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Jurgen Geiser - One of the best experts on this subject based on the ideXlab platform.
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Operator-Splitting methods via the Zassenhaus product formula
Applied Mathematics and Computation, 2011Co-Authors: Jurgen Geiser, Gamze TanoğluAbstract:Abstract In this paper, we contribute an Operator-Splitting method improved by the Zassenhaus product. Zassenhaus products are of fundamental importance for the theory of Lie groups and Lie algebras. While their applications in physics and physical chemistry are important, novel applications in CFD (computational fluid dynamics) arose based on the fact that their sparse matrices can be seen as generators of an underlying Lie algebra. We apply this to classical Splitting and the novel Zassenhaus product formula. The underlying analysis for obtaining higher order Operator-Splitting methods based on the Zassenhaus product is presented. The benefits of dealing with sparse matrices, given by spatial discretization of the underlying partial differential equations, are due to the fact that the higher order commutators are very quickly computable (their matrix structures thin out and become nilpotent). When applying these methods to convection–diffusion-reaction equations, the benefits of balancing time and spatial scales can be used to accelerate these methods and take into account these sparse matrix structures. The verification of the improved Splitting methods is done with numerical examples. Finally, we conclude with higher order Operator-Splitting methods.
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Higher order Operator Splitting methods via Zassenhaus product formula: Theory and applications
Computers & Mathematics with Applications, 2011Co-Authors: Jurgen Geiser, Gamze Tanoğlu, Nurcan GücüyenenAbstract:Abstract In this paper, we contribute higher order Operator Splitting methods improved by Zassenhaus product. We apply the contribution to classical and iterative Splitting methods. The underlying analysis to obtain higher order Operator Splitting methods is presented. While applying the methods to partial differential equations, the benefits of balancing time and spatial scales are discussed to accelerate the methods. The verification of the improved Splitting methods are done with numerical examples. An individual handling of each Operator with adapted standard higher order time-integrators is discussed. Finally, we conclude the higher order Operator Splitting methods.
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Overlapping Operator Splitting methods and applications in stiff differential equations
Neural Parallel & Scientific Computations archive, 2008Co-Authors: Jurgen Geiser, Christos KravvaritisAbstract:In this article we study the stability of an overlapping Operator-Splitting methods based on iterative methods. We discuss the overlapping iterative Operator Splitting method in the context of decoupling the stiff and non-stiff Operators. In the context of stabilisation the stiff Operators, we present the overlapping ideas as extension to the standard iterative Operator Splitting method. The efficiency of considering the overlapping method instead of the standard method whole domain in the is discussed. We apply our theoretical results on model problems in stiff parabolic partial differential equations.
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A domain decomposition method based on the iterative Operator Splitting method
Applied Numerical Mathematics, 2008Co-Authors: Jurgen Geiser, Christos KravvaritisAbstract:In this article a new approach is proposed for constructing a domain decomposition method based on the iterative Operator Splitting method. The convergence properties of such a method are studied. The main feature of the proposed idea is the decoupling of space and time. We present a multi-iterative Operator Splitting method that combines iteratively the space and time Splitting. We confirm with numerical applications the effectiveness of the proposed iterative Operator Splitting method in comparison with the classical Schwarz waveform relaxation method as a standard method for domain decomposition. We provide improved results and convergence rates.
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Iterative Operator-Splitting methods for nonlinear differential equations and applications of deposition processes
2008Co-Authors: Jurgen Geiser, Lena NoackAbstract:In this article we consider iterative Operator-Splitting methods for nonlinear differential equations with bounded and unbounded Operators. The main feature of the proposed idea is the embedding of Newton’s method for solving the split parts of the nonlinear equation at each step. The convergence properties of such a mixed method are studied and demonstrated. We confirm with numerical applications the effectiveness of the proposed scheme in comparison with the standard Operator-Splitting methods by providing improved results and convergence rates. We apply our results to deposition processes. Keyword numerical analysis, Operator-Splitting method, initial value problems, iterative solver method, stability analysis, convection-diffusion-reaction equation. AMS subject classifications. 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50.
Petra Csomos - One of the best experts on this subject based on the ideXlab platform.
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Operator Splitting for abstract Cauchy problems with dynamical boundary condition
arXiv: Analysis of PDEs, 2020Co-Authors: Petra Csomos, Matthias Ehrhardt, Bálint FarkasAbstract:In this work we study Operator Splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of one-sided coupled Operator matrices provides an excellent framework to study the well-posedness of such problems. We show that with this machinery even Operator Splitting methods can be treated conveniently and rather efficiently. We consider three specific examples: the Lie (sequential), the Strang and the weighted Splitting, and prove the convergence of these methods along with error bounds under fairly general assumptions.
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Operator Splitting with spatial temporal discretization
arXiv: Functional Analysis, 2012Co-Authors: Andras Batkai, Petra Csomos, Bálint Farkas, Gregor NickelAbstract:Continuing earlier investigations, we analyze the convergence of Operator Splitting procedures combined with spatial discretization and rational approximations.
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Operator Splitting for delay equations
Computers & Mathematics with Applications, 2008Co-Authors: Petra Csomos, Gregor NickelAbstract:Operator Splitting methods are widely used for partial differential equations. Up until now, they have not been used for delay differential equations. In this paper we introduce Splitting methods for delay equations in an abstract setting. We then prove the convergence of the method and discuss the results of some numerical experiments.
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Operator Splitting procedures for air pollution transport models
Lecture Notes in Computer Science, 2006Co-Authors: Petra CsomosAbstract:In the present paper a simple two dimensional air pollution transport model is investigated applying a sequential Operator Splitting procedure. A comparison is done between the cases when Eulerian and semi-Lagrangian schemes are used for the advection sub-problem.
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LSSC - Operator Splitting procedures for air pollution transport models
Large-Scale Scientific Computing, 2006Co-Authors: Petra CsomosAbstract:In the present paper a simple two dimensional air pollution transport model is investigated applying a sequential Operator Splitting procedure. A comparison is done between the cases when Eulerian and semi-Lagrangian schemes are used for the advection sub-problem.
Victor M. Preciado - One of the best experts on this subject based on the ideXlab platform.
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Robust Convergence Analysis of Three-Operator Splitting
arXiv: Optimization and Control, 2019Co-Authors: Han Wang, Mahyar Fazlyab, Shaoru Chen, Victor M. PreciadoAbstract:Operator Splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-Operator Splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to verify sublinear/linear convergence of the TOS algorithm and guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. We illustrate our results numerically by solving an input-constrained optimal control problem.
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Allerton - Robust Convergence Analysis of Three-Operator Splitting
2019 57th Annual Allerton Conference on Communication Control and Computing (Allerton), 2019Co-Authors: Han Wang, Mahyar Fazlyab, Shaoru Chen, Victor M. PreciadoAbstract:Operator Splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-Operator Splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to verify sublinear/linear convergence of the TOS algorithm and guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. We illustrate our results numerically by solving an input-constrained optimal control problem.
Istvan Farago - One of the best experts on this subject based on the ideXlab platform.
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A modified iterated Operator Splitting method
Applied Mathematical Modelling, 2008Co-Authors: Istvan FaragoAbstract:In this paper we give a short overview of some traditional Operator Splitting methods. Furthermore, we introduce two recently developed methods, namely the additive Splitting and the iterated Splitting. We analyze the iterated Splitting method in detail and give the suitable strategy for the choice of the initial elements in the iterations in order to get higher order discretization.
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iterative Operator Splitting methods for linear problems
Computational Science and Engineering, 2007Co-Authors: Istvan Farago, Jurgen GeiserAbstract:The Operator-Splitting methods are based on Splitting of the complex problem into a sequence of simpler tasks. A useful method is the iterative Splitting method which ensures a consistent approximation in each step. In our paper, we suggest a new method which is based on the combination of the Splitting time interval and the traditional iterative Operator Splitting. We analyse the local Splitting error of the method. Numerical examples are given in order to demonstrate the method.
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Application of the Operator Splitting method for real-life problems
Idojaras, 2006Co-Authors: Istvan FaragoAbstract:In the modeling of real-life complex time-dependent phenomena, the simultaneous effect of several different sub-processes has to be described. The Operators describing the sub-processes are as a rule simpler than the whole spatial differential Operator. Operator Splitting is a widely used procedure in the numerical solution of such problems. The point in Operator Splitting is the replacement of the original model with one in which appropriately chosen groups of the sub-processes, described by the model, take place successively in time. This de-coupling procedure allows us to solve a few simpler problems instead of the whole one. In this paper several Splitting methods are constructed (sequential Splitting, Strang-Marchuk Splitting, weighted Splitting, additive Splitting, iterated Splitting) and analyzed. Application of the Operator Splitting method to real-life problems is investigated, with great emphasis on long-range air pollution transport. The accuracy (local Splitting error) of the methods is discussed, and the main advantages and drawbacks of this approach are listed.
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LSSC - The Mathemathical Background of Operator Splitting and the Effect of Non-commutativity
Large-Scale Scientific Computing, 2001Co-Authors: Istvan Farago, Ágnes HavasiAbstract:Operator Splitting is a widely used procedure in the numerical solution of initial and boundary value problems of partial differential equations. In this paper the error of the Operator Splitting, the so-called Splitting error, is investigated. The mathematical background of Operator Splitting is shortly discussed. Sufficient conditions, under which the Splittingerror vanishes, are formulated for the Splittingmetho d of the Danish Eulerian Model. The study is based on the L-commutativity of the Operators used in the model. Finally, the size of the Splitting error is analysed in the case where the splitted Operators are linear.
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Numerical Methods and Applications - New Operator Splitting methods and their analysis
Numerical Methods and Applications, 1Co-Authors: Istvan FaragoAbstract:In this paper we give a short overview of some traditional Operator Splitting methods. Then we introduce two new methods, namely the additive Splitting and the iterated Splitting. We analyze these methods and compare them to the traditional ones.
Han Wang - One of the best experts on this subject based on the ideXlab platform.
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Robust Convergence Analysis of Three-Operator Splitting
arXiv: Optimization and Control, 2019Co-Authors: Han Wang, Mahyar Fazlyab, Shaoru Chen, Victor M. PreciadoAbstract:Operator Splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-Operator Splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to verify sublinear/linear convergence of the TOS algorithm and guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. We illustrate our results numerically by solving an input-constrained optimal control problem.
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Allerton - Robust Convergence Analysis of Three-Operator Splitting
2019 57th Annual Allerton Conference on Communication Control and Computing (Allerton), 2019Co-Authors: Han Wang, Mahyar Fazlyab, Shaoru Chen, Victor M. PreciadoAbstract:Operator Splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-Operator Splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to verify sublinear/linear convergence of the TOS algorithm and guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. We illustrate our results numerically by solving an input-constrained optimal control problem.
