Stochastic Perturbation Method Research Articles

Response Probability Analysis of Random Acoustic Field Based on Perturbation Stochastic Method and Change-of-Variable Technique

To calculate the probability density function of the response of a random acoustic field, a change-of-variable perturbation stochastic finite element method (CVPSFEM), which integrates the perturbation stochastic finite element method (PSFEM) and the change-of-variable technique in a unified form, is proposed. In the proposed method, the response of a random acoustic field is approximated as a linear function of the random variables based on a first order stochastic perturbation analysis. According to the linear relationship between the response and the random variables, the formal expression of the probability density function of the response of a random acoustic field is obtained by the change-of-variable technique. The numerical examples on a two-dimensional (2D) acoustic tube and a three-dimensional (3D) acoustic cavity of an automobile cabin verify the accuracy and efficiency of the proposed method. Hence, the proposed method can be considered as an effective method to quantify the effects of the parametric randomness of a random acoustic field on the sound pressure response.

Path Transmissibility Analysis Considering Two Types of Correlations in Hydropower Stations

A new vibration model is built by introducing the head-cover vibration transfer path based on a previous analysis of the vertical vibration model for hydropower station units and powerhouses. This research focuses on disturbance- and parameter-related transfer paths in a practical situation. In a complex situation, the application of the stochastic perturbation method is expanded using an algebra synthesis method the Hadamard product, and theoretical analyses, and numerical simulations of transfer paths in the new vibration model are carried out through the expanded perturbation method. The path transfer force, the path transmissibility, and the path disturbance ranges in the frequency domain are provided. The results indicate that the methods proposed in this study can efficiently reduce the disturbance range and can accurately analyze the transfer paths of hydraulic-source vertical vibration in hydropower stations.

A black box method for solving the complex exponentials approximation problem

A common problem, arising in many different applied contexts, consists in estimating the number of exponentially damped sinusoids whose weighted sum best fits a finite set of noisy data and in estimating their parameters. Many different methods exist to this purpose. The best of them are based on approximate Maximum Likelihood estimators, assuming to know the number of damped sinusoids, which can then be estimated by an order selection procedure. As the problem can be severely ill posed, a stochastic perturbation method is proposed which provides better results than Maximum Likelihood based methods when the signal-to-noise ratio is low. The method depends on some hyperparameters which turn out to be essentially independent of the application. Therefore they can be fixed once and for all, giving rise to a black box method.

Towards Providing Automated Feedback on the Quality of Inferences from Synthetic Datasets

When releasing individual-level data to the public, statistical agencies typically alter data values to protect the confidentiality of individuals’ identities and sensitive attributes. When data undergo substantial perturbation, secondary data analysts’ inferences can be distorted in ways that they typically cannot determine from the released data alone. This is problematic, in that analysts have no idea if they should trust the results based on the altered data.To ameliorate this problem, agencies can establish verification servers, which are remote computers that analysts query for measures of the quality of inferences obtained from disclosure-protected data. The reported quality measures reflect the similarity between the analysis done with the altered data and the analysis done with the confidential data. However, quality measures can leak information about the confidential values, so that they too must be subject to disclosure protections. In this article, we discuss several approaches to releasing quality measures for verification servers when the public use data are generated via multiple imputation, also known as synthetic data. The methods can be modified for other stochastic perturbation methods.

Homogenization of fiber-reinforced composites with random properties using the weighted least squares response function approach

The main aim of the paper is a determination of the basic probabilistic characteristics for the effective elasticity tensor of the periodic fiber-reinforced composites, using the generalized stochastic perturbation technique. An evaluation of the generalized stochastic perturbation method of the analytical formulas and the Monte-Carlo simulation technique is provided for the 1D periodic structure with random material parameters. The higher-order terms are determined using numerical determination of the response functions between the effective tensor components and the given random input variables. It is carried out with the use of the Least Squares Method (LSM), applied for the series of computational experiments consisting of the Finite Element Method (FEM) solutions to the cell problems for the randomized input parameters. The key problem is the weighting LSM procedure worked out to speed up the probabilistic convergence of the homogenization results.

Robust parameter design optimization of simulation experiments using stochastic perturbation methods

Stochastic perturbation methods can be applied to problems for which either the objective function is represented analytically, or the objective function is the result of a simulation experiment. The Simultaneous Perturbation Stochastic Approximation (SPSA) method has the advantage over similar methods of requiring only two measurements at each iteration of the search. This feature makes SPSA attractive for robust parameter design (RPD) problems where some factors affect the variance of the response(s) of interest. In this paper, the feasibility of SPSA as a RPD optimizer is presented, first when the objective function is known, and then when the objective function is estimated by means of a discrete-event simulation.

A generalized stochastic perturbation technique for plasticity problems

The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is provided in two different ways: (i) via the straightforward differentiation of the initial incremental equation and (ii) using the modified response surface method. This methodology is illustrated with the analysis of the elasto-plastic plane truss with random Young’s modulus leading to the determination of the probabilistic moments by the hybrid stochastic symbolic-finite element method computations.

An introduction to nonlinear problems with random parameters by the Stochastic perturbation‐based FEM

AbstractThe main aim of this paper is to present an algorithm and the solution to the nonlinear problems with random parameters. The methodology is based on the generalized nth order stochastic perturbation method and, on the other hand, on the Finite Element Method adjacent to the physical and geometrical nonlinearities. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is proposed in two different ways – thanks to the straightforward differentiation of the initial incremental equation and, separately, using the modified Response Function Method. This approach is illustrated with the analysis of the simply supported elastoplastic beam loaded centrally with the concentrated force, where the probabilistic moments for the limit load and the reliability indices are determined via the stochastic symbolic computations. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adjoint-Based Well-Placement Optimization Under Production Constraints

Summary Determining the optimal location of wells with the aid of an automated search method can significantly increase a project's net present value (NPV) as modeled in a reservoir simulator. This paper has two main contributions: first, to determine the effect of production constraints on optimal well locations and, second, to determine optimal well locations using a gradient-based optimization method. Our approach is based on the concept of surrounding the wells whose locations have to be optimized by so-called pseudowells. These pseudowells produce or inject at a very low rate and, thus, have a negligible influence on the overall flow throughout the reservoir. The gradients of NPV over the lifespan of the reservoir with respect to flow rates in the pseudowells are computed using an adjoint method. These gradients are used subsequently to approximate improving directions (i.e., directions to move the wells to achieve an increase in NPV), on the basis of which improving well locations can be determined. The main advantage over previous approaches such as finite-difference or stochastic-perturbation methods is that the method computes improving directions for all wells in only one forward (reservoir) and one backward (adjoint) simulation. The process is repeated until no further improvements are obtained. The method is applied to three waterflooding examples.

Linearized perturbation method for stochastic analysis of a rill erosion model

The linearization and correction method (LCM) proposed by He is a simple and effective perturbation technique to solve nonlinear equations. To analyze the random properties of rill erosion model, a new stochastic perturbation technique called linearized perturbation method is developed by combining the traditional stochastic perturbation method with the LCM. Comparisons between the numerical results obtained by the linearized perturbation method and those obtained by Monte Carlo method indicated an excellent agreement. However, the calculation efficiency of the linearized perturbation method is higher.

Upscaling Hydraulic Properties and Soil Water Flow Processes in Heterogeneous Soils: A Review

This review covers, in a comprehensive manner, the approaches available in the literature to upscale soil water processes and hydraulic parameters in the vadose zone. We distinguish two categories of upscaling methods: forward approaches requiring information about the spatial distribution of hydraulic parameters at a small scale, and inverse modeling approaches requiring information about the spatial and temporal variation of state variables at various scales, including so‐called “soft data”. Geostatistical and scaling approaches are crucial to upscale soil water processes and to derive large‐scale effective fluxes and parameters from small‐scale information. Upscaling approaches include stochastic perturbation methods, the scaleway approach, the stream‐tube approach, the aggregation concept, inverse modeling approaches, and data fusion. With all upscaling methods, the estimated effective parameters depend not only on the properties of the heterogeneous flow field but also on boundary conditions. The use of the Richards equation at the field and watershed scale is based more on pragmatism than on a sound physical basis. There are practically no data sets presently available that provide sufficient information to extensively validate existing upscaling approaches. Use of numerical case studies has therefore been most common. More recently and still under development, hydrogeophysical methods combined with ground‐based remote sensing techniques promise significant contributions toward providing high‐quality data sets. Finally, most of the upscaling literature in vadose zone research has dealt with bare soils or deep vadose zones. There is a need to develop upscaling methods for real world soils, considering root water uptake mechanisms and other soil–plant–atmosphere interactions.

Hydrofoil vibration induced by a random flow: a stochastic perturbation approach

The problem of an elastic lifting hydrofoil in a randomly perturbed flow is considered. It appears that in these conditions the phenomenon of hydroelastic-induced vibrations is controlled by a stochastic differential operator. By using the theory of stochastic perturbation, a technique of solution for this class of problems is proposed, leading to an effective numerical solution. The fundamentals of the method are given and it is applied to the general problem of hydroelastic vibrations. A numerical application to the case of an elastic control surface for a prototype-high- speed marine vehicle is presented. Comparisons between the results obtained by the Stochastic Perturbation Method (SPM) and those provided by standard Monte Carlo simulations (MCS) show the accuracy of the proposed method and a useful saving in computational time. A method is given for comparing the computational time required by the two methods, for a given statistical accuracy.

Stochastic modeling of complex nonstationary groundwater systems

Despite the intensive research over the past two decades in the field of stochastic subsurface hydrology, a substantial gap remains between theory and application. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows. In this paper, we present an efficient, nonstationary spectral approach for modeling complex stochastic flows in moderately heterogeneous media. Specifically, we reformulate the governing stochastic equations and introduce a new transfer function to characterize the propagation of system uncertainty. The new transfer function plays a similar role as the commonly used Green’s functions in classical stochastic perturbation methods but is more amenable to numerical solution. The compact transfer function can be used to construct efficiently various spatial statistics of interest, such as covariances, cross-covariances, variances, and mean closure fluxes. We demonstrate the advantages of the proposed approach by applying it to a number of nonstationary problems, including a large, complex problem that is difficult to solve by traditional methods. In particular, we focus in this paper on demonstrating the new approach’s ability to compute efficiently covariances and cross-covariances critical for measurement conditioning, monitoring network analyses, and stochastic transport modeling in the presence of complex mean flow nonstationarities (caused, e.g., by complex trends in aquifer properties, boundary conditions, and sources and sinks). This paper is an extension of our recent work that illustrated the basic approach for modeling nonlocal and nonstationary scale effects and uncertainty propagation in relatively simple situations.

Quasi-Failure Analysis on Resonant Demolition of Random Structural Systems

Introduction P ARAMETER uncertainty of structural systems is inherent in most engineering problems. Reliability analysis can help the designer to establish acceptable tolerances on structures and to determinethe  uctuationsof the systemparametersfor safeoperations. Up to now, failure problemsof uncertainvibrationstructuresystems have followed two paths. One is failure research on the basis of the responses (displacement, stress, etc.) of forced vibration.1i8 The other is failure research on the basis of the relation between natural frequency and forcing frequency of vibration systems at resonance and nonresonance. In this Note, the reliability analysis method of avoiding resonance for multi-degree-of-freedom structural systems is presented. The failure model and the failure probability of the random systems are deŽ ned. When the stochastic perturbation method and the reliability theory are used, the formulas of quasifailure probability for resonant problems of random structures are obtained.

Dynamic Research of a Nonlinear Stochastic Vibratory Machine

This paper presents the dynamics problems of stochastic vibratory machine systems. The random responses of the vibratory machine systems with stochastic parameters subjected to random excitation are researched using a stochastic perturbation method. The numerical results are obtained. The dynamic characteristics of nonlinear stochastic vibratory machine are analyzed.

Partitioning tracer transport in a hydrogeochemically heterogeneous aquifer

In this paper a stochastic model is developed to predict the behavior of partitioning tracers in a three‐dimensional aquifer subject to spatial heterogeneities of nonaqueous phase liquid (NAPL) and Darcy flux. Stochastic Eulerian perturbation methods are applied to derive a system of six coupled partial‐differential first‐order moment equations from the traditional advection‐dispersion‐retardation equation. These equations describe the transient behavior of the concentration moments that explicitly accounts for the effects of the considered heterogeneities on prediction uncertainty and spatiotemporal correlation between tracer concentration, Darcy flux, and NAPL distribution. A finite difference technique is used to solve these equations numerically. Three different cases are presented to evaluate the impact of various statistical relationships between the random fields of natural log conductivity and natural log volumetric NAPL content. It is found that strong negative correlation between NAPL content and hydraulic conductivity leads to increased prediction uncertainty, whereas strong positive correlation between NAPL content and hydraulic conductivity leads to decreased prediction uncertainty over the uncorrelated case. For all cases, cross correlations between tracer concentration and Darcy flux or NAPL saturation are more extensive in the direction of mean flow than transverse to the mean flow. Furthermore, the magnitude of these correlations is approximately constant over the duration of the tracer experiment, implying that all concentration measurements taken at a particular location contain approximately equal information about Darcy flux or NAPL distribution within the experimental domain. The cross correlation between tracer concentration and Darcy flux is relatively insensitive to the degree of the conductivity‐NAPL correlation. However, the correlation between tracer concentration and NAPL concentration is much larger for strongly (positively or negatively) correlated hydraulic conductivity and NAPL distributions. In a comparison paper [Zhang and Graham, this issue] these correlations, together with tracer measurements, are incorporated into a parameter estimation algorithm to produce site‐specific conditional estimates of NAPL and Darcy flux distribution.

Accuracy of stochastic perturbation methods: The case of asset pricing models

This paper investigates the accuracy of a perturbation method in approximating the solution to stochastic equilibrium models under rational expectations. As a benchmark model, we use a version of asset pricing models proposed by Burnside (1998, Journal of Economic Dynamics and Control 22, 329–340) which admits a closed-form solution while not making the assumption of certainty equivalence. We then check the accuracy of perturbation methods — extended to a stochastic environment — against the closed form solution. Second- and especially fourth-order expansions are then found to be more efficient than standard linear approximation, as they are able to account for higher-order moments of the distribution — which constitutes a major improvement of this stochastic approach to approximation compared to other methods that assume certainty equivalence.

Understanding stochastic perturbation theory: toy models and statistical analysis

The numerical stochastic perturbation method based on Parisi–Wu quantisation is applied to a suite of simple models to test its validity at high orders. Large deviations from normal distribution for the basic estimators are systematically found in all cases (“Pepe effect”). As a consequence one should be very careful in estimating statistical errors. We present some results obtained on Weingarten's “pathological” model where reliable results can be obtained by an application of the bootstrap method. We also present some evidence that in the far less trivial application to lattice gauge theory a similar problem should not arise at moderately high loops (up to O (α 10 ) ).

Adaptive phase-distortion correction based on parallel gradient-descent optimization

We describe an adaptive wave-front control technique based on a parallel stochastic perturbation method that can be applied to a general class of adaptive-optical system. The efficiency of this approach is analyzed numerically and experimentally by use of a white-light adaptive-imaging system with an extended source. To create and compensate for static phase distortions, we use 127-element liquid-crystal phase modulators. Results demonstrate that adaptive wave-front correction by a parallel-perturbation technique can significantly improve image quality.

A stochastic perturbation method for studying inverse problems of dynamics

In this paper we introduce a method for the local reconstruciton of an analytical vector field ƒ in ℝn from a set of observations about the asymptotics of n different brownian motions driven by ƒ. The case n = 2 is studied in details and an inversion algorithm is proposed. Although such algorithm turns out to be unstable, it can be a starting point for the quantitative analysis of some inverse problems in dissipative dynamics.