Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
1
1
2011
MARGINALIZATION OF UNINTERESTING DISTRIBUTED PARAMETERS IN INVERSE PROBLEMS-APPLICATION TO DIFFUSE OPTICAL TOMOGRAPHY
1-17
Ville
Kolehmainen
University of Eastern Finland, Department of Applied Physics, P.O.B. 1627, FI-70211 Kuopio, Finland
Tanja
Tarvainen
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 1627, 70211 Kuopio, Finland
Simon R.
Arridge
Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
Jari P.
Kaipio
Department of Mathematics, University of Auckland, New Zealand; and Department of Physics and Mathematics, University of Eastern Finland
With inverse problems there are often several unknown distributed parameters of which only one may be of interest. Since assigning incorrect fixed values to the uninteresting parameters usually leads to a severely erroneous model, one is forced to estimate all distributed parameters simultaneously. This may increase the computational complexity of the problem significantly. In the Bayesian framework, all unknowns are generally treated as random variables and estimated simultaneously and all uncertainties can be modeled systematically. Recently, the approximation error approach has been proposed for handling uncertainty and model-reduction-related errors in the models. In this approach approximate marginalization of these errors is carried out before the estimation of the interesting variables. In this paper we discuss the adaptation of the approximation error approach to the marginalization of uninteresting distributed parameters. As an example, we consider the marginalization of scattering coefficient in diffuse optical tomography.
ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA
19-33
Raymond S.
Tuminaro
Sandia National Laboratories, PO Box 969, MS 9159, Livermore, CA 94551, USA
Eric T.
Phipps
Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico
87185, USA
Christopher W.
Miller
Department of Applied Mathematics and Scientific Computation, University of Maryland, USA
Howard C.
Elman
Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, USA
We compare the performance of two methods, the stochastic Galerkin method and the stochastic collocation method, for solving partial differential equations (PDEs) with random data. The stochastic Galerkin method requires the solution of a single linear system that is several orders larger than linear systems associated with deterministic PDEs. The stochastic collocation method requires many solves of deterministic PDEs, which allows the use of existing software. However, the total number of degrees of freedom in the stochastic collocation method can be considerably larger than the number of degrees of freedom in the stochastic Galerkin system. We implement both methods using the Trilinos software package and we assess their cost and performance. The implementations in Trilinos are known to be efficient, which allows for a realistic assessment of the computational complexity of the methods. We also develop a cost model for both methods which allows us to examine asymptotic behavior.
PROBABILISTIC PREDICTIONS OF INFILTRATION INTO HETEROGENEOUS MEDIA WITH UNCERTAIN HYDRAULIC PARAMETERS
35-47
Peng
Wang
Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA
Daniel M.
Tartakovsky
Department of Energy Resources Engineering, Stanford University, USA
Soil heterogeneity and the lack of detailed site characterization are two ubiquitous factors that render predictions of flow and transport in the vadose zone inherently uncertain. We employ the Green-Ampt model of infiltration and the Dagan- Bresler statistical parameterization of soil properties to compute probability density functions (PDFs) of infiltration rate and infiltration depth. By going beyond uncertainty quantification approaches based on mean and variance of system states, these PDF solutions enable one to evaluate probabilities of rare events that are required for probabilistic risk assessment. We investigate the temporal evolution of the PDFs of infiltration depth and corresponding infiltration rate, the relative importance of uncertainty in various hydraulic parameters and their cross-correlation, and the impact of the choice of a functional form of the hydraulic function.
ASSIMILATION OF COARSE-SCALEDATAUSINGTHE ENSEMBLE KALMAN FILTER
49-76
Yalchin
Efendiev
Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia; Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
A.
Datta-Gupta
Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843, USA
Santha
Akella
The Johns Hopkins University, Baltimore, MD 21218, USA
Reservoir data is usually scale dependent and exhibits multiscale features. In this paper we use the ensemble Kalman filter (EnKF) to integrate data at different spatial scales for estimating reservoir fine-scale characteristics. Relationships between the various scales is modeled via upscaling techniques. We propose two versions of the EnKF to assimilate the multiscale data, (i) where all the data are assimilated together and (ii) the data are assimilated sequentially in batches. Ensemble members obtained after assimilating one set of data are used as a prior to assimilate the next set of data. Both of these versions are easily implementable with any other upscaling which links the fine to the coarse scales. The numerical results with different methods are presented in a twin experiment setup using a two-dimensional, two-phase (oil and water) flow model. Results are shown with coarse-scale permeability and coarse-scale saturation data. They indicate that additional data provides better fine-scale estimates and fractional flow predictions. We observed that the two versions of the EnKF differed in their estimates when coarse-scale permeability is provided, whereas their results are similar when coarse-scale saturation is used. This behavior is thought to be due to the nonlinearity of the upscaling operator in the case of the former data. We also tested our procedures with various precisions of the coarse-scale data to account for the inexact relationship between the fine and coarse scale data. As expected, the results show that higher precision in the coarse-scale data yielded improved estimates. With better coarse-scale modeling and inversion techniques as more data at multiple coarse scales is made available, the proposed modification to the EnKF could be relevant in future studies.
ON A POLYNOMIAL CHAOS METHOD FOR DIFFERENTIAL EQUATIONS WITH SINGULAR SOURCES
77-98
Yunfei
Song
Department of Mathematics, State University of New York at Buffalo, USA
Jae-Hun
Jung
Department of Mathematics, State University of New York at Buffalo, USA
Singular source terms in the differential equation represented by the Dirac δ-function play a crucial role in determining
the global solution. Due to the singular feature of the δ-function, physical parameters associated with the δ-function are
highly sensitive to random and measurement errors, which makes the uncertainty analysis necessary. In this paper we
use the generalized polynomial chaos method to derive the general solution of the differential equation under uncertainties
associated with the δ-function. For simplicity, we assume the uniform distribution of the random variable and use
the Legendre polynomials to expand the solution in the random space. A simple differential equation with the singular
source term is considered. The polynomial chaos solution is derived. The Gibbs phenomenon and the convergence of
high order moments are discussed. We also consider a direct collocation method which can avoid the Gibbs oscillations
on the collocation points and enhance the accuracy accordingly.