Nonlinear Stochastic Ordinary Differential Equations Research Articles

Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold

This paper introduces a surrogate modeling scheme based on Grassmannian manifold learning to be used for cost-efficient predictions of high-dimensional stochastic systems. The method exploits subspace-structured features of each solution by projecting it onto a Grassmann manifold. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. The method utilizes a solution clustering approach in order to identify regions of the parameter space over which solutions are sufficiently similarly such that they can be interpolated on the Grassmannian. In this clustering, the reduced-order solutions are partitioned into disjoint clusters on the Grassmann manifold using the eigen-structure of properly defined Grassmannian kernels and, the Karcher mean of each cluster is estimated. Then, the points in each cluster are projected onto the tangent space with origin at the corresponding Karcher mean using the exponential mapping. For each cluster, a Gaussian process regression model is trained that maps the input parameters of the system to the reduced solution points of the corresponding cluster projected onto the tangent space. Using this Gaussian process model, the full-field solution can be efficiently predicted at any new point in the parameter space. In certain cases, the solution clusters will span disjoint regions of the parameter space. In such cases, for each of the solution clusters we utilize a second, density-based spatial clustering to group their corresponding input parameter points in the Euclidean space. The proposed method is applied to two numerical examples. The first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.

Chaotic analysis of Kelvin–Voigt viscoelastic plates under combined transverse periodic and white noise excitation: an analytic approach

Due to the use of materials with high structural damping in new applications and the role of plates as one of the basic elements of many engineering and interdisciplinary structures, the study of the stability of viscoelastic plates is very important in real conditions. In this article, the border curves of instability for a nonlinear Kelvin–Voigt viscoelastic plate under combined lateral periodic and white noise excitation are obtained analytically. Firstly, the governing equation of the plate is derived and then transformed into a nonlinear stochastic ordinary differential equation using Galerkin’s method. Secondly, Melnikov’s equation and its modified version in a stochastic sense are evaluated. At last, the border curves of instability for many types of plates and different variations of plate parameters and external excitations are obtained and drawn. The results show how and when the chaotic behavior changes by varying the plate parameters or fluctuating the intensities, magnitudes or frequencies of the external loads. It is shown that in the presence of white noise excitation the chaotic area become larger and this effect is larger at the frequencies far from the natural frequency of the corresponding linear system. It is also shown that under a combination of periodic and white noise excitations the chaotic behavior at low damping values might be completely different from the case there is only a periodic force.

Probabilistic density function method for nonlinear dynamical systems driven by colored noise.

We present a probability density function (PDF) method for a system of nonlinear stochastic ordinary differential equations driven by colored noise. The method provides an integrodifferential equation for the temporal evolution of the joint PDF of the system's state, which we close by means of a modified large-eddy-diffusivity (LED) closure. In contrast to the classical LED closure, the proposed closure accounts for advective transport of the PDF in the approximate temporal deconvolution of the integrodifferential equation. In addition, we introduce the generalized local linearization approximation for deriving a computable PDF equation in the form of a second-order partial differential equation. We demonstrate that the proposed closure and localization accurately describe the dynamics of the PDF in phase space for systems driven by noise with arbitrary autocorrelation time. We apply the proposed PDF method to analyze a set of Kramers equations driven by exponentially autocorrelated Gaussian colored noise to study nonlinear oscillators and the dynamics and stability of a power grid. Numerical experiments show the PDF method is accurate when the noise autocorrelation time is either much shorter or longer than the system's relaxation time, while the accuracy decreases as the ratio of the two timescales approaches unity. Similarly, the PDF method accuracy decreases with increasing standard deviation of the noise.

Bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations

A Bayesian model selection and parameter estimation algorithm is applied to investigate the influence of nonlinear and unsteady aerodynamic loads on the limit cycle oscillation (LCO) of a pitching airfoil in the transitional Reynolds number regime. At small angles of attack, laminar boundary layer trailing edge separation causes negative aerodynamic damping leading to the LCO. The fluid-structure interaction of the rigid, but elastically mounted, airfoil and nonlinear unsteady aerodynamics is represented by two coupled nonlinear stochastic ordinary differential equations containing uncertain parameters and model approximation errors. Several plausible aerodynamic models with increasing complexity are proposed to describe the aeroelastic system leading to LCO. The likelihood in the posterior parameter probability density function (pdf) is available semi-analytically using the extended Kalman filter for the state estimation of the coupled nonlinear structural and unsteady aerodynamic model. The posterior parameter pdf is sampled using a parallel and adaptive Markov Chain Monte Carlo (MCMC) algorithm. The posterior probability of each model is estimated using the Chib-Jeliazkov method that directly uses the posterior MCMC samples for evidence (marginal likelihood) computation. The Bayesian algorithm is validated through a numerical study and then applied to model the nonlinear unsteady aerodynamic loads using wind-tunnel test data at various Reynolds numbers.

Predicting the impact of border control on malaria transmission: a simulated focal screen and treat campaign.

BackgroundSouth Africa is one of many countries committed to malaria elimination with a target of 2018 and all malaria-endemic provinces, including Mpumalanga, are increasing efforts towards this ambitious goal. The reduction of imported infections is a vital element of an elimination strategy, particularly if a country is already experiencing high levels of imported infections. Border control of malaria is one tool that may be considered.MethodsA metapopulation, non-linear stochastic ordinary differential equation model is used to simulate malaria transmission in Mpumalanga and Maputo province, Mozambique (the source of the majority of imported infections) to predict the impact of a focal screen and treat campaign at the Mpumalanga–Maputo border. This campaign is simulated by nesting an individual-based model for the focal screen and treat campaign within the metapopulation transmission model.ResultsThe model predicts that such a campaign, simulated for different levels of resources, coverage and take-up rates with a variety of screening tools, will not eliminate malaria on its own, but will reduce transmission substantially. Making the campaign mandatory decreases transmission further though sub-patent infections are likely to remain undetected if the diagnostic tool is not adequately sensitive. Replacing screening and treating with mass drug administration results in substantially larger decreases as all (including sub-patent) infections are treated before movement into Mpumalanga.ConclusionsThe reduction of imported cases will be vital to any future malaria control or elimination strategy. This simulation predicts that FSAT at the Mpumalanga–Maputo border will be unable to eliminate local malaria on its own, but may still play a key role in detecting and treating imported infections before they enter the country. Thus FSAT may form part of an integrated elimination strategy where a variety of interventions are employed together to achieve malaria elimination.Electronic supplementary materialThe online version of this article (doi:10.1186/s12936-015-0776-2) contains supplementary material, which is available to authorized users.

Stochastic Response of an Inclined Shallow Cable with Linear Viscous Dampers under Stochastic Excitation

Considering the coupling between the in-plane and out-of-plane vibration, the stochastic response of an inclined shallow cable with linear viscous dampers subjected to Gaussian white noise excitation is investigated in this paper. Selecting the static deflection shape due to a concentrated force at the dampers location and the first sine term as shape functions, a reduced four-degree-of-freedom system of nonlinear stochastic ordinary differential equations are derived to describe dynamic response of the cable. Since only polynomial-type terms are contained, the fourth-order cumulant-neglect closure together with the C-type Gram-Charlier expansion with a fourth-order closure are applied to obtain statistical moments, power spectral density and probabilistic density function of the cable response, whose availability is verified by Monte Carlo method. Taking a typical cable as an example, the influence of several factors, which include excitation level and direction as well as damper size, on the dynamic response of the cable is extensively investigated. It is found that the sum of mean square in-plane and out-of-plane displacement is primarily independent of the load direction when the excitation level and viscous coefficient of the damper are fixed. Moreover, the peak frequency and half-band width of the spectra of both the in-plane and the out-of-plane displacements are increasing with excitation level when the damper size is constant. It is also observed that, even though the actual optimal damper size is slightly greater than the one obtained by the complex modal theory, the difference of statistical moment of the cable caused by these two damper size is negligible, so the vibration reduction effect provided by the theoretical optimal viscous coefficient is satisfactory.

Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Modeling Approach

Symmetry methods can be used to transform almost any kind of linear or nonlinear partial differential equation (PDE) that represents a hydrologic process in any dimension to an equivalent ordinary differential equation (ODE). Meanwhile, Kavvas recently shown in 2003 that the conservation equations of hydrologic processes under uncertainty, expressed as linear or nonlinear stochastic ODEs or PDEs, have a one-to-one correspondence to a mixed Eulerian-Lagrangian nonlocal form of the Fokker-Planck equation (FPE) when the underlying process has finite correlation lengths. Under such correspondence, it is possible to obtain a solution for the ensemble behavior of a particular hydrologic process in terms of the solution of its corresponding FPE for the probability distribution function (PDF) of its state variables under appropriate initial and boundary conditions. A major issue with the resulting FPE in the case of conservation equations in PDE form is that spatial gradients of the process state variables appear...

Model predictive control of nonlinear stochastic partial differential equations with application to a sputtering process

A method is developed for model predictive control of nonlinear stochastic partial differential equations (PDEs) to regulate the state variance, which physically represents the roughness of a surface in a thin film growth process, to a desired level. Initially a nonlinear stochastic PDE is formulated into a system of infinite nonlinear stochastic ordinary differential equations by using Galerkin's method. A finite-dimensional approximation is then derived that captures the dominant mode contribution to the state variance. A model predictive control problem is formulated, based on the finite-dimensional approximation, so that the future state variance can be predicted in a computationally efficient way. To demonstrate the method, the model predictive controller is applied to the stochastic Kuramoto-Sivashinsky equation, and the kinetic Monte Carlo model of a sputtering process to regulate the surface roughness at a desired level. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Model parameter estimation and feedback control of surface roughness in a sputtering process

This work focuses on model parameter estimation and model-based output feedback control of surface roughness in a sputtering process which involves two surface micro-processes: atom erosion and surface diffusion. This sputtering process is simulated using a kinetic Monte Carlo (kMC) simulation method and its surface height evolution can be adequately described by the stochastic Kuramoto–Sivashinsky equation (KSE), a fourth-order nonlinear stochastic partial differential equation (PDE). First, we estimate the four parameters of the stochastic KSE so that the expected surface roughness profile predicted by the stochastic KSE is close (in a least-square sense) to the profile of the kMC simulation of the same process. To perform this model parameter estimation task, we initially formulate the nonlinear stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs). A finite-dimensional approximation of the stochastic KSE is then constructed that captures the dominant mode contribution to the state and the evolution of the state covariance of the stochastic ODE system is derived. Then, a kMC simulator is used to generate representative surface snapshots during process evolution to obtain values of the state vector of the stochastic ODE system. Subsequently, the state covariance of the stochastic ODE system that corresponds to the sputtering process is computed based on the kMC simulation results. Finally, the model parameters of the nonlinear stochastic KSE are obtained by using least-squares fitting so that the state covariance computed from the stochastic KSE process model matches that computed from kMC simulations. Subsequently, we use appropriate finite-dimensional approximations of the identified stochastic KSE model to design state and output feedback controllers, which are applied to the kMC model of the sputtering process. Extensive closed-loop system simulations demonstrate that the controllers reduce the expected surface roughness by 55 % compared to the corresponding values under open-loop operation.

PARAMETER IDENTIFICATION FOR NONLINEAR STOCHASTIC PDE MODEL OF A SPUTTERING PROCESS

This work focuses on identification of the parameters of the nonlinear stochastic Kuramoto-Sivashinsky equation (KSE), a fourth order nonlinear stochastic partial differential equation (PDE), that describes the fluctuation of surface height of a sputtering process including two surface micro-processes, diffusion and erosion. To perform the system identification, we initially formulate the nonlinear stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs) by using Galerkin's method. A finite-dimensional approximation of the stochastic KSE is then constructed that captures the dominant mode contribution to the state and the evolution of the state covariance of the stochastic ODE system is derived. Then, a kinetic Monte-Carlo (kMC) simulator is used to generate surface snapshots during process evolution to obtain values of the state vector of the stochastic ODE system. Subsequently, the state covariance of the stochastic ODE system that corresponds to the sputtering process is computed based on the kMC simulation results. Finally, the model parameters of the nonlinear stochastic KSE are obtained by using least-squares fitting so that the state covariance computed from the stochastic KSE process model matches that computed from kMC simulations. Simulations are performed to demonstrate the effectiveness of the proposed parameter identification approach.

Nonlinear Feedback Control of Surface Roughness Using a Stochastic PDE: Design and Application to a Sputtering Process

In this work, we develop a method for nonlinear feedback control of the roughness of a one-dimensional surface whose evolution is described by the stochastic Kuramoto-Sivashinsky equation (KSE), a fourthorder nonlinear stochastic partial differential equation. We initially formulate the stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations by using Galerkin’s method. A finite-dimensional approximation of the stochastic KSE is then derived that captures the dominant mode contribution to the surface roughness. A nonlinear feedback controller is then designed based on the finite-dimensional approximation to control the surface roughness. An analysis of the closed-loop nonlinear infinite-dimensional system is performed to characterize the closed-loop performance enforced by the nonlinear feedback controller in the closed-loop infinite-dimensional system. The effectiveness of the proposed nonlinear controller and the advantages of the nonlinear controller over a linear controller resulting from the linearization of the nonlinear controller around the zero solution are demonstrated through numerical simulations. Finally, a successful application of a stochastic KSE-based nonlinear feedback controller to the kinetic Monte Carlo model of a sputtering process is also demonstrated.

Exact soliton-like solutions for stochastic combined Burgers–KdV equation

Abstract In this paper, by using symmetry reduction, the stochastic Wick-type combined Burgers–KdV equation is reduced to nonlinear stochastic ordinary differential equation (NSODE), which contains the Jacobian elliptic equation and Painleve type II equation. Based on this so obtained NSODE, some stochastic Wick-type exact soliton-like solutions are given via Hermite transformation.

SIMPLE NEURAL NETWORKS THAT OPTIMIZE DECISIONS

We review simple connectionist and firing rate models for mutually inhibiting pools of neurons that discriminate between pairs of stimuli. Both are two-dimensional nonlinear stochastic ordinary differential equations, and although they differ in how inputs and stimuli enter, we show that they are equivalent under state variable and parameter coordinate changes. A key parameter is gain: the maximum slope of the sigmoidal activation function. We develop piecewise-linear and purely linear models, and one-dimensional reductions to Ornstein–Uhlenbeck processes that can be viewed as linear filters, and show that reaction time and error rate statistics are well approximated by these simpler models. We then pose and solve the optimal gain problem for the Ornstein–Uhlenbeck processes, finding explicit gain schedules that minimize error rates for time-varying stimuli. We relate these to time courses of norepinephrine release in cortical areas, and argue that transient firing rate changes in the brainstem nucleus locus coeruleus may be responsible for approximate gain optimization.

Pulse propagation in media with deterministic and random dispersion variations

We obtain a widely valid description of pulse broadening in dispersion-managed fiber with a random component in the dispersion. For short distances, a quasi-linear approach in the form of a simple perturbation expansion suffices to capture the broadening for each realization of the random dispersion. For intermediate distances, over which the noise and nonlinearity interact, we demonstrate that the partial differential equation can be reduced to a relatively simple system of nonlinear stochastic ordinary differential equations . Finally, we demonstrate that, over long distances, the slow evolution of the pulse width can be obtained by applying an appropriate multiple-scales averaging procedure to yield a new, scaled noise process effecting pulse broadening.

Time evolution and fluctuations of the probability density and entropy function for a class of nonlinear stochastic systems in mathematical physics

A mathematical theory is proposed to describe the time evolution and fluctuations for the probability density and the entropy functions related to the state variable of nonlinear stochastic systems. The class of physical systems whose evolution can be modelled by nonlinear stochastic ordinary differential equations is considered and the evolution equation for the above specified functions is derived under suitable regularity conditions. Quantitative results in the applications are supplied on the basis of an analysis using the Adomian's decomposition method.