Matrix formulation of the Picard method for parallel computation

Feagin, T.; Nacozy, P. E.

doi:10.1007/bf01232802

Cited by 26 publications

(5 citation statements)

References 6 publications

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“…Feagin published his PhD dissertation [9] in 1972 on Picard iteration using Chebyshev approximation. He established the first vector-matrix version of Picard iteration utilising orthogonal basis functions [10]. In 1980 Shaver wrote a related dissertation giving insights on parallel computation using Picard Iteration and Chebyshev approximation [11].…”

Section: Modified Chebyshev-picard Iterationmentioning

confidence: 99%

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

et al. 2017

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We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified ChebyshevPicard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions, however we show that using modified Chebyshev-Picard iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert's problem using the method of particular solutions with modified Chebyshev-Picard iteration is about an order of magnitude faster compared with the classical shooting method and a tenthtwelfth order Runge-Kutta integrator. It is well known that the solution to Lambert's problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.

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Section: Modified Chebyshev-picard Iterationmentioning

confidence: 99%

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

et al. 2017

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“…This approach is sometimes referred to as a collocation method, and has been used in astrodynamics by several authors. 11,[16][17][18] We emphasize that it is mathematically equivalent to an s-stage collocation-based IRK method having the same collocation points and basis functions, 14,15 since this fact does not appear to be well known. Since collocation methods are a subset of IRK methods, we prefer to describe the new propagator in the broader context of IRK.…”

Section: B Construction Of Collocation-based Runge-kutta Methodsmentioning

confidence: 99%

Implicit Runge-Kutta Methods for Orbit Propagation

Aristoff

Poore

2012

AIAA/AAS Astrodynamics Specialist Conference

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Accurate and efficient orbital propagators are critical for space situational awareness because they drive uncertainty propagation which is necessary for tracking, conjunction analysis, and maneuver detection. We have developed an adaptive, implicit Runge-Kuttabased method for orbit propagation that is superior to existing explicit methods, even before the algorithm is potentially parallelized. Specifically, we demonstrate a significant reduction in the computational cost of propagating objects in low-Earth orbit, geosynchronous orbit, and highly elliptic orbit. The new propagator is applicable to all regimes of space, and additional features include its ability to estimate and control the truncation error, exploit analytic and semi-analytic methods, and provide accurate ephemeris data via built-in interpolation. Finally, we point out the relationship between collocation-based implicit Runge-Kutta and Modified Chebyshev-Picard Iteration.

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“…Since Chebyshev function approximation is orthogonal, Clenshaw and Norton found it beneficial to combine it with Picard iteration in a simultaneous manner to provide a solution to non-linear ordinary differential equations [8]. The MCPI algorithm is quite prone to parallel processing, and several early studies proposed different approaches [3,9,10,11] to the method. Additionally, [3] demonstrates the efficiency of MCPI in non-linear IVP and BVP when compared to other solvers, as will be discussed in the next section.…”

Section: Introductionmentioning

confidence: 99%

Lambert probleminin modifiye Chebyshev-Picard yineleme yöntemini kullanarak paralel çözümü

AJROUDİ

Torun

2022

NÖHÜ Müh. Bilim. Derg.

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Lambert problemi, yörünge belirlemede çoklu devir problemini çözmek için kullanılan klasik yöntemlerden biridir. Uzay araştırma programlarına ve uydu ağlarının kullanımına olan ilginin artmasıyla, ağ kontrol merkezine ağdaki her bir uydunun yörüngesine ilişkin bilgileri sağlayacak ve uyduların yönlendirme kararlarını iyileştirmesine yardımcı olacak doğru ve hızlı bir yöntemin sağlanması önemlidir. Lambert problemi, bu problemi yinelemeli olarak çözen yöntemlerden biridir ve bu yineleme önceki yıllarda Newton'un yineleme yöntemi kullanılarak yapılmaktaydı. Daha güncel araştırmalarda bu problemi çözmek için Chebyshev-Picard yineleme yöntemi kullanılması önerilmektedir. Önerilen metot çözüm süresinde iyileştirmeler sunmasına rağmen büyük problemlerde çözüm çok uzun süreler alabilmektedir. Bu çalışmada, Lambert problemini paralel programlama teknikleri kullanarak daha hızlı çözen yeni bir paralel algoritma önerilmiştir. Ayrıca algoritmanın paralel ölçeklenebilirliğini göstermek için 2 farklı paralel sistemde; paylaşımlı ve dağıtık bellek mimarilerinde deneyler yapılmıştır. Deneysel sonuçlar, paralel algoritmanın dağıtık bellek ve paylaşımlı bellek mimarilerinde sırasıyla 8.26 ve 3.94 kat daha hızlı çözüm süresine ulaştığını göstermektedir.

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Matrix formulation of the Picard method for parallel computation

Cited by 26 publications

References 6 publications

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

Implicit Runge-Kutta Methods for Orbit Propagation

Lambert probleminin modifiye Chebyshev-Picard yineleme yöntemini kullanarak paralel çözümü

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