Numerical instabilities and convergence control for convex approximation methods

Yang, Dixiong; Yang, Ping

doi:10.1007/s11071-010-9674-x

Cited by 21 publications

(8 citation statements)

References 28 publications

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“…Failure of convergence that is due to the existence of a range of diverging initial conditions, however, will persist, even with the decay. The results are similar for a broad parameter range (e.g., decay rate v ∈ [0.65, 0.99] and decay kick-in time M ∈ [10,100]). For a decay rate too close to 100% or a too large decay kick-in time, it will take many iterations to reach the convergent interval.…”

Section: Numerical Resultsmentioning

confidence: 59%

“…This feedback control is noninvasive, i.e., the control strength vanishes upon convergence, and is extremely easy to implement. It is a special case of a recent effort to stabilize all periodic points of a discrete time dynamical system [8,9] which is also closely related to nonlinear successive over-relaxation methods [10,11]. It has been extensively studied [12][13][14][15][16] and extended [17,18] with respect to its original purpose as a tool for examining the structure of chaotic attractors.…”

Section: Introductionmentioning

confidence: 99%

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Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed

Bick¹,

Timme²,

Kolodziejski³

2012

SIAM J. Appl. Dyn. Syst.

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Stabilizing unstable periodic orbits in a chaotic invariant set not only reveals information about its structure but also leads to various interesting applications. For the successful application of a chaos control scheme, convergence speed is of crucial importance. Here we present a predictive feedback chaos control method that adapts a control parameter online to yield optimal asymptotic convergence speed. We study the adaptive control map both analytically and numerically and prove that it converges at least linearly to a value determined by the spectral radius of the control map at the periodic orbit to be stabilized. The method is easy to implement algorithmically and may find applications for adaptive online control of biological and engineering systems

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Section: Numerical Resultsmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed

Bick¹,

Timme²,

Kolodziejski³

2012

SIAM J. Appl. Dyn. Syst.

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“…Therefore, inspired by the idea of stress-constraint corrections, 8 the SSCS is developed to achieve the better solution quality for stress-based TO under varying temperature field. By virtue of the stability transformation method (STM) that is used to realize chaos control of nonlinear dynamical system, 28 the SSCS can overcome iterative oscillation for the high nonlinearity of stress constraint.…”

Section: Stress Stabilizing Control Schemementioning

confidence: 99%

“…In fact, the STM contains twofold meanings about mathematical programming and dynamical system and it can effectively alleviate iteration oscillation in optimization procedure. 28 Hence, the SSCS based on STM is also able to control numerical oscillation for structural TO with stress and temperature constraints.…”

Section: Stress Stabilizing Control Schemementioning

confidence: 99%

Thermo‐elastic topology optimization with stress and temperature constraints

Meng

Wang

et al. 2021

Numerical Meth Engineering

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A thermo-elastic topology optimization with stress and temperature constraints is proposed to attack the complex multiphysics problem in this paper. Based on the rational approximation of material properties (RAMP), the coupled equations of mechanic and temperature field are solved. Two optimization problems, volume minimization with temperature and stress constraints, and traditional compliance minimization with volume and temperature constraints, are discussed for comparison. The stress stabilizing control scheme (SSCS) combined with global stress measure is presented to tackle highly nonlinear and local nature of stress with thermal expansion in varying temperature field. The adjoint method is applied to achieve the sensitivity of multiphysics field and the density function is updated utilizing the method of moving asymptotes (MMA).Representative examples are investigated to demonstrate the effectiveness and utility of the proposed method. Clear topology and stable iterative process can be obtained for complex coupled problem by means of the SSCS. Meanwhile, the topology with stress and temperature constraints has obvious sensitivity to even subtle change in temperature. The optimization design considering several stress constraints under multithermal conditions can work well in different temperature ranges.

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“…Floquet systems, time-periodically driven systems, have attracted great deal of attention [8,9] to explore non-trivial topological phases induced by periodically driving external fields [10][11][12][13][14][15][16][17][18][19][20], analyze the stability of systems itself or limit cycles [21][22][23][24][25][26][27][28][29][30][31], to name a few. Floquet systems with high tunability have been realized in various experimental setups.…”

Section: Introductionmentioning

confidence: 99%

Stability of topologically protected edge states in nonlinear quantum walks: additional bifurcations unique to Floquet systems

Mochizuki

Kawakami

Obuse

2020

J. Phys. A: Math. Theor.

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Quantum walk, a kind of systems with time-periodic driving (Floquet systems), is defined by a time-evolution operator, and can possess non-trivial topological phases. Recently, the stability of topologically protected edge states in a nonlinear quantum walk has been studied, in terms of an effective time-indepedent non-Hermitian Hamiltonian, by applying a continuum limit to the nonlinear quantum walk. In this paper, we study the stability of the edge states by treating a nonunitary time-evolution operator, which is derived from the time-evolution operator of nonlinear quantum walks without the continuum limit. As a result, we find additional bifurcations at which edge states change from stable attractors to unstable repellers with increasing the strength of nonlinearity. The additional bifurcation we shall show is unique to Floquet nonlinear systems, since the origin of the bifurcations is that a stable region for eigenvalues of nonunitary time-evolution operators is bounded, while that of effective non-Hermitian Hamiltonians is unbouneded.

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Numerical instabilities and convergence control for convex approximation methods

Cited by 21 publications

References 28 publications

Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed

Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed

Thermo‐elastic topology optimization with stress and temperature constraints

Stability of topologically protected edge states in nonlinear quantum walks: additional bifurcations unique to Floquet systems

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