Spectral Stochastic Method Research Articles

Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics

The simulation of dynamical systems involving random coefficients bymeans of stochastic spectral methods (Polynomial Chaos or othertypes of orthogonal stochastic expansions) is faced with well knowncomputational difficulties, arising in particular due to thebroadening of the solution spectrum as time evolves. The simulationof such systems thus requires increasing the basis dimension andcomputational resources for long time integration. This paper dealswith systems having almost surely a stable limit cycles. It isproposed to reformulate the problem at hand in a rescaled timeframework such that the spectrum of the rescaled solution remainsnarrow-banded. Two variants of this approach are considered andevaluated. The first relies on an explicit expression of atime-dependent, uncertain, time scale related to some distancebetween the corresponding solution and a reference deterministicsystem. The time scale is adjusted at each time step so that thedistance from the reference system solution remains small, mimicking'in phase'' behavior. The second variant achieves the sameobjective by borrowing concepts from optimal control theory, andyields more precise time-scale estimates at the price of a higherCPU cost. It is thus more appropriate for uncertain systemsexhibiting a stiff behavior and complex limit cycles. The method isapplied to the case of a linear oscillator with uncertainproperties, and to a stiff nonlinear chemical system involvinguncertain reaction constants. The tests demonstrate theeffectiveness of the proposed approaches, at least in situationswhere the topology of the limit cycle does not change when theuncertain system parameters vary.

Robust Simulation Methodology for Surface-Roughness Loss in Interconnect and Package Modelings

In multigigahertz integrated-circuit design, the extra energy loss caused by conductor surface roughness in metallic interconnects and packagings is more evident than ever before and demands explicit consideration for accurate prediction of signal integrity and energy consumption. Existing techniques based on analytical approximation, despite simple formulations, suffer from restrictive valid ranges, namely, either small or large roughness/frequencies. In this paper, we propose a robust and efficient numerical-simulation methodology applicable to evaluating general surface roughness, described by parameterized stochastic processes, across a wide frequency band. Traditional computation-intensive electromagnetic simulation is avoided via a tailored scalar-wave modeling to capture the power loss due to surface roughness. The spectral stochastic collocation method is applied to construct the complete statistical model. Comparisons with full wave simulation as well as existing methods in their respective valid ranges then verify the effectiveness of the proposed approach.

Statistical modeling and analysis of chip-level leakage power by spectral stochastic method

In this paper, we present a novel statistical full-chip leakage power analysis method. The new method can provide a general framework to derive the full-chip leakage current or power in a closed form in terms of the variational parameters, such as the channel length, the gate oxide thickness, etc. It can accommodate various spatial correlations. The new method employs the orthogonal polynomials to represent the variational gate-level leakages in a closed form first, which is generated by a fast multi-dimensional Gaussian quadrature method. The total leakage currents then are computed by simply summing up the resulting orthogonal polynomials (their coefficients). Unlike many existing approaches, no grid-based partitioning and approximation are required. Instead, the spatial correlations are naturally handled by orthogonal decompositions. The proposed method is very efficient and it becomes linear in the presence of strong spatial correlations. Experimental results show that the proposed method is about 16 × faster than the recently proposed method (Chang and Sapatnekar, 2005 [1]) with constant better accuracy.

Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations

Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the last two decades, a growing interest has been devoted to a new family of methods, called spectral stochastic methods, for the propagation of uncertainties through physical models governed by stochastic partial differential equations. These approaches rely on a fruitful marriage of probability theory and approximation theory in functional analysis. This paper provides a review of some recent developments in computational stochastic methods, with a particular emphasis on spectral stochastic approaches. After a review of different choices for the functional representation of random variables, we provide an overview of various numerical methods for the computation of these functional representations: projection, collocation, Galerkin approaches…. A detailed presentation of Galerkin-type spectral stochastic approaches and related computational issues is provided. Recent developments on model reduction techniques in the context of spectral stochastic methods are also discussed. The aim of these techniques is to circumvent several drawbacks of spectral stochastic approaches (computing time, memory requirements, intrusive character) and to allow their use for large scale applications. We particularly focus on model reduction techniques based on spectral decomposition techniques and their generalizations.

Speeding Up SSFEM Computation Using Kronecker Tensor Products

The spectral stochastic finite-element method makes it possible to convey some random aspects of input data to the output data. However, the system size dramatically increases with the number of input random variables. Using matrix Kronecker tensor products for system solving noticeably reduces the computation time and the storage requirements.

COMPARISON OF METHODS FOR MODELING UNCERTAINTIES IN A 2D HYPERTHERMIA PROBLEM

Uncertainties in biological tissue properties are weighed in the case of a hyperthermia problem. Statistic methods, experimental design and kriging technique, and stochastic methods, spectral and collocation approaches, are applied to analyze the impact of these uncertainties on the distribution of the electromagnetic power absorbed inside the body of a patient. The sensitivity and uncertainty analyses made with the different methods show that experimental designs are not suitable in this kind of problem and that the spectral stochastic method is the most efficient method only when using an adaptative algorithm.

Uncertainty quantification for systems of conservation laws

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. In this paper, we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with Polynomial Chaos (PC) methods. The idea is to introduce a new variable, the entropic variable, in bijection with our vector of unknowns, which we develop on the polynomial basis: by performing a Galerkin projection, we obtain a deterministic system of conservation laws. We state several properties of this deterministic system in the case of a general uncertain system of conservation laws. We then apply the method to the case of the inviscid Burgers’ equation with random initial conditions and we present some preliminary results for the Euler system. We systematically compare results from our new approach to results from the stochastic Galerkin method. In the vicinity of discontinuities, the new method bounds the oscillations due to Gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations. It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.

The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications

Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multi-element probabilistic collocation method (ME-PCM), which is a generalized form of the probabilistic collocation method. In the ME-PCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and Clenshaw-Curtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive ME-PCM are shown, including low-regularity problems and long-time integration. We test the ME-PCM on two-dimensional Navier-Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of ME-PCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of ME-PCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasi-random sequence methods.

Stochastic simulation method for a 2D elasticity problem with random loads

We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper [Sabelfeld KK, Shalimova IA, Levykin AI. Discrete random walk over large spherical grids generated by spherical means for PDEs. Monte Carlo Methods and Applications 2006; 12(1): 55–93]. The vector random field of loads which stands on the right-hand side of the system of elasticity equations is simulated by the Randomization Spectral method presented in [Sabelfeld KK. Monte Carlo methods in boundary value problems. Berlin (Heidelberg, New York): Springer-Verlag; 1991] and recently revised and generalized in [Kurbanmuradov O, Sabelfeld KK. Stochastic spectral and Fourier-wavelet methods for vector Gaussian random field. Monte Carlo Methods and Applications 2006; 12(5–6): 395–445]. Comparative analysis of the RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of the determination of the elastic constants from the known longitudinal and transverse correlations of the loads, and give some relevant numerical illustrations.

Evaluation of Nonlinear System Parameters by Stochastic Spectral Method

AbstractA stochastic spectral method is proposed to estimate the nonlinear parameters of a system subjected to random excitation. Sometimes, it is impossible to measure the excitation process; therefore, estimation must then be based on the response measurement only, in conjunction with a stochastic model of the excitation. The concept of this method is to establish an objecting function involving the unknown parameters through frequency domain analysis, and then we can minimize the objecting function to estimate the unknown parameters by using standard nonlinear optimization algorithm. Finally, the validation of the proposed method is accomplished through many simulations and it can be concluded that the proposed method is applicable to a variety of different problems.

Guest Editors' Introduction: Stochastic Modeling of Complex Systems

This special issue of CiSE highlights various aspects of and outstanding problems in stochastic modeling of complex systems. The techniques covered include stochastic spectral methods, the information-theoretic approach, and sensitivity analysis for uncertainty quantification. Two application articles-one in the field of ecology and the other in the field of manufacturing design-emphasize the diversity of research disciplines in which the ability to handle uncertainty is crucial to obtaining reliable science-based predictions

Sparse grid collocation schemes for stochastic natural convection problems

In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are made.

Stochastic spectral methods for efficient Bayesian solution of inverse problems

We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation, the choice of PC basis, and the decomposition of the support of the prior. The computational cost of the new scheme shows significant gains over direct sampling.

Stochastic finite element method for random temperature in concrete structures

This paper proposes numerical methods for random temperature field in mass concrete structures to research various kinds of random variability including both random environmental temperature and random material properties. First presented, based on local average method of random field, are the formulas of random variational principle as well as the corresponding formulas of stochastic FEM. Then discussed is the stochastic spectral response finite element method, given the complex frequency functions as random functions of material parameters, to resolve the effect of random environmental temperature. Finally, the Taylor expansion series is referred to simplify the non-stationary process and suggest the corresponding formulas for the effect of random hydration heat of concrete. The paper demonstrates that these proposed methods can help to study conveniently the random variability of temperature, which could appear as both random processes and random fields in mass concrete structures. Numerical examples calculated by the methods, theoretical and Monte-Carlo simulation are also provided.

Stochastic Finite Elements with Multiple Random Non-Gaussian Properties

The spectral formulation of the stochastic finite-element method is applied to the problem of heat conduction in a random medium. Specifically, the conductivity of the medium, as well as its heat capacity are treated as uncorrelated random processes with spatial random fluctuations. Using the spectral stochastic finite-element method, this paper analyzes the sensitivity of heat conduction problems to probabilistic models of random data. In particular, both the thermal conductivity and the heat capacity of the medium are assumed to be uncertain. The implementation of the method is demonstrated for both Gaussian and lognormal material properties, modeled either as random variables or random processes.

Random fields of broken clouds and their associated direct solar radiation, scattered transmission and albedo

To model fields of broken clouds, we consider two methods based on Gaussian random fields and one method based on truncated random paraboloids. These models are determined by a few parameters only: the amount of cloud, the mean cloud base diameter and the mean cloud height. A field of broken clouds is a realization of such a random field, it is obtained by stochastic simulation and special spectral methods for Gaussian fields. The direct solar radiation S, the scattered transmission T and the albedo A of such a field of broken clouds are obtained by solving the stationary radiation transfer equation. Numerically we do this by variance reduction Monte Carlo methods. Varying the cloud parameters, the angle of incidence of the solar radiation we get interesting new functional relations between S, T and A for visible and infrared wavelengths.