Showing papers on "Polynomial chaos published in 2006"
TL;DR: ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.
Abstract: We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.
497 citations
TL;DR: In this article, a non-intrusive method based on a least-squares minimization procedure is presented to solve stochastic boundary value problems where material properties and loads are random.
Abstract: The stochastic finite element method allows to solve stochastic boundary value problems where material properties and loads are random. The method is based on the expansion of the mechanical response onto the so-called polynomial chaos. In this paper, a non intrusive method based on a least-squares minimization procedure is presented. This method is illustrated by the study of the settlement of a foundation. Different analysis are proposed: the computation of the statistical moments of the response, a reliability analysis and a parametric sensitivity analysis.
378 citations
17 May 2006
TL;DR: Algorithmic complexity Ball lightning Biological evolution Boundary value problems Butterfly effect Cardiac arrhythmias and electrocardiogram Cellular automata Chaos vs. turbulence Controlling chaos Determinism Emergence Fractals Game of life Laboratory models of nonlinear waves Monte-Carlo methods Multidimensional solitons Neural network models Nonequilibrium statistical mechanics Nonlinear optics Nonlinear Schrodinger equations Nonlinear toys Numerical methods Order from chaos Partial differential equations, nonlinear period doubling Perturbation theory Population dynamics Quantum chaos Random matrix theory Reaction diffusion systems Sandpile model Sp
Abstract: Algorithmic complexity Ball lightning Biological evolution Boundary value problems Butterfly effect Cardiac arrhythmias and electrocardiogram Cellular automata Chaos vs. turbulence Controlling chaos Determinism Emergence Fractals Game of life Laboratory models of nonlinear waves Monte-Carlo methods Multidimensional solitons Neural network models Nonequilibrium statistical mechanics Nonlinear optics Nonlinear Schrodinger equations Nonlinear toys Numerical methods Order from chaos Partial differential equations, nonlinear Period doubling Perturbation theory Population dynamics Quantum chaos Random matrix theory Reaction diffusion systems Sandpile model Spatio-temporal chaos Sine--Gordon (SG) equation Stochastic analyses of neural systems Symmetry groups Tacoma Narrows Bridge collapse Threshold phenomena Universality Vortex dominated flows Wave stability and instability
321 citations
09 Jan 2006
TL;DR: In this article, a non-intrusive Polynomial Chaos (PC) method was proposed for the propagation of input uncertainty in Computational Fluid Dynamics (CFD) simulations, which does not modify the original deterministic code used in the simulations.
Abstract: In this extended abstract, we present a Non-Intrusive Polynomial Chaos (PC) method for the propagation of input uncertainty in Computational Fluid Dynamics (CFD) simulations. By the “non-intrusive” term, we specify a method which does not modify the original deterministic code used in the simulations. In our proposed paper, we focus on investigating such a method which uses deterministic solutions in a stochastic model to simulate the propagation of the input uncertainties for obtaining various statistics of output variables. In a previous study [1], an intrusive PC formulation was implemented to a deterministic Euler code. This intrusive method involves substituting the PC expansions into fluxes and Jacobians, and projecting them onto a random basis function in the form of Hermite Polynomials. Despite its power in uncertainty propagation modeling, this intrusive method can be relatively difficult, expensive, and time consuming to implement to complex problems such as the full Navier-Stokes simulation of 3-D, viscous, turbulent flows around realistic aerospace vehicles or multi-system level simulations which include the interaction of many different codes from different disciplines. In this abstract, we give a brief description of a non-intrusive PC method, which is relatively easy to implement. Our preliminary results with this method includes two stochastic flow problems: (a) steady, subsonic, 2-D, zero-pressure gradient laminar boundary layer over a flat plate, which has the free-stream dynamic viscosity as the uncertain parameter and (b) inviscid, steady, supersonic, 2-D flow over a wedge which has an uncertainty in the wedge
261 citations
TL;DR: This paper focuses on recent application of PC methods for uncertainty representation and propagation in CFD computations, focusing exclusively on applications involving the unreduced Navier–Stokes equations.
257 citations
TL;DR: A numerical method based on Wiener Chaos expansion is proposed and applied to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion and it is demonstrated that for short time solutions the numerical methods are more efficient and accurate than those based on the Monte Carlo simulations.
221 citations
TL;DR: A formulation is presented for the impact of data limitations associated with the calibration of parameters for these models, on their overall predictive accuracy and a new method for the characterization of stochastic processes from corresponding experimental observations is obtained.
202 citations
TL;DR: Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior.
172 citations
TL;DR: In this paper, the identification of probabilistic models of the random coefficients in stochastic boundary value problems (SBVP) is addressed, where the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing and an inverse problem is formulated to calculate the corresponding, optimal realization of the coefficients of the unknown random field on the adapted basis.
Abstract: This paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems (SBVP). The data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing. Starting with a particular mathematical model for the mechanical behaviour of the specimen, the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization. For each set of measurements of the displacement field along the boundary, an inverse problem is formulated to calculate the corresponding, optimal realization of the coefficients of the unknown random field on the adapted basis. Realizations of these coefficients are then used, in conjunction with the maximum likelihood principle, to set-up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the SBVP.
165 citations
TL;DR: In this paper, the authors explore the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty.
Abstract: 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.
164 citations
TL;DR: Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures and generalized Polynomial Chaos (gPC) locally.
Abstract: In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures. Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each element and subsequently implement generalized Polynomial Chaos (gPC) locally. Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures
Book•
01 Jan 2006
TL;DR: In this paper, a model of chaotic growth with the Logistic Model of P.-F. Verhulst's final triumph is presented, where the authors study the influence of advection and nonlocal effects on the evolution of complex networks of oscillators.
Abstract: General and Historical Introduction.- Chaotic Growth with the Logistic Model of P.-F. Verhulst.- Pierre-Francois Verhulst's Final Triumph.- Limits to Success. The Iron Law of Verhulst.- Recurrent Generation of Verhulst Chaos Maps at Any Order and Their Stabilization Diagram by Anticipative Control.- Coherence in Complex Networks of Oscillators.- Growth of Random Sequences.- Life Relevant Physics.- Logistic Population Growth and Beyond: The Influence of Advection and Nonlocal Effects.- Predator-Prey Encounters Studied as Relative Particle Diffusion.- Extinction Dynamics in Lotka-Volterra Ecosystems on Evolving Networks.- Exact Law of Live Nature.- Manifestation of Chaos in Real Complex Systems: Case of Parkinson's Disease.- Monte Carlo Simulations of Ageing and Speciation.- Econophysics.- Influence of Information Flow in the Formation of Economic Cycles.- Logistic Function in Large Financial Crashes.- Agent Based Approaches to Income Distributions and the Impact of Memory.- Condensed Matter.- Agglomeration/Aggregation and Chaotic Behaviour in d-Dimensional Spatio-Temporal Matter Rearrangements Number-Theoretic Aspects.- A Chaos and Fractal Dynamic Approach to the Fracture Mechanics.- Nonlinear Dynamics and Fractal Avalanches in a Pile of Rice.- Miscellaneous.- A Recent Appreciation of the Singular Dynamics at the Edge of Chaos.- Quantum Chaos Versus Classical Chaos: Why is Quantum Chaos Weaker?.- On the Prediction of Chaos in the Restricted Three-Body Problem.- Order and Chaos in Some Hamiltonian Systems of Interest in Plasma Physics.
09 Jan 2006
TL;DR: It is shown that stability of a system can be inferred from the evolution of modal amplitudes, covering nearly the full support of the uncertain parameters with a finite series.
TL;DR: In this article, generalized polynomial chaos theory is applied to model complex nonlinear multibody dynamic systems operating in the presence of parametric and external uncertainty, and theoretical and computational aspects of this methodology are discussed in the companion paper "Modeling Multibody Dynamic Systems with Uncertainties".
Abstract: This study applies generalized polynomial chaos theory to model complex nonlinear multibody dynamic systems operating in the presence of parametric and external uncertainty. Theoretical and computational aspects of this methodology are discussed in the companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects”.
01 Jan 2006
TL;DR: In this article, a numerical method based on the Wiener chaos expansion (WCE) for solving SPDEs driven by Brownian motion forcing is proposed, which can reduce a stochastic PDE into a system of deterministic PDEs and separate the randomness from the computation.
Abstract: Stochastic partial differential equations (SPDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Developing efficient numerical methods for simulating SPDEs is a very important while challenging research topic. In this thesis, we study a numerical method based on the Wiener chaos expansion (WCE) for solving SPDEs driven by Brownian motion forcing. WCE represents a stochastic solution as a spectral expansion with respect to a set of random basis. By deriving a governing equation for the expansion coefficients, we can reduce a stochastic PDE into a system of deterministic PDEs and separate the randomness from the computation. All the statistical information of the solution can be recovered from the deterministic coefficients using very simple formulae. We apply the WCE-based method to solve stochastic Burgers equations, Navier-Stokes equations and nonlinear reaction-diffusion equations with either additive or multiplicative random forcing. Our numerical results demonstrate convincingly that the new method is much more efficient and accurate than MC simulations for solutions in short to moderate time. For a class of model equations, we prove the convergence rate of the WCE method. The analysis also reveals precisely how the convergence constants depend on the size of the time intervals and the variability of the random forcing. Based on the error analysis, we design a sparse truncation strategy for the Wiener chaos expansion. The sparse truncation can reduce the dimension of the resulting PDE system substantially while retaining the same asymptotic convergence rates. For long time solutions, we propose a new computational strategy where MC simulations are used to correct the unresolved small scales in the sparse Wiener chaos solutions. Numerical experiments demonstrate that the WCE-MC hybrid method can handle SPDEs in much longer time intervals than the direct WCE method can. The new method is shown to be much more efficient than the WCE method or the MC simulation alone in relatively long time intervals. However, the limitation of this method is also pointed out. Using the sparse WCE truncation, we can resolve the probability distributions of a stochastic Burgers equation numerically and provide direct evidence for the existence of a unique stationary measure. Using the WCE-MC hybrid method, we can simulate the long time front propagation for a reaction-diffusion equation in random shear flows. Our numerical results confirm the conjecture by Jack Xin that the front propagation speed obeys a quadratic enhancing law. Using the machinery we have developed for the Wiener chaos method, we resolve a few technical difficulties in solving stochastic elliptic equations by Karhunen-Loeve-based polynomial chaos method. We further derive an upscaling formulation for the elliptic system of the Wiener chaos coefficients. Eventually, we apply the upscaled Wiener chaos method for uncertainty quantification in subsurface modeling, combined with a two-stage Markov chain Monte Carlo sampling method we have developed recently.
TL;DR: Different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit-cycle oscillation, subject to parametric variability, and are compared in terms of computational cost, convergence properties, ease of implementation, and potential for application to complex aeroElastic systems.
TL;DR: A hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of Stochastic partial differential equations is proposed.
TL;DR: It is shown that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach, particularly for large-scale problems with many random variables.
TL;DR: In this paper, a method based on the Cameron-Martin version of the Wiener Chaos expansion is described for constructing a generalized solution for stochastic differential equations, and a detailed analysis is presented for the various forms of the passive scalar equation.
Abstract: A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise with either adapted or anticipating input. Existence, uniqueness, regularity, and probabilistic representation of this Wiener Chaos solution is established for a large class of equations. A number of examples are presented to illustrate the general constructions. A detailed analysis is presented for the various forms of the passive scalar equation and for the first-order It\^{o} stochastic partial differential equation. Applications to nonlinear filtering if diffusion processes and to the stochastic Navier-Stokes equation are also discussed.
TL;DR: In this paper, the Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems.
TL;DR: In this paper, a new method for constructing a generalized solution of a stochastic evolution equation is described, and existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations.
Abstract: A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.
TL;DR: In this paper, a quantitative measure of incomplete environmental knowledge or information (i.e., environmental uncertainty) is included in any simulation-based predictions linked to acoustic wave propagation. But this measure is not considered in this paper.
Abstract: It is argued that a quantitative measure of incomplete environmental knowledge or information (i.e., environmental uncertainty) should be included in any simulation-based predictions linked to acoustic wave propagation. A method is then proposed to incorporate environmental uncertainty directly into the computation of acoustic wave propagation in ocean waveguides. In this regard, polynomial chaos expansions are chosen to represent uncertainty in both the environment and acoustic field. The sound-speed distribution and acoustic field are therefore generalized to stochastic processes, where uncertainty in the field is interpreted in terms of its statistical moments. Starting from the narrow angle parabolic approximation, a set of coupled differential equations is derived in which the coupling term links incomplete environmental information to the corresponding uncertainty in the acoustic field. Propagation of both the field and its uncertainty in an isospeed waveguide is considered as an example, where the ...
TL;DR: The results presented show that the variance of the location of perturbed shock grows quadratically with the distance from the wedge apex for steady randomness, however, for a time-dependent random process the dependence is quadratic only close to the apex and linear for larger distances.
TL;DR: In this article, a new stochastic finite element procedure (SFEP) in the tradition of Ghanem's work is presented, which allows to deal with any number of input random variables of any type that can model both material properties and loading.
Abstract: A new stochastic finite element procedure (SFEP) in the tradition of Ghanem’s work is presented. It allows to deal with any number of input random variables of any type that can model both material properties and loading. The method makes use of Hermite series expansion of the input random variables and polynomial chaos expansion of the response, for which an original implementation is proposed. The link with reliability analysis is also established. Three application examples in geotechnical engineering are given for the sake of illustration. The accuracy and efficiency of SFEP is thoroughly investigated by comparison with well-established approaches.
TL;DR: In this paper, an original, robust, multi-level dynamic condensation method of stochastic models is proposed, which is based on a discretization technique of random fields that was established using the Karhunen-Loeve procedure.
TL;DR: In this paper, the effects of a nonlinear energy sink during the instationary regime are analyzed by introducing uncertain parameters to verify the robustness of the transient spatial energy transfer when parameters are not well known.
Abstract: The effects of a nonlinear energy sink during the instationary regime are analyzed by introducing uncertain parameters to verify the robustness of the transient spatial energy transfer when parameters are not well known. It was shown that it is possible to passively absorb energy from a linear nonconservative system (damped) structure to a nonlinear attachment weakly coupled to the linear one. This rapid and irreversible transfer of energy, named energy pumping, is studied by taking into account uncertainties on parameters, especially damping (since damping plays a great role and there is a lack of knowledge about it). In essence, the nonlinear subsystem acts as a passive nonlinear energy sink for impulsively applied external vibrational disturbances. The aim is to be able to apply energy pumping in practice where the nonlinear attachment realization will never perfectly reflect the design. Since strong nonlinearities are involved, polynomial chaos expansions are used to obtain information about random displacements. Not only are numerical investigations done, but nonlinear normal modes and the role of damping are also analytically studied, which confirms the numerical studies and shows the supplementary information obtained compared to a parametrical study.
TL;DR: The notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system.
Abstract: In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.
TL;DR: In this article, it was shown that an annihilation operator of the unforced quantum harmonic oscillator exhibits distributional chaos as introduced in B Schweizer and J Smital (1994 Trans. Am. Math. Soc. 344 737−54).
Abstract: It is known that many physical systems which do not exhibit deterministic chaos when treated classically may exhibit such behaviour if treated from the quantum mechanics point of view. In this paper, we will show that an annihilation operator of the unforced quantum harmonic oscillator exhibits distributional chaos as introduced in B Schweizer and J Smital (1994 Trans. Am. Math. Soc. 344 737–54). Our approach strengthens previous results on chaos in this model and provides a very powerful tool to measure chaos in other (quantum or classical) models.