An efficient computational cost reduction strategy for the population-based intelligent optimization of nonlinear dynamical systems

Abstract This paper addresses the issue of offline and online computational cost reduction of the proper orthogonal decomposition (POD) which is a popular nonlinear model order reduction (MOR) technique. Online computational cost is reduced by using the discrete empirical interpolation method (DEIM), which reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD: this is the POD-DEIM approach. Offline computational cost is reduced by generating an approximate snapshot-ensemble of the nonlinear dynamical system, consequently, completely avoiding the need to simulate the full-order system. Two snapshot ensembles: one of the states and the other of the nonlinear function are obtained by simulating the successive linearization of the original nonlinear system. The proposed technique is applied to two benchmark large-scale nonlinear dynamical systems and clearly demonstrates comprehensive savings in computational cost and time with insignificant or no deterioration in performance.

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Computational Schemes for Optimizing Domains of Attraction in Dynamical Systems

Volume 1: 23rd Biennial Conference on Mechanical Vibration and Noise, Parts A and B ◽

10.1115/detc2011-48158 ◽

2011 ◽

Author(s):

A´ngela Castillo ◽

Pedro J. Zufiria

Keyword(s):

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Low Cost ◽

Computational Cost ◽

Approximation Procedure ◽

Domains Of Attraction ◽

Nonlinear Dynamical ◽

Computational Schemes ◽

Stochastic Approximation Procedure ◽

High Computational Cost

In this paper, several computational schemes are presented for the optimal tuning of the global behavior of nonlinear dynamical systems. Specifically, the maximization of the size of domains of attraction associated with invariants in parametrized dynamical systems is addressed. Cell Mapping (CM) techniques are used to estimate the size of the domains, and such size is then maximized via different optimization tools. First, a genetic algorithm is tested whose performance shows to be good for determining global maxima at the expense of high computational cost. Secondly, an iterative scheme based on a Stochastic Approximation procedure (the Kiefer-Wolfowitz algorithm) is evaluated showing acceptable performance at low cost. Finally, several schemes combining neural network based estimations and optimization procedures are addressed with promising results. The performance of the methods is illustrated with two applications: first on the well-known van der Pol equation with standard parametrization, and second the tuning of a controller for saturated systems.

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Sequential sampling strategy for extreme event statistics in nonlinear dynamical systems

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1813263115 ◽

2018 ◽

Vol 115 (44) ◽

pp. 11138-11143 ◽

Cited By ~ 20

Author(s):

Mustafa A. Mohamad ◽

Themistoklis P. Sapsis

Keyword(s):

Dynamical Systems ◽

Parameter Space ◽

Extreme Event ◽

Nonlinear Dynamical Systems ◽

Computational Cost ◽

Optimization Procedure ◽

Irregular Waves ◽

Dimensional System ◽

Nonlinear Dynamical ◽

Quantity Of Interest

We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems from a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the “next-best” data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach uses Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds using the posterior distribution of the inferred map. The next-best design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process uses only information from the inferred map, it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size and for problems where each sample is very expensive to obtain. We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to 3D irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using a limited number of samples.

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