Stochastic Reduced Order Model Research Articles

Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems

AbstractData-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

Effective conductivity by stochastic reduced order models (SROMs)

A method is developed for calculating statistics of effective conductivity Σ eff for microstructures with conductivity Σ varying randomly in space. The method is based on the representation of Σ by a stochastic reduced order model (SROM) Σ ∼ , that is, a random field with a finite and small number m of samples that, generally, are not equally likely. An optimization algorithm with objective function quantifying the discrepancy between the probability laws of Σ ∼ and Σ is used to construct Σ ∼ . Samples of Σ eff corresponding to those of Σ ∼ and their probabilities define a SROM Σ ∼ eff for Σ eff. Bounds are develop to quantify the discrepancy between Σ and Σ ∼ and between Σ eff and Σ ∼ eff . The method is applied to find statistics of effective conductivity for a two-dimensional specimen with conductivity described by a non-Gaussian homogeneous field. Statistics of effective conductivity obtained from SROMs Σ ∼ eff with m samples are accurate and stable, in contrast to Monte Carlo estimates based on the same number of samples that are unstable and can have large errors.

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations

A computational methodology is developed to address the solution of high-dimensional stochastic problems. It utilizes high-dimensional model representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. HDMR is efficient at capturing the high-dimensional input–output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms using only the important dimensions. The newly developed adaptive sparse grid collocation (ASGC) method is incorporated into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output. Several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.

Linear random vibration by stochastic reduced‐order models

AbstractA practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non‐Gaussian noise and systems with uncertain properties subjected to Gaussian and non‐Gaussian noise. These classes of problems are relevant as most systems have uncertain properties, physical noise is rarely Gaussian, and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non‐Gaussian.The method (1) is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced‐order models (SROMs), that is, simple random elements having a finite number of outcomes of unequal probabilities, (2) can be used to calculate statistics of a system state beyond its first two moments, and (3) establishes bounds on the discrepancy between exact and SROM‐based solutions of linear random vibration problems. The implementation of the method has required to integrate existing and new numerical algorithms. Examples are presented to illustrate the application of the proposed method and assess its accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

Stochastic reduced order models for uncertain geometrically nonlinear dynamical systems

A general methodology is presented for the consideration of both parameter and model uncertainty in the determination of the response of geometrically nonlinear structural dynamic systems. The approach is rooted in the availability of reduced order models of these nonlinear systems with a deterministic basis extracted from a reference model (the mean model). Uncertainty, both from parameters and model, is introduced by randomizing the coefficients of the reduced order model in a manner that guarantees the physical appropriateness of every realization of the reduced order model, i.e. while maintaining the fundamental properties of symmetry and positive definiteness of every such reduced order model. This randomization is achieved not by postulating a specific joint statistical distribution of the reduced order model coefficients but rather by deriving this distribution through the principle of maximization of the entropy constrained to satisfy the necessary symmetry and positive definiteness properties. Several desirable features of this approach are that the uncertainty can be characterized by a single measure of dispersion, affects all coefficients of the reduced order model, and is computationally easily achieved. The reduced order modeling strategy and this stochastic modeling of its coefficients are presented in details and several applications to a beam undergoing large displacement are presented. These applications demonstrate the appropriateness and computational efficiency of the method to the broad class of uncertain geometrically nonlinear dynamic systems.

Direction of arrival estimation using stochastic model order reduction via state space modeling

In this paper, we present a class of high resolution direction finding techniques for processing of passive, uniform linear arrays. These are based on the state space parameterization of a linear array data. Directions of arrival (DOAs) are obtained as the eigenvalues of the state transition matrix φ. Since there exist an infinite number of state variable characterizations for a model, a number of possible methods can be devised depending on the manner in which the state transition matrix is identified. These techniques are also applicable to the situation of source coherence in array processing problems, which have required special attention in the past. Simulation results ascertaining the performance of these methods are also presented.