Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
1
3
2011
A STOCHASTIC FINITE-ELEMENT METHOD FOR TRANSFORMED NORMAL RANDOM PARAMETER FIELDS
189-201
10.1615/Int.J.UncertaintyQuantification.v1.i3.10
Carsten
Proppe
Institute of Engineering Mechanics, Karlsruhe Institute of Technology, Germany
polynomial chaos
stochastic finite-element method
stochastic sparse grid collocation
Smolyak algorithm
stochastic response surface method
adaptivity
random fields
Transformed normal random fields are convenient models, e.g., for random material property fields obtained from microstructure analysis. In the context of the stochastic finite-element (FE) method, discretization of non-normal random fields by polynomial chaos expansions has been frequently employed. This introduces a non-linear relationship between the system matrix and normal random variables. For transformed normal random fields, the truncated Karhunen-Loeve expansion of the transformed field is introduced into the stochastic FE formulation. This leads to a linear dependence of the system matrix on non-normal random variables. These non-normal random variables are then utilized to represent the discretized solution of the stochastic boundary value problem. Introduction of the approximations into the variational formulation of the stochastic boundary value problem and application of a collocation scheme yields a nonintrusive algorithm that allows coupling of reliability estimation procedures and existing FE solvers.
RECONSTRUCTION OF DOMAIN BOUNDARY AND CONDUCTIVITY IN ELECTRICAL IMPEDANCE TOMOGRAPHY USING THE APPROXIMATION ERROR APPROACH
203-222
10.1615/Int.J.UncertaintyQuantification.v1.i3.20
Antti
Nissinen
Department of Applied Physics, University of Eastern Finland, Kuopio, Finland
Ville
Kolehmainen
Department of Applied Physics
University of Kuopio
P.O.B. 1627, FI-70211 Kuopio, Finland
Jari P.
Kaipio
Department of Mathematics, University of Auckland, New Zealand; and Department of Physics and Mathematics, University of Eastern Finland
inverse problems
electrical impedance tomography
statistical inversion
modelling errors
inaccurately known boundary
model reduction
Electrical impedance tomography (EIT) is a highly unstable problem with respect to measurement and modeling errors.
With clinical measurements, knowledge about the body shape is usually uncertain. Since the use of an incorrect model
domain in the measurement model is bound to lead to severe estimation errors, one possibility is to estimate both the
conductivity and parametrization of the domain boundary. This could in principle be carried out using the Bayesian
inversion paradigm and Markov chain Monte Carlo sampling, but such an approach would lead in clinical situation to
an impractical solution because of the excessive computational complexity. In this paper, we adapt the so-called approximation error approach for approximate recovery of the domain boundary and the conductivity. In the approximation
error approach, the modeling error caused by an inaccurately known boundary is treated as an auxiliary noise process
in the measurement model and sample statistics for the noise process are estimated based on the prior models of the
conductivity and boundary shape. Using the approximation error model, we reconstruct the conductivity and a low
rank approximation for the realization of the modeling error, and then recover an approximation for the domain boundary using the joint distribution of the modeling error and the boundary parametrization. We also compute approximate
spread estimates for the reconstructed boundary. We evaluate the approach with simulated examples of thorax imaging
and also with experimental data from a laboratory setting. The reconstructed boundaries and posterior uncertainty are
feasible; in particular, the actual domain boundaries are essentially within the posterior spread estimates.
POLYNOMIAL CHAOS FOR LINEAR DIFFERENTIAL ALGEBRAIC EQUATIONS WITH RANDOM PARAMETERS
223-240
10.1615/Int.J.UncertaintyQuantification.v1.i3.30
Roland
Pulch
Institute for Mathematics and Computer Science, University of Greifswald,
Walther-Rathenau-Str. 47, D-17489 Greifswald, Germany
polynomial chaos
differential algebraic equations
index
consistency of initial values
stochastic Galerkin method
uncertainty quantification
Technical applications are often modeled by systems of differential algebraic equations. The systems may include parameters that involve some uncertainties. We arrange a stochastic model for uncertainty quantification in the case of linear systems of differential algebraic equations. The generalized polynomial chaos yields a larger linear system of differential algebraic equations, whose solution represents an approximation of the corresponding random process. We prove sufficient conditions such that the larger system inherits the index of the original system. Furthermore, the choice of consistent initial values is discussed. Finally, we present numerical simulations of this stochastic model.
ORTHOGONAL POLYNOMIAL EXPANSIONS FOR SOLVING RANDOM EIGENVALUE PROBLEMS
241-255
10.1615/Int.J.UncertaintyQuantification.v1.i3.40
George N.
Karystinos
Department of Electronic and Computer Engineering, Technical University of Crete, Kounoupidiana, Chania, 73100, Greece
Keith R.
Dalbey
Department of Optimization and Uncertainty Quantification, Sandia National Laboratories, Albuquerque, New Mexico 87123, USA
uncertainty quantification
Monte Carlo
Latin hypercube sampling
space-filling
computational design
high-dimensional methods
regularity detection
In the literature, space-filling Latin hypercube sample designs typically are generated by optimizing some criteria such as maximizing the minimum distance between points or minimizing discrepancy. However, such methods are time consuming and frequently produce designs that are highly regular, which can bias results. A fast way to generate irregular space-filling Latin hypercube sample designs is to randomly distribute the sample points to a pre-selected set of well-spaced bins. Such designs are said to be ”binning optimal” and are shown to be irregular. Specifically, Fourier analysis reveals regular patterns in the multi-dimensional spacing of points for the Sobol sequence but not for Binning optimal symmetric Latin hypercube sampling. For M = 2r ≤ 8 dimensions and N = 2s ≥ 2M points, where r and s are non-negative integers, simple patterns can be used to create a list of maximally spaced bins. Good Latin hypercube sample designs for non-power of two dimensions can be generated by discarding excess dimensions. Since the octants/bins containing the 2M end points of an ”orientation” (a rotated set of orthogonal axes) are maximally spaced, the process of generating the list of octants simplifies to finding a list of maximally spaced orientations. Even with this simplification, the ”patterns” for maximally spaced bins in M ≥ 16 dimensions are not so simple. In this paper, we use group theory to generate 2M / (2M) disjoint orientations, and present an algorithm to sort these into maximally spaced order. Conceptually, the procedure works for arbitrarily large numbers of dimensions. However, memory requirements currently preclude even listing the 2M / (2M) orientation leaders for M ≥ 32 dimensions. In anticipation of overcoming this obstacle, we outline a variant of the sorting algorithm with a low memory requirement for use in M ≥ 32 dimensions.
PINK NOISE, 1/fα NOISE, AND THEIR EFFECT ON SOLUTIONS OF DIFFERENTIAL EQUATIONS
257-278
10.1615/Int.J.UncertaintyQuantification.2011003089
Miroslav
Stoyanov
Applied Mathematics Group, Computer Science and Mathematics Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, P.O. Box 2008, Oak Ridge TN 37831-6164
Max
Gunzburger
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306-4120
John
Burkardt
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306-4120, USA
random fields
colored noise
pink noise
stochastic finite element method
White noise is a very common way to account for randomness in the inputs to partial differential equations, especially in cases where little is know about those inputs. On the other hand, pink noise, or more generally, colored noise having a power spectrum that decays as 1/fα, where f denotes the frequency and α Є (0; 2] has been found to accurately model many natural, social, economic, and other phenomena. Our goal in this paper is to study, in the context of simple linear and nonlinear two-point boundary-value problems, the effects of modeling random inputs as 1/fα random fields, including the white noise (α = 0), pink noise (α = 1), and brown noise (α = 2) cases. We show how such random fields can be approximated so that they can be used in computer simulations. We then show that the solutions of the differential equations exhibit a strong dependence on α, indicating that further examination of how randomness in partial differential equations is modeled and simulated is warranted.