Quantifying Uncertainty
Xiu, D. and Karniadakis, G.E.. "Modeling uncertainty in flow simulations via generalized Polynomial Chaos" J. Comput. Phys.. 187
(1).
2003.
pp. 137--167.
We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach can be considered as a generalization of the original polynomial chaos expansion, first introduced by Wiener [Am. J. Math. 60 (1938) 897]. The original method employs the Hermite polynomials (one of the 13 members of the Askey scheme) as the basis in random space. The algorithm is applied to micro-channel flows with random wall boundary conditions, and to external flows with random freestream. Efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by Monte Carlo simulations. It is shown that the generalized polynomial chaos method promises a substantial speed-up compared with the Monte Carlo method. The utilization of different type orthogonal polynomials from the Askey scheme also provides a more efficient way to represent general non-Gaussian processes compared with the original Wiener-Hermite expansions.
Le Maitre, O.P., Knio, O.M., Debusschere, B.J., Najm, H.N., and Ghanem, R.G.. "A multigrid solver for two-dimensional stochastic diffusion equations" Methods in Applied Mechanics and Engineering.
vol. 192.
2003.
pp. 4723--4744.
Schwab, C. and Todor, R.A.. "Sparse finite elements for stochastic elliptic problems: higher order moments" Computing. 71
(1).
2003.
pp. 43--63.
We define the higher order moments associated to the stochastic solution of an elliptic BVP in D â Rd with stochastic input data. We prove that the k-th moment solves a deterministic problem in Dk â Rdk, for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting system.
Ghanem, R.G. and Sarkar, A.. "Reduced models for the medium-frequency dynamics of stochastic systems" Journal of the Acoustical Society of America. 113
(2).
FEB 2003.
pp. 834--846.
In this paper, a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium-frequency range. In this frequency range, both traditional modal analysis and statistical energy analysis (SEA) procedures well-suited for low- and high-frequency vibration analysis respectively, lead to computational and conceptual difficulties. The uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical. The methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods. In particular, the Karhunen-Loeve and Polynomial Chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest. (C) 2003 Acoustical Society of America.
Lucor, D., Xiu, D., Su, C.-H., and Karniadakis, G.E.. "Predictability and uncertainty in CFD" International Journal for Numerical Methods in Fluids. 43
(5).
OCT 20 2003.
pp. 483--505.
CFD has reached some degree of maturity today, but the new question is how to construct simulation error bars that reflect uncertainties of the physical problem, in addition to the usual numerical inaccuracies. We present a fast Polynomial Chaos algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by Wiener-Hermite functionals, and the governing equations are formulated by employing Galerkin projections. The resulted system is deterministic, and therefore existing solvers can be used in this new context of stochastic simulations. The algorithm is applied to a second-order oscillator and to a flow-structure interaction problems. Open issues and extensions to general random distributions are presented. Copyright (C) 2003 John Wiley Sons, Ltd.
Gerstner, T. and Griebel, M.. "Dimension-adaptive tensor-product quadrature" Computing. 71
(1).
SEP 2003.
pp. 65--87.
We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself. The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm. The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.
Bungartz, H.J. and Dirnstorfer, S.. "Multivariate quadrature on adaptive sparse grids" Computing. 71
(1).
SEP 2003.
pp. 89--114.
In this paper, we study the potential of adaptive sparse grids for multivariate numerical quadrature in the moderate or high dimensional case, i.e. for a number of dimensions beyond three and up to several hundreds. There, conventional methods typically suffer from the curse of dimension or are unsatisfactory with respect to accuracy. Our sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points. The presented algorithm is applied to some test problems and compared with other existing methods.
Anile, A.M., Spinella, S., and Rinaudo, S.. "Stochastic response surface method and tolerance analysis in microelectronics" COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering. 22
(2).
2003.
pp. 314--327.
Tolerance analysis is a very important tool for chip design in the microelectronics industry. The usual method for tolerance analysis is Monte Carlo simulation, which, however, is extremely CPU intensive, because in order to yield statistically significant results, it needs to generate a large sample of function values. Here we report on another method, recently introduced in several fields, caged stochastic response surface method, which might be a viable alternative to Monte Carlo simulation for some classes of problems. The application considered here is on the tolerance analysis of the current of a submicrometer n(+)-n-n(+) diode as a function of the channel length and the channel doping. The numerical simulator for calculating the current is based on the energy transport hydrodynamical model introduced by Stratton, which is one of the most widely used in this field.