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Exponential integrators
, 2010
"... In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential eq ..."
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Cited by 67 (5 self)
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in
On algebraic structures of numerical integration on vector spaces and manifolds
 IRMA Lectures in Mathematics and Theoretical Physics
, 2012
"... Abstract. Numerical analysis of timeintegration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra first appeared in Butcher’s work on composition of integration method ..."
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Cited by 8 (1 self)
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Abstract. Numerical analysis of timeintegration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra first appeared in Butcher’s work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey algebraic structures that have found applications within these areas. This includes preLie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent postLie and Dalgebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of nonautonomous flows and in backward error analysis. Noncommutative Bell polynomials and a noncommutative Faa ̀ di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.
Multiproduct expansion, Suzuki’s method and the Magnus integrator for solving timedependent problems
, 2009
"... Abstract. In this paper we discuss the extention to exponential splitting methods with respect to timedependent operators. For such extensions, the Magnus integration, see [3], [4] and [5] and the Suzuki’s method are incorporating ideas to the timeordered exponential, see [22], [2], [7] and [8]. ..."
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Abstract. In this paper we discuss the extention to exponential splitting methods with respect to timedependent operators. For such extensions, the Magnus integration, see [3], [4] and [5] and the Suzuki’s method are incorporating ideas to the timeordered exponential, see [22], [2], [7] and [8]. We formulate each methods and present their advantages to special timedependent harmonic oscillator problems. An decisive and comprehensive comparison on the Magnus expansion with Suzuki’s method on some problems are given. Here classical and also quantum mechanical can be treated to present the solving in timedependent problems. We choose a radial Schrodinger equation as a classical timedependent harmonic oscillation which combine classical and quantum calculations simultaneously. Here we present the different schemes of the integrator based Magnus scheme and the differential based Suzuki’s method. Based on the spiked harmonic oscillator case we could analyze the differences.
An RBFGalerkin Approach to the TimeDependent Schrödinger Equation
, 2012
"... In this article, we consider the discretization of the timedependent Schrödinger equation using radial basis functions (RBF). We formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions. Since we can show that timestability of the discretization is ..."
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In this article, we consider the discretization of the timedependent Schrödinger equation using radial basis functions (RBF). We formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions. Since we can show that timestability of the discretization is not guaranteed for an RBFcollocation method, we propose to employ a Galerkin ansatz instead. For Gaussians, it is shown that exponential convergence is obtained up to a point where a systematic error from the domain where no basis functions are centered takes over. The choice of the shape parameter and of the resolved region is studied numerically. Compared to the Fourier method with periodic boundary conditions, the basis functions can be centered in a smaller domain which gives increased accuracy with the same number of points.
Geometric integrators with applications to optimal control Lectures by Philipp Bader
"... During the last decades, the community of numerical analysts has shifted from allpurpose methods to schemes that are designed to solve a particular problem efficiently by putting as much information about the system as possible into the numerical integrator. Geometric numerical integrators obtained ..."
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During the last decades, the community of numerical analysts has shifted from allpurpose methods to schemes that are designed to solve a particular problem efficiently by putting as much information about the system as possible into the numerical integrator. Geometric numerical integrators obtained their name after it was noticed that they preserved certain geometric properties of the exact solution of a differential equation and could be considered prototypes of this paradigm. In this series of lectures, an introduction to geometric numerical integrators will be given, the benefits of such methods will be discussed and applications to problems in optimal control will be presented. It is useful to have a rough idea about the numerical integration of ordinary differential equations and a basic knowledge of Lie algebras and Lie groups. The treated examples will come from classical and quantum mechanics and a rudimentary understanding of Hamiltonian will be helpful.
Volume 106
"... Splitting methods in the numerical integration of nonautonomous dynamical systems ..."
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Splitting methods in the numerical integration of nonautonomous dynamical systems
Energétique et transferts
"... Simulations of flame stabilization and stability in highpressure propulsion systems mercredi 5 juin 2013 Romain GARBY ..."
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Simulations of flame stabilization and stability in highpressure propulsion systems mercredi 5 juin 2013 Romain GARBY
Information Amidst Noise: Preserved Codes, Error Correction, and Fault Tolerance in a Quantum World
, 2010
"... For my dad, whose secret ambition has always been to be a physicist, for my mum, for being the strong woman that she is, and of course, for my husband, for always being here. ivv Acknowledgments Looking back at the five years I have spent in Caltech as a part of the Institute of Quantum Information ..."
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For my dad, whose secret ambition has always been to be a physicist, for my mum, for being the strong woman that she is, and of course, for my husband, for always being here. ivv Acknowledgments Looking back at the five years I have spent in Caltech as a part of the Institute of Quantum Information (IQI), I cannot help but consider myself incredibly fortunate to have had guidance from so many people. The first person I have to thank is of course my advisor John Preskill. He has put together the amazing place that is IQI, with its constant flux of visitors, postdocs and students, offering me an invaluable chance to meet so many talented people in the field. Along with sharing his insights in physics and always extending a willing hand in times of difficulty, he taught me a most valuable