The Experts below are selected from a list of 231 Experts worldwide ranked by ideXlab platform
E Gerlach - One of the best experts on this subject based on the ideXlab platform.
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EFFICIENT INTEGRATION OF THE VARIATIONAL EQUATIONS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS: APPLICATION TO THE FERMI–PASTA–ULAM LATTICE
International Journal of Bifurcation and Chaos, 2012Co-Authors: E Gerlach, Siegfried Eggl, Charalampos SkokosAbstract:We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.
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Numerical integration of variational equations.
Physical review. E Statistical nonlinear and soft matter physics, 2010Co-Authors: Ch Skokos, E GerlachAbstract:We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the "Tangent Map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic Map S , while the corresponding Tangent Map TS is used for the integration of the variational equations. A simple and systematic technique to construct TS is also presented.
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Numerical integration of variational equations
Physical Review E, 2010Co-Authors: Ch Skokos, E GerlachAbstract:We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the \textit{`Tangent Map (TM) method'}, a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic Map $S$, while the corresponding Tangent Map $TS$, is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.Comment: 27 pages, 11 figures, to appear in Phys. Rev.
Steven W. Zucker - One of the best experts on this subject based on the ideXlab platform.
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CVPR Workshops - The differential geometry of shape from shading: Biology reveals curvature structure
2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2012Co-Authors: Benjamin Kunsberg, Steven W. ZuckerAbstract:Shape from shading is a classical inverse problem in computer vision. We introduce a novel mathematical formulation for calculating local surface shape based on covariant derivatives, rather than the customary integral minimization or P.D.E approaches. Motivated by neurobiology, we introduce the shading flow field (the Tangent Map to the image isophotes) between the image and the surface levels. Just as in the perceptual organization of texture, we use the parallel transport of our shading flow field to move the isophote field at different points on the unknown surface to a single point, amassing restrictions on our surface curvatures. Under simplifying assumptions we solve exactly for the light source/surface pairs needed for a local image patch to have a given shading flow. The magnitude of the brightness gradient then restricts this family to a single light source and surface estimate pair, up to the concave/convex ambiguity and an additional elliptical/saddle ambiguity. Example calculations illustrate our approach.
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ICCV - Folds and cuts: how shading flows into edges
Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001, 1Co-Authors: Patrick S. Huggins, Steven W. ZuckerAbstract:We consider the interactions between edges and intensity distributions in semi-open image neighborhoods surrounding them. Locally this amounts to a kind of figure-ground problem, and we analyze the case of smooth surface occluding one another. Techniques from differential topology permits a classification of edges based on what we call folds and cuts. Intuititively, folds arise when a surface "folds" out of sight, which in turn may "cut" another surface from view. The classification depends on tangency between an edge Tangent Map and a shading flow field. Examples are included.
Ch Skokos - One of the best experts on this subject based on the ideXlab platform.
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Numerical integration of variational equations.
Physical review. E Statistical nonlinear and soft matter physics, 2010Co-Authors: Ch Skokos, E GerlachAbstract:We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the "Tangent Map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic Map S , while the corresponding Tangent Map TS is used for the integration of the variational equations. A simple and systematic technique to construct TS is also presented.
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Numerical integration of variational equations
Physical Review E, 2010Co-Authors: Ch Skokos, E GerlachAbstract:We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the \textit{`Tangent Map (TM) method'}, a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton's equations of motion by the repeated action of a symplectic Map $S$, while the corresponding Tangent Map $TS$, is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.Comment: 27 pages, 11 figures, to appear in Phys. Rev.
Ted Belytschko - One of the best experts on this subject based on the ideXlab platform.
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An atomistic-based finite deformation membrane for single layer crystalline films
Journal of the Mechanics and Physics of Solids, 2002Co-Authors: Marino Arroyo, Ted BelytschkoAbstract:A general methodology to develop hyper-elastic membrane models applicable to crystalline films one-atom thick is presented. In this method, an extension of the Born rule based on the exponential Map is proposed. The exponential Map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a Tangent Map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential Map is locally approximated. The effectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed configurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by finite elements.Peer ReviewedPostprint (author’s final draft
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An atomistic-based finite deformation membrane for crystalline films one atom thick
2001Co-Authors: Marino Arroyo, Ted BelytschkoAbstract:A general methodology to develop hyper-elastic membrane models equivalent to crystalline films one atom thick is presented. This membrane theory is used to study the mechanics of carbon nanotubes. The method relies on an extension of the Born based on the exponential Map. The exponential Map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a Tangent Map like the standard Born rule is inadequate. In order to obtain practical models, the exponential Map is locally approximated. Numerical simulations of a membrane for carbon nanotubes discretized by finite elements are given.
Benjamin Kunsberg - One of the best experts on this subject based on the ideXlab platform.
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CVPR Workshops - The differential geometry of shape from shading: Biology reveals curvature structure
2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, 2012Co-Authors: Benjamin Kunsberg, Steven W. ZuckerAbstract:Shape from shading is a classical inverse problem in computer vision. We introduce a novel mathematical formulation for calculating local surface shape based on covariant derivatives, rather than the customary integral minimization or P.D.E approaches. Motivated by neurobiology, we introduce the shading flow field (the Tangent Map to the image isophotes) between the image and the surface levels. Just as in the perceptual organization of texture, we use the parallel transport of our shading flow field to move the isophote field at different points on the unknown surface to a single point, amassing restrictions on our surface curvatures. Under simplifying assumptions we solve exactly for the light source/surface pairs needed for a local image patch to have a given shading flow. The magnitude of the brightness gradient then restricts this family to a single light source and surface estimate pair, up to the concave/convex ambiguity and an additional elliptical/saddle ambiguity. Example calculations illustrate our approach.
