Mathematical methods of preconditioning in nonlinear physical models  Page description

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Details of project

  
Identifier
43765
Type K
Principal investigator Faragó, István
Title in Hungarian A prekondicionálás matematikai módszerei nemlineáris fizikai modellekben
Title in English Mathematical methods of preconditioning in nonlinear physical models
Panel Mathematics and Computing Science
Department or equivalent Applied Analysis and Computational Mathematics (Eötvös Loránd University)
Participants Csomós, Petra
Havasi, Ágnes
Horváth, Róbert
Izsák, Ferenc
Karátson, János
Lóczi, Lajos
Starting date 2003-01-01
Closing date 2007-12-31
Funding (in million HUF) 3.625
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
Operátorszeletelések és alkalmazásaik témájában elméleti és számítógépes kutatásokat folytattunk a hagyományos szeletelésekre. További új szeletelések kidolgozása mellett alkalmaztuk a módszereket valós feladatokra. Prekondicionált iterációs módszerek és alkalmazásaik területén nemlineáris modellfeladatokra dolgoztunk ki iterációs eljárásokat (a prekondicionáló operátorok elve alapján), valamint a prekondicionált konjugált gradiens-módszer szuperlineáris konvergenciájával és annak rácsfüggetlenségével foglalkoztunk. Numerikus módszerek kvalitatív tulajdonságai témakörében elliptikus és parabolikus feladatokra tanulmányoztuk a diszkrét maximum-elvet. Az időfüggő feladatokra megadtuk a legfontosabb diszkrét kvalitatív tulajdonságok közti kapcsolatokat, és hasznos elégséges feltételeket igazoltunk. Nemhiperbolikus esetben részletesen tanulmányoztuk a strukturális stabilitást. Egy fontos kémiai alkalmazás kapcsán a Liesegang-jelenség modelljeinek kvalitatív tulajdonságait vizsgáltuk. Fizikai és kémiai feladatok numerikus megoldása területén az operátorszeletelési eljárás segítségével új és a korábbi FDTD típusú módszereknél hatékonyabb módszereket konstruáltunk. Eredményeinkből 97 közleményt készítettünk. Kb. 50 nemzetközi és hazai konferencián tartottunk előadást, ill. workshopokat szerveztünk. Kutatásaink nagy része nemzetközi együttműködés keretében valósult meg. A résztvevők közül ketten doktoráltak, ketten idén védenek, és a témavezető a pályázati idő alatt habilitált.
Results in English
In the topic of operator splitting and its applications we investigated theoretically and numerically the traditional splitting methods. We constructed further new splitting techniques, and applied them in real-life model problems. In the field of preconditioned iterative methods and their applications we developed iterative solution methods for various nonlinear problems (based on the principle of preconditioning operators). We also studied the superlinear convergence and mesh-independence of the preconditioned conjugate gradient method. For the qualitative properties of the numerical methods we studied the discrete maximum principle for elliptic and parabolic problems. We gave the relations between the most important discrete qualitative properties and proved sufficient conditions that can be used in practice. In the non-hyperbolic case we gave a detailed study of the structural stability. We investigated the qualitative properties of mathematical models of the Liesegang phenomenon. For physical and chemical problems, by the use of the operator splitting technique, we constructed new time integration methods that are more efficient than the classical FDTD-method. We prepared 97 scientific publications. We participated in about 50 conferences, organized workshops. Most of the research was done within international cooperation. Two of the participants have completed their PhD studies, two are to defend this year, and the project coordinator has made his habilitation.
Full text http://real.mtak.hu/1218/
Decision
Yes





 

List of publications

  
I. Dimov, I. Faragó, A. Z. Zlatev: Parallel computations with large-scale air pollution models, Problems in Programming 3, 44-52., 2003
L. Lóczi: A normal form for the fold bifurcation and its discretization, Preprint 03/006, DFG research group ''Spectral analysis, asymptotic distributions and stochastic dynamics'', Univ. of Bielefeld, www.mathematik.uni-b, 2003
I. Faragó, Á. Havasi: Relationship between vanishing splitting errors and pairwise commutativity, Applied Math. Letters (közlésre elfogadva), 2007
L. Lóczi: A thesaurus.maths.org többnyelvű internetes matematikai fogalomtár, Matematika Tanári Kincsestár, 13. kieg. kötet, 2004. szept., I 4.2. 1-8., 2004
L. Lóczi: Könyvismertető: Prem K. Kythe, Pratap Puri, Michael R. Schäferkotter: Partial Differential Equations and Mathematica, Alkalmazott Matematikai Lapok, 21 (2003) 177-179., 2003
L. Lóczi: Two centuries of the equations of commutativity and associativity of exponentiation, Teaching Math. and Comp. Sci., 1/2 (2003) 219-233., 2003
P. Csomós, Á. Havasi, I. Faragó: Weighted sequential splittings and their analysis, Comp. Math. Appl., 50, 1017-1031., 2005
P. Csomós: Some aspects of interaction between operator splitting procedures and numerical methods, Adv. in Air Pollution Modeling for Environmental Sec., NATO Science Series, 54, Springer Verlag, 77-91., 2005
I. Dimov, I. Faragó, Á. Havasi, Z. Zlatev: Operator splitting and commutativity analysis in the Danish Eulerian Model, Math. Comp. Sim. V. 67 (2004) 217-233., 2004
M. Botchev, I. Faragó, A. Havasi: Testing weighted splitting schemes on a one-column transport-chemistry model, Int. J. Environmental Pollution 22, 3-16., 2004
I. Faragó: Operator splittings and numerical methods, Lect. Notes Comp. Sci. 3743, Springer Verlag, Berlin, 347-354., 2006
R. Horváth, I. Faragó, W.H.A. Schilders: Iterative Solution Methods of the Maxwell Equations Using Staggered Grid Spatial Discretization, : Fifty Years of Conjugate Gradient, Springer Verlag Berlin, 2004, 211-220., 2004
F. Izsák, I. Lagzi: Precipitate pattern formation in fluctuating media, J. Chem. Phys., V.120 (2004) 1837-1840., 2004
J. Karátson: Constructive Sobolev gradient preconditioning for semilinear elliptic systems, Electron. J. Diff. Eqns. (2004), N 75, 1-26., 2004
M. Botchev, I. Faragó, R. Horváth: Application of the operator splitting to the Maxwell equations with the source term, Memorandum 1818, Department of Applied Mathematics, University of Twente, Enschede. ISSN 0169-2690. http://eprints.eemcs.utwente.nl/9206/, 2007
O. Axelsson, J. Karátson,: , Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems, Comput. Math. Simul., .64 (2004), 649-668., 2004
O. Axelsson, I. Faragó, J. Karátson: On the application of preconditioning operators for nonlinear elliptic problems, Conjugate Gradient Algorithms and Finite Element Methods, Springer, 2004, 247-261., 2004
I. Faragó, S. Korotov: Nonnegativity preservation in the numerical solution of linear partial differntial equations, Conjugate Gradient Algorithms and Finite Element Methods, Springer Verlag, Berlin, 2004, 185-196., 2004
I. Faragó, R. Horváth, S. Korotov: Discrete maximum principle for Galerkin finite element solutions to parabolic problems on rectangular meshes, Feistauer, M. et all ed; Numerical Mathematics and Advanced Applications; Springer Verlag, Berlin, 2004, 298-307., 2004
I. Faragó: Splitting methods for abstract Cauchy problems, in: Numerical Analysis and Its Application, , Lect. Notes Comp.Sci. 3401, Springer Verlag, Berlin, 2005, 35-45., 2004
B. M. Garay, L. Lóczi: Monotone delay equations and Runge-Kutta discretization, Functional Differential Equations, Vol. 11, (2004) 59-67., 2004
I. Lagzi, F. Izsák: Stabilization and destabilization effects of the electric field on stochastic precipitate pattern formation, Chem. Phys., V.303 (2004) 151-155., 2004
I. Lagzi, F. Izsák, S.C. Muller, J. Ross: , Comment on ``Precipitate pattern formation in fluctuating media\'\', J. Chem. Phys., V. 121(2004) 3943-3943., 2004
L. Lóczi: Az ötödfokú egyenlet megoldása, POLYGON folyóirat (közlésre elfogadva), 2007
J. Karátson, S.Korotov: Continuous and discrete maximum principles for nonlinear elliptic problems, Computational and Mathematical Methods in Science and Engineering, (2004) 191-200., 2004
I. Faragó, Á. Havasi: On the convergence and local splitting error of different splitting schemes, Progress in Computational Fluid Dynamics, 5 (2005) 495-504., 2005
P. Csomós: Analysis of a transport model applying operator splitting and semi-Lagrangian method, International Journal of Computational Science and Engineering (közlésre elfogadva), 2007
P. Csomós: Analyticcal solutions and numerical experiments for optimizing operator splitting procedures, Időjárás, Quarterly Journal of Hungarian Meteorological Service (közlésre elfogadva), 2007
I. Faragó: New operator splittings and their applications, Lect. Notes Comp. Sci. 4310, Springer Verlag, Berlin, 443-450., 2007
P. Csomós: Operator splitting procedures for air pollution transport models, Proceedings of the 5th International Conference on Large-Scale Scientific Computations, June 6-10, Sozopol (2005) (to appear)., 2005
I. Faragó, J. Karátson: Variable preconditioning for nonlinear elliptic problems via inexact Newton methods in Hilbert spaces, SIAM J. Numer. Anal. 41, 1242-1262., 2003
F. Izsák: An existence theorem for a type of functional differential equation with infinite delay, Acta Mathematica Hungarica, 108 (2005) 137-153., 2005
I. Faragó, R. Horváth, S. Korotov: Discrete maximum principle for linear parabolic problems solved on hybrid meshes, Appl. Numer. Math. 43, 249-264., 2005
J. Karátson: Sobolev regularity of the second biharmonic problem on a rectangle, Acta Math. Hung., 109 (2005) 269-273., 2005
M. Ripszam, D. Nagy, A. Wolford, I. Lagzi, F. Izsák: The Liesegang eyes phenomenon, Chemical Physics Letters, 414 (2005).384-388., 2005
I. Faragó, S. Korotov, P. Neittaanmäki: Galerkin approximations for the linear parabolic equation with the third boundary condition, Appl. Math. 48, 111-128., 2003
J. Karátson, S. Korotov: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math., 99 (2005) 669-698., 2005
L. Lóczi: Conjugacy in the discretized transcritical bifurcation, Preprint 05/016, DFG research group ''Spectral analysis, asymptotic distributions and stochastic dynamics'', Univ. of Bielefeld, www.mathematik.uni-b, 2005
I. Faragó, R. Horváth, W. Schilders: Investigation of numerical time integrations of the Maxwell equations using the staggered grid spatial discretization, Int. J. Num. Modelling, 18, 149-169., 2005
I. Faragó: Lax equivalence theorem in Banach spaces with parameter, Tübinger Berichte zur Funktionalanalysis, 14, Univesität Tübingen 89-95., 2005
I. Faragó: Splitting methods for abstract Cauchy problems, in: Z. Li, L. Vulkov, J. Wa¶niewski eds. Numerical Analysis and Its Application, Lect. Notes Comp. Sci. 3401, Springer Verlag, Berlin, 35-45., 2005
Á. Havasi: Dispersion analysis of operator splittings in the linearized shallow water equations, Proceedings of the 5th International Conference on Large-Scale Scientific Computations, June 6-10, 2005, Sozopol (2005) (to appear), 2005
R. Horváth: On the Sign-Stability of the Finite Difference Solutions of One-Dimensional Parabolic Problems, Lecture Notes in Computer Science (közlésre elfogadva), 2007
F. Izsák: Discontinuous Galerkin Methods for Partial Differential Equations in the Atmospheric Modeling, Időjárás, Quarterly Journal of Hungarian Meteorological Service (közlésre elfogadva), 2007
J.J.W. van der Vegt, F. Izsák, O. Bokhove: Error analysis of a continuous-discontinous Galerkin finite element model for generalized 2D vorticity dynamics, SIAM Journal on Numerical Analysis (közlésre elfogadva), 2007
F. Izsák, I. Lagzi: A new universal law for the Liesegang pattern formation, Journal of Chemical Physics, 122 (2005) (184707 – Article number)., 2005
F. Izsák, I. Lagzi: Simulation of a crossover from the precipitation wave to moving Liesegang pattern formation, Journal of Physical Chemistry A, 109 (2005) 730-733., 2005
J. Karátson, L. Lóczi: Sobolev gradient preconditioning for the electrostatic potential equation, Comput. Math. Appl., 50 (2005) 1093-1104., 2005
J. Karátson, I. Faragó: Preconditioning operators and Sobolev gradients for nonlinear elliptic problems, Comput. Math. Appl. 50, 1077-1092., 2005
J. Karátson: Mesh independent superlinear convergence estimates of the conjugate gradient method for some equivalent self-adjoint operators, Appl. Math. (Prága) 50 (2005) 277-290., 2005
L. Lóczi: Construction of a conjugacy and closeness estimates in the discretized fold bifurcation, Preprint 04/022, DFG research group ''Spectral analysis, asymptotic distributions and stochastic dynamics'', Univ. of Bielefeld, www.mathematik.uni-b, 2004
I. Faragó, Á. Havasi, K. Georgiev: Advances in Air Pollution Modeling for Environmental Security, NATO Science Series, 54, Springer Verlag, 2005
O. Axelsson, J. Karátson,: Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators, Numer. Math., 99, 197-223., 2004
J. Karátson, S. Korotov: Discrete maximum principles for finite element solutions of mixed nonlinear elliptic problems using quadratures, J. Comp. Appl. Math. 192, No. 1, pp. 75-88, 2006
J. Karátson, T. Kurics: On Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations, Lecture Notes Comp Sci (közlésre elküldve), preprint: ELTE Dept. Appl. Anal., www.cs.elte.hu/applanal/preprints, 2006-13, 2006
O. Axelsson, J. Karátson: Mesh independent superlinear PCG rates via compact-equivalent operators, SIAM J Numer Anal (közlésre elfogadva), 2007
I. Antal, J. Karátson: A mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems, Comput. Math. Appl. (közlésre elfogadva), 2007
J. Karátson, S. Korotov: Discrete maximum principles for FEM solutions of some nonlinear elliptic interface problems, Helsinki University of Technology, Institute of Mathematics, Research Report A510, 2006
J. Karátson, S. Korotov, M. Krizek: On Discrete Maximum Principles for Nonlinear Elliptic Problems, Math. Comp. Simul. (közlésre elfogadva), 2006
O. Axelsson, J. Karátson,: Symmetric part preconditioning of the CGM for Stokes type saddle-point systems, Numer. Funct. Anal. (közlésre elküldve), preprint: Technical reports from the Dept. of Inf. Tech., Uppsala, No. 2006-004, 2006
J. Karátson,: On the superlinear convergence rate of the preconditioned CGM for some nonsymmetric elliptic problems, Numer. Funct. Anal. (közlésre elküldve), preprint: ELTE Dept. Appl. Anal., www.cs.elte.hu/applanal/preprints, 2006-11, 2006
P. Csomós. I. Faragó: Error analysis of the numerical solution of split differential equations, Mathematical and Computer Modelling (közlésre elfogadva), 2007
I. Faragó, J. Geiser: Iterative operator- splitting methods for linear problems, Weierstass Institute für Angewandte Analysis, 1043, 1-18., 2005
I. Dimov, I. Faragó, Á. Havasi, Z. Zlatev: Different splitting techniques with application to air pollution models, Int. J. Environmental. Pollution (közlésre elfogadva), 2007
I. Faragó: On the efficiency of the operator splitting method, Problems in Programming 7, 654-658., 2006
A. Dorosenko, I. Faragó, Á. Havasi, V. Prussov: On the numerical solution of the three-dimensional advection-diffusion equation, Problems in Programming 7, 641-647., 2006
I. Faragó, Á. Havasi: Consistency analysis of operator splitting methods for C0-semigroups, Semigroup Forum, online at http://www.springerlink.com/content/u0h415q559527u6x/fulltext.pdf., 2006
I. Faragó, R. Horváth: A review of reliable numerical models for three-dimensional linear parabolic problems, Int.. J. Numer. Meth. Engng. (közlésre elfogadva), 2007
I. Faragó, R. Horváth: On the connections between the qualitative properties of the numerical solutions of linear parabolic problems, SIAM Scientific Computing 28, 2313-2336., 2006
I. Faragó: Application of the operator splitting method for real-life problems, Időjárás, Quarterly Journal of Hungarian Meteorological Service (közlésre elfogadva), 2007
Csomós, I. Dimov, I. Faragó, Á. Havasi, Tz. Ostromsky: Computational complexity of weighted splitting scheme on parallel computers, International Journal of Parallel, Emergent and Distributed Systems (közlésre elfogadva), 2007
I. Faragó, B. Gnandt, Á. Havasi: Additive and iterative splitting methods and their numerical investigation, Comput. Math. Appl. (közlésre elfogadva), 2007
P. Csomós, I. Faragó, Á. Havasi: Operator splitting and global error analysis, NATO Advenced Research Workshop "Air Pollution Processes in Regional Scale", Kallithea, Greece, NATO Science Series, Kluwer Academic Publishers 37-44., 2003
I. Faragó, R. Horváth: Qualitative analysis of discrete mesh operators, Lect. Notes Comp. Sci. (közlésre elküldve), 2007
I. Faragó: Modified iterated operator splitting method, Applied Mathematical Modelling (közlésre elküldve), 2007
I. Faragó, P. Thomsen, Z. Zlatev: On the additive splitting procedures and their computer realization, Applied Mathematical Modelling (közlésre elküldve), 2007
I. Faragó: Operator splitting procedures and their analysis, Alkalmazott Matematikai Lapok (közlésre elküldve), 2007
I. Faragó, Á. Havasi: Higher order adaptive operator splitting schemes, J. Comp. Appl. Math. (közlésre elküldve), 2007
Á. Havasi: Dispersion analysis of operator splittings in the linearized shallow water equations, Lecture Notes in Computer Science, Springer 3743/2006, 355-362., 2006
J. Karátson, J. W. Neuberger: Newton's method in the context of gradients, Trans. AMS (közlésre elküldve), 2006
O. Axelsson, J. Karátson,: Symmetric part preconditioning for the conjugate gradient method, Numer Funct Anal 24 (2003), No. 5-6, 455-474, 2003
J. Karátson, P.L.Simon: Exact multiplicity for degenerate two-point boundary value problems with p-convex nonlinearity, Nonlin. Anal 52 (2003), No. 6., 1569-1590., 2003
J. Karátson, T. Kurics: Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems, J. Comp. Appl. Math (közlésre elfogadva), 2006
I. Lagzi, F. Izsák: Stochastic description of precipitate pattern formation in an electric field, PHYSICAL CHEMISTRY CHEMICAL PHYSICS 5 (19): 4144-4148, 2003
R. Horváth: Uniform Treatment of the Numerical Time-Integration of the Maxwell Equations, Lect. Notes Comp. Sci., 2003
F. Izsak, I. Lagzi: Simulation of Liesegang pattern formation using a discrete stochastic model, CHEMICAL PHYSICS LETTERS 371 (3-4): 321-326, 2003
I. Lagzi, F. Izsak: Regular Liesegang patterns and precipitation waves in an open system, PHYSICAL CHEMISTRY CHEMICAL PHYSICS 7 (22): 3845-3850, 2005
R. Horváth: On the Maximum-Minimum Principle for Advection-Diffusion Equations, Problems in Programming, Nr. 2-3, 2006
A. Volford, F. Izsak, M Ripszam: Systematic front distortion and presence of consecutive fronts in a precipitation system, JOURNAL OF PHYSICAL CHEMISTRY B 110 (10): 4535-4537, 2006
F. Izsak: Maximum likelihood estimation for constrained parameters of multinomial distributions - Application to Zipf-Mandelbrot models, COMPUTATIONAL STATISTICS & DATA ANALYSIS 51 (3): 1575-1583, 2006
R. Horváth: Operator Splittings for the Numerical Solution of Maxwell's Equations, Lect. Notes Comp. Sci., 2006
A. Volford, F. Izsak, Ripszam, M., I. Lagzi: Pattern Formation and Self-Organization in a Simple Precipitation System, Langmuir; (Letter); 23(3); 961-964., 2007
R. Horváth: New Unconditionally Stable Numerical Schemes for Maxwell's Equations, Int. J. Comput. Sci. Eng. (közlésre elfogadva), 2007
F. Izsák, D. Harutyunyan, J.J.W. van der Vegt: Implicit a posteriori error estimates for the Maxwell equations, Memorandum 1816, Department of Applied Mathematics, University of Twente, Enschede. ISSN 0169-2690. http://eprints.eemcs.utwente.nl/8448/, 2007
D. Harutyunyan, F. Izsák, M. A. Botchev, J.J.W. van der Vegt: Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates., Memorandum 1810, Department of Applied Mathematics, University of Twente, Enschede. ISSN 0169-2690. http://eprints.eemcs.utwente.nl/8541/, 2006
R. Horváth: On the Sign-Stability of Numerical Solutions of One-Dimensional Parabolic Problems, Applied Mathematical Modelling (közlésre elküldve), 2007
R. Horváth: Sufficient Conditions of the Discrete Maximum-Minimum Principle for Parabolic Problems on Rectangular Meshes, Int. J. Comput. Sci. Eng. (közlésre elfogadva), 2007



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