Statistical analysis via generalized decoupled polynomial chaos

A multiresolution approach is presented to study the uncertainty propagation problem in dynamic systems. The proposed approach is an extension of the generalized polynomial chaos and involves a separation of random variables from deterministic ones in the solution algorithm for a stochastic dierential equation. The random variables are expanded in various local polynomial expansions which are further blended together to obtain a global approximation for the random state variables. These polynomials are associated with the assumed probability density functions for the input variables. Galerkin projection is used to generate a system of deterministic dierential equations for the expansion coecients. We rst present this method to linear systems with parametric uncertainties, and subsequently we generalize it to nonlinear systems with parametric and initial condition uncertainties. The eectiveness of the proposed method is demonstrated by considering dierent numerical examples.

Discretization of a random field by the generalized polynomial chaos (GPC) begins with selecting a specific type of orthogonal polynomial (e.g. Legendre and Hermite polynomials) (e.g. Ghanem and Spanos, 1991). This selection of a type of orthogonal polynomial can be performed based on the reported experiences (e.g. Xiu and Karniadakis, 2003) or data revealing the distribution of a random field to be discretized. If such data or reported experiences are unavailable, a third way may be generating some pilot tests to study the performance of a specific type of orthogonal polynomial in discretizing this random field. This study tries to develop an evolutionary algorithm-based auxiliary tool for the implementation of such pilot tests. A similar tool (Allaix and Carbone, 2009), which is based on the singleobjective evolutionary algorithm, had been developed for constructing the Karhunen-Loeve (KL) representation of a random field. Both KL and GPC expansions are two of the popular random field discretization methods (Ghanem and Spanos, 1991). But, the KL expansion should be applied under a prerequisite of knowing the covariance matrix of a random field to be discretized (e.g. Ghanem and Spanos, 1991); while, the GPC expansions can be applied without similar prerequisites (e.g. Xiu and Karniadakis, 2003). Therefore, the development of an auxiliary tool for constructing a GPC representation of a random field would be necessary. The succeeding research considers the derivation of an GPC representation of a random field as a multi-objective (MO) problem having two goals: (a) limiting the computational efforts spent in applying the resulting GPC representation; and (b) keeping the resulting GPC representation satisfying all accuracy standards (e.g. getting the sufficiently accurate prediction of statistical parameters of a random field). The former goal will be attained by limiting the highest order of polynomial term and total number of uncorrelated random variables used to construct a GPC representation; while the latter goal will be attained by minimizing multiple error estimators. Since there are multiple goals to be attained, a multiple objective evolutionary algorithm (MOEA) is required. Among all available MOEAs, the strength Pareto evolutionary algorithm II (SPEA 2) (Zitzler, et al., 2001) is chosen. The highest order of polynomial term and total number of uncorrelated random variables used to construct an GPC representation are considered as two parameters to be identified. The remainder of this study is organized into four sections. In Sec. 2, the theoretical backgrounds of GPC expansions (e.g. Xiu and Karniadakis, 2003) are briefly reviewed. In Sec. 3, the SPEA 2 (Zitzler, et al., 2001) is used to construct a parameter identification procedure to identify the aforementioned highest order of orthogonal polynomial and total

Generalized Polynomial Chaos (gPC) is known to fail for problems involving strong nonlinear dependencies on stochastic inputs, which especially arise in the context of long term integration or stochastic discontinuities. There are various attempts in the literature which address these difficulties, such as the time-dependent generalized Polynomial Chaos (TD-gPC) and the multi-element generalized Polynomial Chaos (ME-gPC) both leading to higher accuracies but higher numerical costs in comparison to the classical gPC approach. A combination of these methods is introduced, which leads to a powerful solution method since high accuracies can be maintained and computational cost can be distributed by utilizing parallel computation. However, to be able to apply the hybrid method to all types of ordinary differential equations subject to random inputs, new modifications with respect to TD-gPC are carried out by creating an orthogonal tensor basis consisting of the random input variable as well as the solution itself. Such modifications allow TD-gPC to capture the dynamics of the solution by increasing the approximation quality of its time derivatives.

Numerical analysis of the Burgers’ equation in the presence of uncertainty

Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos

Generalized polynomial chaos (gPC), often combined with mono-implicit Runge--Kutta (MIRK) methods, is widely used to solve random ODEs. We investigate the impact of stiffness of random ODEs on the gPC performance. We start by extending pragmatic definitions of stiffness used in deterministic ODEs to their random counterparts. Then we introduce gPC with parallel MIRK schemes to solve random stiff ODEs, in which a suitable parallelism partially alleviates the curse of dimensionality. Our stiffness analysis comprises two parts: (i) the relationship between Jacobians of random ODEs and the corresponding gPC equations and (ii) stiffness of the gPC equations. It provides a direct way to determine whether a random ODE and/or the corresponding gPC equations are stiff. This theoretical analysis plays a key role in designing numerical implementations not only of gPC but also of other methods of stochastic computation, e.g., Monte Carlo simulations and stochastic collocation. A series of computational experiments is used to demonstrate the agreement between our theoretical analysis and numerical results and to establish gPC with parallel MIRK as a feasible and effective tool for solving stiff random ODEs.

Probabilistic safety margin characterization of an integrated small modular reactor using MFM and adaptive polynomial chaos

<p>A robust adaptive controller is developed for a chemical process using a generalized Polynomial Chaos (gPC) expansion-based Markov decision model, which can account for time-invariant probabilistic uncertainty and overcome computational challenge for building Markov models. To calculate the transition probability, a gPC model is used to iteratively predict probability density functions (PDFs) of system’s states including controlled and manipulated variables. For controller tuning, these PDFs and controller parameters are discretized to a finite number of discrete states for building a Markov model. The key idea is to predict the transition probability of controlled and manipulated variables over a finite future control horizon, which can be further used to calculate an optimal sequence of control actions. This approach can be used to optimally tune a controller for set point tracking within a finite future control horizon. The proposed method is illustrated by a continuous stirred tank reactor (CSTR) system with stochastic perturbations in the inlet concentration. The efficiency of the proposed algorithm is quantified in terms of control performance and transient decay.</p> <p> </p>

Error analysis of the Wiener–Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input

High-dimensional uncertainty quantification for Mars atmospheric entry using adaptive generalized polynomial chaos

In this paper a novel improvement to the polynomial chaos (PC) approach for the uncertainty analysis of high speed circuits is presented. The key feature of this work is the development of an alternative hyperbolic truncation scheme to replace the conventional linear truncation scheme used in generation of PC expansions. This hyperbolic truncation scheme results in a sparse PC expansion for marginal loss of accuracy. The computational effort required to evaluate the coefficients of the resultant sparse expansion is only a small fraction of that required for full-blown PC expansions. A greedy adaptive methodology to determine the number of basis terms and evaluate the corresponding coefficients for the sparse expansion is also presented. This approach is validated on a nonlinear radio-frequency (RF) circuit against conventional full-blown PC methods.

Polynomial Chaos for random fractional order differential equations

SENSITIVITY ANALYSIS OF IMAGE CLASSIFICATION MODELS USING GENERALIZED POLYNOMIAL CHAOS - 6th International Conference on Uncertainty Quantification in Computational Sciences and Engineering - 15-18 June 2025, Rhodes Island, Greece

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. Additionally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.