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http://cmcampos.xyz/publications
My Feed DescriptionSat, 29 Apr 2017 00:00:00 +0100Palindromic 3-stage splitting integrators, a roadmap
http://cmcampos.xyz/publications/s3map
http://cmcampos.xyz/publications/s3mapSat, 29 Apr 2017 00:00:00 +0100
The implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling.
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paperHigh order variational integrators in the optimal control of mechanical systems
http://cmcampos.xyz/publications/hovi-ocms
http://cmcampos.xyz/publications/hovi-ocmsWed, 01 Feb 2017 00:00:00 +0000
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.
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paperHamilton-Jacobi theory in Cauchy data space
http://cmcampos.xyz/publications/cft-hj
http://cmcampos.xyz/publications/cft-hjFri, 14 Nov 2014 00:00:00 +0000
Recently, M. de León el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.
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paperExtra Chance Generalized Hybrid Monte Carlo
http://cmcampos.xyz/publications/xhmc
http://cmcampos.xyz/publications/xhmcWed, 30 Jul 2014 00:00:00 +0100
We study a method, Extra Chance Generalized Hybrid Monte Carlo, to avoid rejections in the Hybrid Monte Carlo method and related algorithms. In the spirit of delayed rejection, whenever a rejection would occur, extra work is done to find a fresh proposal that, hopefully, may be accepted. We present experiments that clearly indicate that the additional work per sample carried out in the extra chance approach clearly pays in terms of the quality of the samples generated.
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paperHigher Order Variational Integrators: a polynomial approach
http://cmcampos.xyz/publications/hovi-poly
http://cmcampos.xyz/publications/hovi-polyTue, 23 Jul 2013 00:00:00 +0100
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the structural properties of these systems, like the symplectic form, the evolution of the momentum maps or the energy behaviour. Also they are easily applicable to optimal control problems based on mechanical systems as proposed in Ober-Blöbaum et al. [2011]. Following the same approach, we develop a family of variational integrators to which we refer as symplectic Galerkin schemes in contrast to symplectic partitioned Runge-Kutta. These two families of integrators are, in principle and by construction, different one from the other. Furthermore, the symplectic Galerkin family can as easily be applied in optimal control problems, for which Campos et al. [2012b] is a particular case.
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paperHigher order variational time discretization of optimal control problems
http://cmcampos.xyz/publications/hovtd-ocp
http://cmcampos.xyz/publications/hovtd-ocpFri, 27 Apr 2012 00:00:00 +0100
We reconsider the variational integration of optimal control problems for mechanical systems based on a direct discretization of the Lagrange-d'Alembert principle. This approach yields discrete dynamical constraints which by construction preserve important structural properties of the system, like the evolution of the momentum maps or the energy behavior. Here, we employ higher order quadrature rules based on polynomial collocation. The resulting variational time discretization decreases the overall computational effort.
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paperClassical field theories of first order and lagrangian submanifolds of premultisymplectic manifolds
http://cmcampos.xyz/publications/cft-t3
http://cmcampos.xyz/publications/cft-t3Fri, 21 Oct 2011 00:00:00 +0100
A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl equations are the local equations defining Lagrangian submanifolds of a premultisymplectic manifold.
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paperConstrained Variational Calculus for Higher Order Classical Field Theories
http://cmcampos.xyz/publications/cft-ho-cons
http://cmcampos.xyz/publications/cft-ho-consWed, 12 May 2010 00:00:00 +0100
We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of application are studied, in particular, applications to the geometrical description of optimal control theory for partial differential equations.
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paperUnambiguous Formalism for Higher-Order Lagrangian Field Theories
http://cmcampos.xyz/publications/cft-ho-ulf
http://cmcampos.xyz/publications/cft-ho-ulfMon, 16 Nov 2009 00:00:00 +0000
The aim of this paper is to propose an unambiguous intrinsic formalism for higher-order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, which implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geometric setting for the dynamics of a higher-order field theory, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both, the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher-order jet bundle and the canonical multisymplectic form on its dual. As both of these objects are uniquely defined, the Skinner-Rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions. The result is that we obtain a unique and global intrinsic version of the Euler-Lagrange equations for higher-order field theories. Several examples illustrate our construction.
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paperFunctionally Graded Media
http://cmcampos.xyz/publications/fgm
http://cmcampos.xyz/publications/fgmFri, 16 Nov 2007 00:00:00 +0000
The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case Functionally Graded Media, which allows us to obtain some homogeneity conditions.
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