13:59:32
Résztvevők változása



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