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85
Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics  A review with some applications
, 1995
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Solving Differential Equations with Radial Basis Functions: Multilevel Methods and Smoothing
 Advances in Comp. Math
"... . Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a ..."
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Cited by 29 (7 self)
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. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using socalled meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and coworker's results [3] using the socalled elementfree Galerkin method, Duarte and Oden's work [11] using hp clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...
Multistep Approximation Algorithms: Improved Convergence Rates through Postconditioning with Smoothing Kernels
 Advances in Comp. Math. 10
, 1999
"... . We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suff ..."
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Cited by 23 (11 self)
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. We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a "loss of derivatives", and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of NashMoser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step. 1. Introduction It has been only very recently that the idea of multistep (or multilevel) interpolation ...
The KolmogorovArnoldMoser theorem
 Math. Phys. Electron. J
, 2004
"... This paper gives a self contained proof of the perturbation theorem for invariant tori in Hamiltonian systems by Kolmogorov, Arnold, and Moser with sharp differentiablility hypotheses. The proof follows an idea outlined by Moser in [16] and, as byproducts, gives rise to uniqueness and regularity the ..."
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Cited by 14 (0 self)
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This paper gives a self contained proof of the perturbation theorem for invariant tori in Hamiltonian systems by Kolmogorov, Arnold, and Moser with sharp differentiablility hypotheses. The proof follows an idea outlined by Moser in [16] and, as byproducts, gives rise to uniqueness and regularity theorems for invariant tori. 1 1
A nonlinear analysis of the averaged Euler equations
 Fields Inst. Comm., Arnold
, 1998
"... This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter α; one interpretation is that they are obtained by ensemble ..."
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Cited by 8 (7 self)
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This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter α; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order α. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold’s theorem), but with respect to a right invariant H 1 metric instead of the L 2 metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the EbinMarsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, even in the
Multibump orbits homoclinic to resonance bands
 Tran. Amer. Math. Soc
, 1996
"... We establish the existence of several classes of multibump orbits homoclinic to resonance bands for completelyintegrable Hamiltonian systems subject to smallamplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are intersper ..."
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Cited by 8 (2 self)
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We establish the existence of several classes of multibump orbits homoclinic to resonance bands for completelyintegrable Hamiltonian systems subject to smallamplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multibump Šilnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands. 1
INFINITE DIMENSIONAL LIE GROUPS WITH APPLICATIONS TO MATHEMATICAL PHYSICS
"... We give a survey of infinite dimensional Lie groups and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume preserving, symplectic and contact transformations, as well as gauge groups, quantomorphisms and loop gro ..."
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Cited by 7 (0 self)
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We give a survey of infinite dimensional Lie groups and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume preserving, symplectic and contact transformations, as well as gauge groups, quantomorphisms and loop groups. Applications include fluid dynamics, Maxwell’s equations, plasma physics and BRST symmetries in quantum field theory. We discuss the Lie group structures of pseudodifferential and Fourier integral operators, both on compact and noncompact manifolds and give applications to the KdV equation and quantization. 1 INTRODUCTION: Lie Groups play an important role in physical systems both as phase spaces and as symmetry groups. Infinite dimensional Lie groups occur in the study of dynamical systems with an infinite number
Algorithms defined by Nash iteration: some implementations via multilevel collocation and smoothing
 J. Comp. Appl. Math
"... We describe the general algorithms of Nash iteration in numerical analysis. We make a particular choice of algorithm involving multilevel collocation and smoothing. Our test case is that of a linear dierential equation, although the theory allows for the approximate solution of nonlinear dierential ..."
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Cited by 7 (2 self)
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We describe the general algorithms of Nash iteration in numerical analysis. We make a particular choice of algorithm involving multilevel collocation and smoothing. Our test case is that of a linear dierential equation, although the theory allows for the approximate solution of nonlinear dierential equations. We describe the general situation completely, and employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps. We had earlier shown how these ideas apply to scattered data approximation, but in this work we are interested in the application of the ideas to the numerical solution of dierential equations. We make use of approximate smoothers, involving the solution of evolution equations with calibrated time steps.
Construction of invariant whiskered tori by a parameterization method. part I: Maps . . .
, 2009
"... We present theorems which provide the existence of invariant whiskered tori in finitedimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We show that, given an approximate solution of the invariance eq ..."
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Cited by 6 (1 self)
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We present theorems which provide the existence of invariant whiskered tori in finitedimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We show that, given an approximate solution of the invariance equation which satisfies some nondegeneracy conditions, there is a true solution nearby. We call this an a posteriori approach. The proof of the main theorems is based on an iterative method to solve the functional equation. The theorems do not assume that the system is close to integrable nor that it is written in actionangle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant. The a posteriori formulation allows us to justify approximate solutions produced by many nonrigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on succesive corrections. This makes it possible to adapt the method almost verbatim to several infinitedimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.