The partition of unity finite element approach with hp-refinement for the stationary Fokker–Planck equation

Kumar, Mrinal; Chakravorty, Suman; Singla, Puneet; Junkins, John L.

doi:10.1016/j.jsv.2009.05.033

Cited by 46 publications

(34 citation statements)

References 23 publications

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“…Work in formation flying by How et al [26], Alfriend and Carpenter [27] 5 and Carpenter and Schiesser [28] make use of the transformation of variables formula to characterize the navigation error in spacecraft formation flying. One of the chief problems in astrodynamics in general and navigation in particular is the choice of co-ordinates for batch and sequential orbit determination methods.…”

Section: Applications Of the Transformation Theoremmentioning

confidence: 99%

“…The measurement equations of range, azimuth, and elevation can therefore be written as 5 This was pointed out to the first author by Dr. Alfriend himself! functions of the state variables in each of the coordinate system choices under consideration.…”

Section: Sequential Orbit Determination Problemsmentioning

confidence: 99%

“…This metric is used to investigate the nonlinearity of orbital and attitude dynamics models. Recent work [3,4,5] in the area of uncertainty propagation in dynamical systems made significant progress in understanding the evolution of the probability density function governing the state uncertainty. Progress has also been made to incorporate the propagated state and measurement uncertainty in nonlinear estimators that make use of analytic, semi-analytic and numerical methods (cf.…”

Section: Introductionmentioning

confidence: 99%

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Measurement Model Nonlinearity in Estimation of Dynamical Systems

2012

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The role of nonlinearity of the measurement model and its interactions with the uncertainty of measurements and geometry of the problem is studied in this paper. An examination of the transformations of the probability density function in various coordinate systems is presented for several astrodynamics applications. Smooth and analytic nonlinear functions are considered for the studies on the exact transformation of uncertainty. Special emphasis is given to understanding the role of change of variables in the calculus of random variables. The transformation of probability density functions through mappings is shown to provide insight in to understanding the evolution of uncertainty in nonlinear systems. Examples are presented to highlight salient aspects of the discussion. A sequential orbit determination problem is analyzed, where the transformation formula provides useful insights for making the choice of coordinates for estimation of dynamic systems.

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Section: Applications Of the Transformation Theoremmentioning

confidence: 99%

Section: Sequential Orbit Determination Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Measurement Model Nonlinearity in Estimation of Dynamical Systems

2012

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“…Consequently, the n-dimensional PDF of the stochastic closed-curve attractor can be expressed by production of a [11] gives the most detailed description of PDF evolution, but it is difficult to get close-form solutions of FPE for systems without detailed balance except for some special systems [12]. Thus, until now, many approximate methods have been raised to solve FPE under various specific circumstances, such as the stochastic average method [13], path-integral method [14], finite element method [15], Galerkin-Ulam-like method [16], cell-mapping method [8], etc. Most of these methods are proposed to solve the response of systems with weak nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

Approximation of Stochastic Quasi-Periodic Responses of Limit Cycles in Non-Equilibrium Systems under Periodic Excitations and Weak Fluctuations

Guo

2017

Entropy

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Abstract:A semi-analytical method is proposed to calculate stochastic quasi-periodic responses of limit cycles in non-equilibrium dynamical systems excited by periodic forces and weak random fluctuations, approximately. First, a kind of 1/N-stroboscopic map is introduced to discretize the quasi-periodic torus into closed curves, which are then approximated by periodic points. Using a stochastic sensitivity function of discrete time systems, the transverse dispersion of these circles can be quantified. Furthermore, combined with the longitudinal distribution of the circles, the probability density function of these closed curves in stroboscopic sections can be determined. The validity of this approach is shown through a van der Pol oscillator and Brusselator.

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“…FPEs of dimension up to d = 6 are solved approximately on finite rectangular domains. [8] uses a partition-of-unity finite element approach to solve problems of dimension four, applying higherorder polynomials to locally achieve higher resolutions.…”

Section: Introductionmentioning

confidence: 99%

Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems

2011

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Technical systems are subjected to a variety of excitations that cannot generally be described in deterministic ways. External disturbances like wind gusts or road roughness as well as uncertainties in system parameters can be described by random variables, with statistical parameters identified through measurements, for instance.For general systems the statistical characteristics such as the probability density function (pdf) may be difficult to calculate. In addition to numerical simulation methods (Monte Carlo Simulations, MCS) there are differential equations for the pdf that can be solved to obtain such characteristics, most prominently the Fokker-Planck equation (FPE).A variety of different approaches for solving FPEs for nonlinear systems have been investigated in the last decades. Most of these are limited to considerably low dimensions to avoid high numerical costs due to the "curse of dimension". Problems of higher dimension, such as d = 6, have been solved only rarely.In this paper we present results for stationary pdfs of nonlinear mechanical systems with dimensions up W. Martens ( ) · U. to d = 10 using a Galerkin method, which expands approximative solutions (weighting functions) into orthogonal polynomials.

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The partition of unity finite element approach with hp-refinement for the stationary Fokker–Planck equation

Cited by 46 publications

References 23 publications

Measurement Model Nonlinearity in Estimation of Dynamical Systems

Measurement Model Nonlinearity in Estimation of Dynamical Systems

Approximation of Stochastic Quasi-Periodic Responses of Limit Cycles in Non-Equilibrium Systems under Periodic Excitations and Weak Fluctuations

Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems

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