A collocation formulation of multistep methods for variable step-size extensions

Arévalo, Carmen Pagés; Führer, Claus; Selva, M.

doi:10.1016/s0168-9274(01)00138-6

Cited by 12 publications

(6 citation statements)

References 4 publications

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“…, which violates (6). In contrast, using Gauss-Legendre quadrature of order 5 yields satisfaction of (6) with q = 5.…”

Section: G Numerical Quadraturementioning

confidence: 96%

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Dynamic Optimization with Convergence Guarantees

Neuenhofen,

Kerrigan

2018

Preprint

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We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. In order to provide numerical convergence guarantees, it is sufficient for the functions that define the problem to satisfy boundedness and Lipschitz conditions. Our assumptions are the most general to date; we do not require uniqueness, differentiability or constraint qualifications to hold and we avoid the use of Lagrange multipliers. Our approach differs fundamentally from state-of-theart methods based on collocation. We follow a leastsquares approach to finding approximate solutions to the differential equations. The objective is augmented with the integral of a quadratic penalty on the differential equation residual and a logarithmic barrier for the inequality constraints, as well as a quadratic penalty on the point constraint residual. The resulting unconstrained infinite-dimensional optimization problem is discretized using finite elements, while integrals are replaced by quadrature approximations if they cannot be evaluated analytically. Order of convergence results are derived, even if components of solutions are discontinuous.

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“…, which violates (6). In contrast, using Gauss-Legendre quadrature of order 5 yields satisfaction of (6) with q = 5.…”

Section: G Numerical Quadraturementioning

confidence: 96%

“…variants of Gauss schemes, can be interpreted as collocation methods. Collocation methods include certain classes of implicit Runge-Kutta methods, pseudospectral methods, as well as Adams and backward differentiation formula (BDF) methods [1], [3], [5], [6].…”

Section: Introduction a An Important Class Of Dynamic Optimization Pr...mentioning

confidence: 99%

Dynamic Optimization with Convergence Guarantees

Neuenhofen,

Kerrigan

2018

Preprint

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“…The former argument never holds except possibly for ODEs with only a very few equations; the latter is not very significant either, as a small stepsize variation can easily be accommodated in the Newton iterative process, see [6] or the actual strategy used in Dassl [2], which is based on an over/under-relaxation, compensating for the changed stepsize. The only more significant argument for a cut-out seems to be that there are some multistep method implementations whose coefficients have singularities for certain small stepsize ratios, [1,9]. This is however not the case for BDFs.…”

Section: Common Stepsize Strategies and Their Effectsmentioning

confidence: 99%

Adaptive time-stepping and computational stability

Söderlind

Wang

2006

Journal of Computational and Applied Mathematics

130

105

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We investigate the effects of adaptive time-stepping and other algorithmic strategies on the computational stability of ODE codes. We show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability. A series of computational experiments with the standard implementation of Dassl and a modified version, including stepsize control based on digital filters, is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense. The inherent performance and stability of Dassl are therefore much greater than the standard implementation seems to suggest.The objective of this paper is to investigate the effect of adaptive time-stepping and other algorithmic strategies on what we refer to as the computational stability of an ODE solver. All stability notions are concerned with a continuous data dependence. As for computational stability, we ask that the complete algorithm, as well as its implementation (including all internal decision making of the code), is a "continuous" map from data to computed results, provided that the given ODE being solved is smooth enough. In other words, a small change of parameters should only have a small effect on the computed output. Alternatively, this could be expressed by saying that the computational process is well-conditioned.Ideally, this should hold for any parameter, regardless of whether it is a parameter of the ODE itself, a parameter in the algorithmic specification of the code, or a parameter in the calling sequence of the software. As it concerns software, the notion of computational stability goes beyond the classical well-conditioning requirement of an algorithm.

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“…Other authors are, [1,2,3,4,6,9,10,12,13,14,16,19,20,21,22]. In fact, the numerical solution of (1) by ( 2) through collocation and interpolation methods have been well studied in the literature, see for example [23], [25], [5], [7,8], [26], [16], [24], and [17,18]. The interval of absolute stability of the CHLMM is investigated using the root locus method discussed in [20,21] and [3], whose application can be found in [16] and [24] instead of the equivalent boundary locus plot in [7] and [9].…”

Section: Introductionmentioning

confidence: 99%

A class of continuous hybrid linear multistep methods for stiff IVPs in ODEs

Okuonghae¹

2012

Annals of the Alexandru Ioan Cuza University - Mathematics

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In this paper, we present a class of continuous hybrid linear multistep methods (CHLMM) for stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The constuction of these methods are based on the approach of collocation and interpolation. The interval of absolute stability of the method is investigated, using the root locus method. Numerical results of the methods solving stiff IVPs in ODEs are compared with that from the state-of-the-art Ode15s Matlab ODEs code.

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A collocation formulation of multistep methods for variable step-size extensions

Cited by 12 publications

References 4 publications

Dynamic Optimization with Convergence Guarantees

Dynamic Optimization with Convergence Guarantees

Adaptive time-stepping and computational stability

A class of continuous hybrid linear multistep methods for stiff IVPs in ODEs

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