A Discrete Lagrangian Algorithm for Optimal Routing Problems - Kosmas, Odysseas; Vlachos, Dimitros S; Simos, Theodoros E. (AIP Conference Proceedings, 2007)
The ideas of discrete <named-content content-type="sem:AIPTh1.2" rid="kwd1.1" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">Lagrangian methods for conservative systems are exploited for the construction of algorithms applicable in optimal ship routing problems. The algorithm presented here is based on the discretisation of Hamilton’s principle of stationary action <named-content content-type="sem:AIPTh1.2" rid="kwd1.1" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">Lagrangian and specifically on the direct discretization of the Lagrange‐Hamilton principle for a conservative system. Since, in contrast to the <named-content content-type="sem:AIPTh1.2" rid="kwd1.2" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">differential equations, the discrete Euler‐Lagrange <named-content content-type="sem:AIPTh1.2" rid="kwd1.1 kwd1.2" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">equations serve as constrains for the optimization of a given cost functional, in the present work we utilize this feature in order to minimize the cost function for optimal ship routing.
UnivIS-Import:2015-04-16:Pub.2007.tech.FT.ltd.adiscr

Analysis of higher order phase fitted variational integrators - Kosmas, Odysseas; Leyendecker, Sigrid (Advances in Computational Mathematics, 2016)
In the present, the linear stability properties of the higher order phase fitted variational integrators are investigated. Towards this purpose, at first we calculate the eigenvalues of the amplification matrix for each method. Then, since the proposed integrators are derived specifically for the numerical integration of systems with oscillatory solutions, the linear stability analysis verifies their good behavior, when used for these problems. Finally, we test the proposed methods on several numerical examples, first with regard to their stability and secondly concerning the behavior in long term integration of highly oscillatory problems.
Keywords: Variational integrators Discrete variational mechanics Stability analysis Phase lag analysis

A new multistep Integrator based on Discrete Lagrangian Formulation - Kosmas, Odysseas; Vlachos, Dimitros S; Simos, Theodoros E. (AIP Conference Proceedings, 2008)
A multi‐step numerical method is formulated for the <named-content content-type="sem:AIPTh1.2" rid="kwd1.3" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">solution of discrete <named-content content-type="sem:AIPTh1.2" rid="kwd1.1" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">Lagrangian and Hamilton <named-content content-type="sem:AIPTh1.2" rid="kwd1.2" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">differential equations presented in previous works. As an application, the relative motion of the two body problem is treated with a multi‐step algorithm and its possibility to be used in other physics problems (e.g. Schrödinger <named-content content-type="sem:AIPTh1.2" rid="kwd1.1" style="color: rgb(0, 0, 0); font-family: Lora, serif; font-size: 20px;" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://www.niso.org/standards/z39-96/ns/oasis-exchange/table" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">equation) is discussed.
UnivIS-Import:2015-04-16:Pub.2008.tech.FT.ltd.anewmu

Charged particle in an electromagnetic field using variational integrators, International Conference of Numerical Analysis and Applied Mathematics - Kosmas, Odysseas (AIP Conference Proceedings, 2011)
In the present work we extend the discrete Lagrangian integrator method presented in Ref. [1] to derive appropriate numerical maps for the solution of mechanical problems in which the potential energy depends on the velocity of the system. As a representative concrete example and simulated experiment of the method presented here, we examine the motion of a charged particlemoving in an electromagnetic fiel
UnivIS-Import:2015-04-16:Pub.2011.tech.FT.ltd.charge

Deriving numerical techniques with zero phase-lag and derivatives for initial value problems of second order - Papadopoulos, Dim F; Kosmas, Odysseas; Simos, Theodoros E. (AIP Conference Proceedings, 2012)
In the present we investigate the advantages of the phase lag analysis for the derivation of phase-fitted techniques on several numerical schemes. Relying on the main characteristics of the phase lag we evaluate the parameters needed firstly for Runge-Kutta methods and secondly for high order variational integration methods, so that the phase lag and its derivatives are zero. The proposed methods are tested for the solution of initial value problems on ordinary differential equations of second order, like the Hénon-Heiles model. © 2012 American Institute of Physic
UnivIS-Import:2015-04-16:Pub.2012.tech.FT.ltd.derivi
Keywords: initial value problems; orbital problems; phase-lag analysis; Runge-Kutta methods; variational integrators

Discrete Algorithms for Optimization in Ship Routing Problems - Kosmas, Odysseas; Vlachos, Dimitros S; Simos, Theodoros E. (AIP Conference Proceedings, 2007)
Within the ideas of discretization of Lagrangian dynamics for conservative systems, we have constructed a discrete algorithm appropriate for optimal ship routing problems. This algorithm is an iterative optimization which produces optimal routes and optimal cost functions, by exploiting the advantages of the discrete Lagrangian methods. © 2007 American Institute of Physics.
UnivIS-Import:2015-04-16:Pub.2007.tech.FT.ltd.discre_3
Keywords: Discrete Lagrangian dynamics; Optimal ship routing; Variational integrators

Family of higher order exponential integrators for split potential systems - Kosmas, Odysseas; Leyendecker, Sigrid (International Conference on Mathematical Modeling in Physical Sciences, 2014)
In the present work, we derive a family of higher order exponential variational integrators for the numerical integration of systems containing slow and fast potential forces. To increase the order of variational integrators, first the discrete Lagrangian in a time interval is defined as a weighted sum of the evaluation of the continuous Lagrangian at intermediate time nodes while expressions for configurations and velocities are obtained using interpolating functions that can depend on free parameters. Secondly, in order to choose those parameters appropriately, exponential integration techniques are embedded. When the potential can be split into a fast and a slow component, we use dierent quadrature rules for the approximation of the dierent parts in the discrete action. Finally, we study the behavior of this family of integrators in numerical test

Local path fitting: a new approach to variational integrators - Kosmas, Odysseas; Vlachos, Dimitros S (Journal of Computational and Applied Mathematics, 2012)
A new approach for constructing variational integrators is presented. In the general case, the estimation of the action integral in a time interval [ tk,tk+ 1] is used to construct a symplectic map ( qk,qk+ 1)→(qk+ 1,qk+ 2). The basic idea, is that only the partial derivatives of the estimated action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, gives rise to a new integral that depends on the EulerLagrange vector itself (which in the continuous and exact case vanishes) and not on the Lagrangian. Since this new integral can only be computed through a numerical method based on some internal grid points, we can locally fit the exact curve by demanding the EulerLagrange vector to vanish at these grid points. Thus, the integral vanishes, and the process dramatically simplifies the calculation of high order approximations. The new technique is tested in high order solutions of the two-body problem with high eccentricity (up to 0.99) and of the outer planets of the solar system. © 2012 Elsevier B.V. All rights reserve
UnivIS-Import:2015-03-09:Pub.2012.tech.FT.ltd.localp
Keywords: Bernstein basis polynomials; Discrete variational mechanics; Symplectic integrators; Variational integrators

Obstacle Bypassing in Optimal Ship Routing Using Simulated Annealing - Kosmas, Odysseas; Vlachos, Dimitros S; Simos, Theodoros E. (AIP Conference Proceedings, 2007)
In this paper we are going to discuss a variation on the problem of finding the shortest path between two points in optimal ship routing problems consisting of obstacles that are not allowed to be crossed by the path. Our main goal are going to be the construction of an appropriate algorithm, based in an earlier work [16] by computing the shortest path between two points in the plane that avoids a set of polygonal obstacles.
UnivIS-Import:2015-04-16:Pub.2007.tech.FT.ltd.obstac

On frequency estimations in phase fitted variational integrators - Kosmas, Odysseas; Leyendecker, Sigrid (Proc. Appl. Math. Mech (PAMM), 2013)

Optimized Derivative Kernels for Gamma Ray Spectroscopy - Kosmas, Odysseas; Vlachos, Dimitros S; Simos, Theodoros E. (AIP Conference Proceedings, 2007)
In gamma ray spectroscopy, the photon detectors measure the number of photons with energy that lies in an interval which is called a channel. This accumulation of counts produce a measuring function that its deviation from the ideal one may produce high noise in the unfolded spectrum. In order to deal with this problem, the ideal accumulation function is interpolated with the use of special designed derivative kernels. Simulation results are presented which show that this approach is very effective even in spectra with low statistics. © 2007 American Institute of Physics.
UnivIS-Import:2015-04-16:Pub.2007.tech.FT.ltd.optimi
Keywords: Derivative kernels; Gamma spectroscopy

Phase-fitted discrete Lagrangian integrators - Kosmas, Odysseas; Vlachos, Dimitros S (Computer Physics Communications, 2010)
Phase fitting has been extensively used during the last years to improve the behavior of numerical integrators on oscillatory problems. In this work, the benefits of the phase fitting technique are embedded in discrete Lagrangian integrators. The results show improved accuracy and total energy behavior in Hamiltonian systems. Numerical tests on the long term integration (105 periods) of the 2-body problem with eccentricity even up to 0.95 show the efficiency of the proposed approach. Finally, based on a geometrical evaluation of the frequency of the problem, a new technique for adaptive error control is presented. © 2009 Elsevier B.V. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2010.tech.FT.ltd.phasef
Keywords: Discrete Lagrangian integrators; Exponential fitting; Phase fitting

Phase fitted variational integrators using interpolation techniques on non regular grids - Kosmas, Odysseas; Leyendecker, Sigrid (AIP Conference Proceedings, 2012)
The possibility of deriving a high order variational integrator that utilizes intermediate nodes within one time interval time to approximate the action integral is investigated. To this purpose, we consider time nodes chosen through linear or exponential expressions and through the roots of Chebyshev polynomial of the first kind in order to approximate the configurations and velocities at those nodes. Then, by defining the Lagrange function as a weighted sum over the discrete Lagrangians corresponding to the curve segments, we apply the phase fitted technique to obtain an exponentially fitted numerical scheme. The resulting integrators are tested for the numerical simulation of the planar two body problem with high eccentricity and of the three-body orbital motion within a solar syste
UnivIS-Import:2015-04-16:Pub.2012.tech.FT.ltd.phasef
Keywords: Chebyshev nodes; phase lag; phase-fitted methods; variational integrators

Phase lag analysis of variational integrators using interpolation techniques - Kosmas, Odysseas; Leyendecker, Sigrid (Proc. Appl. Math. Mech. (PAMM), 2012)

Simulated annealing for optimal ship routing - Kosmas, Odysseas; Vlachos, Dimitros S (Computers & Operations Research, 2012)
In this article we present a simulated annealing based algorithm for the determination of optimal ship routes through the minimization of a cost function defined as a weighted sum of the time of voyage and the voyage comfort (safety is taken into account too). This cost function is dependent on the wind speed and its direction as well as on the wave height and its direction. The constructed algorithm at the beginning discretizes an initial route and then optimizes it by considering small deviations, which are accepted or rejected by utilizing the simulated annealing technique. Using calculus of variations, we prove a key theorem which tremendously accelerates the convergence of the proposed algorithm. For an illustration of the advantages of the constructed method, both computational and real experiments have been carried out which are presented and discussed. © 2011 Elsevier Ltd. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2012.tech.FT.ltd.simula
Keywords: Calculus of variations; Discrete mechanics; Optimal ship routing; Simulated annealing

Stability analysis of high order phase fitted variational integrators - Kosmas, Odysseas; Leyendecker, Sigrid (Proceedings of WCCM XI - ECCM V - ECFD VI, 2014)
The linear stability properties of higher order variational integration methods that use phase lag technique are investigated. Towards this purpose, at first we calculate the eigenvalues of the amplification matrix for each method. Phase fitted integrators are derived specifically for the numerical integration of systems with oscillatory solutions. A linear stability analysis shows their good behavior, when simulating these problems. Finally, we test the proposed methods in numerical examples, first with regard to their stability and secondly the behavior in long term integration is illustrate
UnivIS-Import:2015-04-16:Pub.2014.tech.FT.ltd.stabil_1
Keywords: Discrete variational mechanics; Phase lag analysis; Variational integrators

Variational integrators for orbital problems using frequency estimation - Kosmas, Odysseas; Leyendecker, Sigrid (Advances in Computational Mathematics, 2018)
In this work, we present a new derivation of higher order variational integration methods that exploit the phase lag properties for numerical integrations of systems with oscillatory solutions. More specifically, for the derivation of these integrators, the action integral along any curve segment is defined using a discrete Lagrangian that depends on the endpoints of the segment and on a number of intermediate points of interpolation. High order integrators are then obtained by writing down the discrete Lagrangian at any time interval as a weighted sum of the Lagrangians corresponding to a set of the chosen intermediate points. The respective positions and velocities are interpolated using trigonometric functions. The methods derived this way depend on a frequency, which in general needs to be accurately estimated. The new methods, which improve the phase lag characteristics by re-estimating the frequency at every time step, are presented and tested on the general N-body problem as numerical example
Keywords: Variational integrators Discrete variational mechanics Highly oscillatory problems Phase lag analysis General N-body problem