Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
2
3
2012
PREFACE
vii
10.1615/Int.J.UncertaintyQuantification.v2.i3.10
Nicholas
Zabaras
Department of Mechanical and Aerospace Engineering, Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA; University of Warwick, Coventry CV4 7AL, UK
USA/SOUTH AMERICA SYMPOSIUM ON STOCHASTIC MODELING AND UNCERTAINTY QUANTIFICATION, LEBLON BEACH, RIO DE JANEIRO, BRAZIL, AUGUST 1-5, 2011
DIFFERENTIAL CONSTRAINTS FOR THE PROBABILITY DENSITY FUNCTION OF STOCHASTIC SOLUTIONS TO THE WAVE EQUATION
195-213
10.1615/Int.J.UncertaintyQuantification.2011003485
Daniele
Venturi
Department of Applied Mathematics, University of California Santa Cruz, Santa Cruz, CA 95064, USA
George Em
Karniadakis
School of Engineering, Brown University, Providence, Rhode Island, 02912,
USA; Division of Applied Mathematics, Brown University, Providence, Rhode Island,
02912, USA
stochastic partial differential equations
high-dimensional methods
random fields
By using functional integral methods we determine new types of differential constraints satisfied by the joint probability density function of stochastic solutions to the wave equation subject to uncertain boundary and initial conditions. These differential constraints involve unusual limit partial differential operators and, in general, they can be grouped into two main classes: the first one depends on the specific field equation under consideration (i.e., on the stochastic wave equation), the second class includes a set of intrinsic relations determined by the structure of the joint probability density function of the wave and its derivatives. Preliminary results we have obtained for stochastic dynamical systems and first-order nonlinear stochastic particle differential equations (PDEs) suggest that the set of differential constraints is complete and, therefore, it allows determining uniquely the probability density function of the solution to the stochastic problem. The proposed new approach can be extended to arbitrary nonlinear stochastic PDEs and it could be an effective way to overcome the curse of dimensionality for random boundary and initial conditions. An application of the theory developed is presented and discussed for a simple random wave in one spatial dimension.
PARALLEL ADAPTIVE MULTILEVEL SAMPLING ALGORITHMS FOR THE BAYESIAN ANALYSIS OF MATHEMATICAL MODELS
215-237
10.1615/Int.J.UncertaintyQuantification.2011003499
Ernesto
Prudencio
Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, Texas, USA
Sai Hung
Cheung
School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore
computational statistics
Markov chain Monte Carlo
adaptivity
Bayesian inference
stochastic modeling
model calibration
In recent years, Bayesian model updating techniques based on measured data have been applied to many engineering and applied science problems. At the same time, parallel computational platforms are becoming increasingly more powerful and are being used more frequently by the engineering and scientific communities. Bayesian techniques usually require the evaluation of multi-dimensional integrals related to the posterior probability density function (PDF) of uncertain model parameters. The fact that such integrals cannot be computed analytically motivates the research of stochastic simulation methods for sampling posterior PDFs. One such algorithm is the adaptive multilevel stochastic simulation algorithm (AMSSA). In this paper we discuss the parallelization of AMSSA, formulating the necessary load balancing step as a binary integer programming problem. We present a variety of results showing the effectiveness of load balancing on the overall performance of AMSSA in a parallel computational environment.
STATE ESTIMATION PROBLEMS IN HEAT TRANSFER
239-258
10.1615/Int.J.UncertaintyQuantification.2012003582
Helcio R. B.
Orlande
Department of Mechanical Engineering, Politecnica/COPPE, Federal University of Rio de Janeiro - UFRJ, Caixa Postal: 68503, Cidade Universitaria, Rio de Janeiro, RJ, 21941-972, Brazil; Nanotechnology Eng. Program, COPPE, Federal University of Rio de Janeiro, Brazil
Marcelo Jose
Colaço
Department of Mechanical Engineering, Federal University of Rio de Janeiro, UFRJ, Technology Center, Bloco G – Cidade Universitaria da UFRJ, Ilha do Governador, Rio de Janeiro/RJ, 21941-909, Brazil
George S.
Dulikravich
none (retired)
Flavio
Vianna
Department of Subsea Technology, Petrobras Research and Development Center−CENPES, Av. Horacio Macedo, 950, Cidade Universitaria, Ilha do Fundao, 21941-915, Rio de Janeiro, RJ, Brazil
Wellington Betencurte
da Silva
Department of Mechanical Engineering, Politecnica/COPPE, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, RJ, 21941-972, Brazil; Universite de Toulouse, Mines Albi, CNRS, Centre RAPSODEE, Campus Jarlard, F-81013 Albi cedex 09, France; Mechanical Engineering Department, Federal University of Espirito Santo, UFES. Alto Universitario, Guararema, Alegre/ES, 29.500-999, Brazil
Henrique M.
Fonseca
Department of Mechanical Engineering, Politecnica/COPPE, Federal University of Rio de Janeiro, UFRJ, Cid. Universitaria, Cx. Postal: 68503, Rio de Janeiro, RJ, 21941-972, Brazil; Department of Mechanical Engineering, Fluminense Federal University - Volta Redonda, Brazil
Olivier
Fudym
Universite de Toulouse, Mines Albi, CNRS, Centre RAPSODEE, Campus Jarlard, F-81013
Albi cedex 09, France
Kalman filter
inverse problems
particle filters
fluid mechanics
heat transfer
The objective of this paper is to introduce applications of Bayesian filters to state estimation problems in heat transfer. A brief description of state estimation problems within the Bayesian framework is presented. The Kalman filter, as well as the following algorithms of the particle filter: sampling importance resampling and auxiliary sampling importance resampling, are discussed and applied to practical problems in heat transfer.
EFFECTIVE PARAMETRIZATION FOR RELIABLE RESERVOIR PERFORMANCE PREDICTIONS
259-278
10.1615/Int.J.UncertaintyQuantification.2012003765
Xiao-Hui
Wu
ExxonMobil Upstream Research Company, P. O. Box 2189, Houston, TX 77252, USA
Linfeng
Bi
ExxonMobil Upstream Research Company, P. O. Box 2189, Houston, TX 77252, USA
Subhash
Kalla
ExxonMobil Upstream Research Company, P. O. Box 2189, Houston, TX 77252, USA
heterogeneous random media
uncertainty quantification
multiscale modeling
porous media flow
reservoir engineering
multiscale estimation
The purpose of reservoir modeling and simulation is to predict reservoir performance for development and depletion planning. Despite decades of research, efficient and reliable reservoir performance predictions are still a challenge in practice. In this paper, we present an overview of reservoir modeling as it is commonly practiced today and the challenges it faces. More specifically, we focus on the challenges posed by the large amount of uncertainty inherent in the characterization of reservoirs that are heterogeneous at multiple scales. We discuss the practical implications of these challenges and recent developments toward addressing them. In particular, we examine the need for effective parametrization of geologic concepts and related recent advances in parametrization and parameter reduction techniques, including their advantages and limitations. Using numerical examples from two different depositional environments, we show that effective parametrization can be achieved by taking advantage of the geologic hierarchy underlying most geologic concepts and a general understanding of the impact of geologic features on fluid flow. Finally, we propose an approach to systematically derive fit-for-purpose parametrization for practical reservoir modeling problems.
STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁1-MINIMIZATION
279-293
10.1615/Int.J.UncertaintyQuantification.2012003925
Liang
Yan
Department of Mathematics, Southeast University, Nanjing, 210096, China
Ling
Guo
Department of Mathematics, Shanghai Normal University, No. 100 Guilin Road, Shanghai 200234, China
Dongbin
Xiu
Ohio State University
stochastic collocation
Legendre polynomials
𝓁1-minimization
multi-dimensional interpolation
The idea of 𝓁1-minimization is the basis of the widely adopted compressive sensing method for function approximation.
In this paper, we extend its application to high-dimensional stochastic collocation methods. To facilitate practical
implementation, we employ orthogonal polynomials, particularly Legendre polynomials, as basis functions, and focus
on the cases where the dimensionality is high such that one can not afford to construct high-degree polynomial approximations.
We provide theoretical analysis on the validity of the approach. The analysis also suggests that using
the Chebyshev measure to precondition the 𝓁1-minimization, which has been shown to be numerically advantageous in
one dimension in the literature, may in fact become less efficient in high dimensions. Numerical tests are provided to
examine the performance of the methods and validate the theoretical findings.
SENSITIVITY ANALYSIS FOR THE OPTIMIZATION OF RADIOFREQUENCY ABLATION IN THE PRESENCE OF MATERIAL PARAMETER UNCERTAINTY
295-321
10.1615/Int.J.UncertaintyQuantification.2012004135
Inga
Altrogge
Center of Complex Systems and Visualization, University of Bremen, Germany
Tobias
Preusser
Fraunhofer MEVIS; and School of Engineering and Science, Jacobs University, Bremen, Germany
Tim
Kroger
Georg Simon Ohm University of Applied Sciences, Nuremberg, Germany
Sabrina
Haase
Fraunhofer Institute for Medical Image Computing MEVIS, Bremen, Germany
Torben
Patz
School of Engineering and Science, Jacobs University Bremen; Fraunhofer Institute for Medical Image Computing MEVIS, Bremen, Germany
Robert Mike
Kirby
School of Computing, and Scientific Computing and Imaging (SCI) Institute, Salt Lake City,
Utah, 84112, USA
stochastic sensitivity analysis
stochastic partial differential equations
adaptive sparse grid
heat transfer
multiscale modeling
representation of uncertainty
We present a sensitivity analysis of the optimization of the probe placement in radiofrequency (RF) ablation which takes the uncertainty associated with biophysical tissue properties (electrical and thermal conductivity) into account. Our forward simulation of RF ablation is based upon a system of partial differential equations (PDEs) that describe the electric potential of the probe and the steady state of the induced heat. The probe placement is optimized by minimizing a temperature-based objective function such that the volume of destroyed tumor tissue is maximized. The resulting optimality system is solved with a multilevel gradient descent approach. By evaluating the corresponding optimality system for certain realizations of tissue parameters (i.e., at certain, well-chosen points in the stochastic space) the sensitivity of the system can be analyzed with respect to variations in the tissue parameters. For the interpolation in the stochastic space we use an adaptive sparse grid collocation (ASGC) approach presented by Ma and Zabaras. We underscore the significance of the approach by applying the optimization to CT data obtained from a real RF ablation case.