A general discussion of the quantification of uncertainty in numerical simulations is presented. A principal conclusion is that the distribution of solution errors is the leading term in the assessment of the validity of a simulation and its associated uncertainty in the Bayesian framework. Key issues that arise in uncertainty quantification are discussed for two examples drawn from shock wave physics and modeling of petroleum reservoirs. Solution error models, confidence intervals and Gaussian error statistics based on simulation studies are presented.

Issue Section:

Technical Papers
1.

S. French and J. Q. Smith, eds., 1997,

*The Practice of Bayesian Analysis*, Arnold, London.2.

Hadamard

, J.

, 1936

, “Equations aux de´rive´es partielles. les conditions de´finies en ge´ne´ral. le cas hyperboliqaue

,” Enseignement Math.

, 35

, pp. 5

–42

.3.

R. Courant and D. Hilbert, 1962,

*Methods of Mathematical Physics II*. Interscience, New York.4.

J. Glimm and D. H. Sharp, 1997, “Stochastic partial differential equations: Selected applications in continuum physics,” R. A. Carmona and B. L. Rozovskii, eds.,

*Stochastic Partial Differential Equations: Six Perspectives*, Mathematical Surveys and Monographs, American Mathematical Society, Providence.5.

Glimm, J., and Sharp, D. H., 1997, “Multiscale science,” SIAM News, Oct.

6.

Glimm

, J.

, and Sharp

, D. H.

, 1998

, “Stochastic methods for the prediction of complex multiscale phenomena

,” Q. Appl. Math.

, 56

, pp. 741

–765

.7.

Glimm

, J.

, and Sharp

, D. H.

, 1999

, “Prediction and the quantification of uncertainty

,” Physica D

, 133

, pp, 152

–170

.8.

I. Babuska, A. Miller, and M. Vogelius, 1983, “Adaptive methods and error estimation for elliptic problems of structural mechanics,” In I. Babuska, J. Chandra, and J. Flaherty, eds.,

*Adaptive Computational Methods for Partial Differential Equations*, pp. 57–73. SIAM, Philadelphia.9.

Babuska

, I.

, and Rheinboldt

, W.

, 1978

, “Error estimates in adaptive finite element computations

,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

, 15

, pp. 736

–754

.10.

Cockburn

, B.

, 1998

, “A simple introduction to error estimation for nonlinear hyperbolic conservation laws,” Technical report, University of Minnesota.

11.

W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, ed. 1996,

*Markov Chain Monte Carlo in Practice*, Chapman and Hall, London and New York.12.

Chu

, K.

, Deng

, Y.

, and Reinitz

, J.

, 1999

, “Parallel simulated annealing by mixing of states

,” J. Comput. Phys.

, 148

, pp. 646

–662

.13.

McKay

, M. D.

, Conover

, W. J.

, and Beckman

, R. J.

, 1979

, “A comparison of three methods for selecting values of INput variables in the analysis of output from a computer code

,” Technometrics

, 21

, pp. 239

–245

.14.

Stein

, M.

, 1987

, “Large sample properties of simulations using latin hypercube sampling

,” Technometrics

, 29

, pp. 143

–151

.15.

Owen

, A. B.

, 1992

, “Orthogonal arrays for computer experiments, integration and visualization

,” Statistica Sinica

, 2

, pp. 439

–452

.16.

Tanveer

, S.

, 1993

, “Singularities in the classical Rayleigh-Taylor flow: Formation and subsequent motion

,” Proc. R. Soc. London, Ser. A

, 441

, pp. 501

–525

.17.

Ye

, K. Q.

, 1998

, “Column orthogonal latin hypercube design and their application in computer experiments

,” J. Am. Stat. Assoc.

, 93

, pp. 1430

–1439

.18.

Owen

, A. B.

, 1994

, “Lattice sampling revised: Monte carlo variance of means over randomized orthogonal arrays

,” Appl. Ocean. Res.

, 22

, pp. 930

–945

.19.

Harald Niederreiter, 1992,

*Random number generation and quasi-Monte Carlo methods*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.20.

J. R. Koehler and A. B. Owen, 1996, “Computer experiments,”

*Design and analysis of experiments*, pp. 261–308, North-Holland, Amsterdam.21.

Sacks

, J.

, Welch

, W.

, Mitchell

, T.

, and Wynn

, P.

, 1989

, “Design and analysis of computer experiments

,” Stat. Sci.

, 4

, pp. 409

–435

.22.

Cheng

, B.

, Glimm

, J.

, and Sharp

, D. H.

, 2000

, “Density dependence of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts

,” Phys. Lett. A

, 268

, pp. 366

–374

.23.

Wallstrom

, T.

, Hou

, S.

, Christie

, M. A.

, Durlofsky

, L. J.

, and Sharp

, D. H.

, 1999

, “Accurate scale up of two phase flow using renormalization and nonuniform coarsening

,” Comput. Geosci.

, 3

, pp. 69

–87

.24.

T. Wallstrom, S. Hou, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, 1999, “Application of a new two-phase upscaling technique to realistic reservoir cross sections,”

*Proceedings of the SPE 15th Symposium on Reservoir Simulation*, pp. 451–462, SPE 51939.25.

Wallstrom, T., Hou, S., Christie, M. A., Durlofsky, L. J., Sharp, D. H., and Zou, Q., 2001, “Effective medium boundary conditions for upscaling relative permeabilities,” Transp. Porous Media, Accepted for Publication.

26.

M. A. Christie, T. C. Wallstrom, L. J. Durlofsky S. Hou, D. H. Sharp, and Q. Zou, 2000, “Effective medium boundary conditions in upscaling,”

*Proceedings of the 7th European Conference on the Mathematics of Oil Recovery, Baveno, Italy, Sept. 5–8*.27.

J. Glimm, S. Hou, H. Kim, D. Sharp, and K. Ye, 1999, “A probability model for errors in the numerical solutions of a partial differential equation,”

*CFD Journal*, 9, 2000. Proceedings of the 8th International Symposium on Computational Fluid Dynamics, Bremen, Germany, SUNY-SB Preprint Number SUNYSB-AMS-99-11.28.

Glimm, J., Hou, S., Kim, H., Lee, Y., Sharp, D., Ye, K., and Zou, Q., 2000, “Risk management for petroleum reservoir production: A simulation-based study of prediction,” Report No. SUNYSB-00-12, State University of New York at Stony Brook, J. Comp. Geosciences (to appear).

29.

J. Glimm, S. Hou, Y. Lee, D. Sharp, and K. Ye, 2001. “Prediction of oil production with confidence intervals,” SPE 66350, SPE Reservoir Simulation Symposium held in Houston, Texas, 11–14 Feb.

30.

R. Richtmyer and K. Morton, 1967,

*Difference Methods for Initial Value Problems*, Interscience, New York, second edition.31.

E. Godlewski and P. A. Raviart, 1991,

*Numerical Approximation of Hyperbolic Systems of Conservation Laws*, Springer Verlag, New York.32.

Riemann

, B.

, 1958

, “Uber die fortpflanzung ebener luftwellen von endlicher schwingungsweite

,” Go¨ttingen Abhandlungen

, 8

, pp. 43

43

.33.

Glimm

, J.

, 1965

, “Solutions in the large for nonlinear hyperbolic systems of equations

,” Commun. Pure Appl. Math.

, 18

, pp. 697

–715

.34.

Glimm

, J.

, and Sharp

, D.

, 1986

, “An S-matrix theory for classical nonlinear physics

,” Found. Phys.

, 16

, pp. 125

–141

.35.

Goduno

, S. K.

, 1959

, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics

,” Mat. Sb.

, 47

, pp. 271

–306

.36.

van Leer

, B.

, 1979

, “Towards the ultimate conservative difference scheme: V. A second order sequel to Godunov’s method

,” J. Comput. Phys.

, 32

, pp. 101

–136

.37.

R. J. LeVeque, J. O. Langseth, M. Berger, and S. Mitran, “CLAWPACK, conservation law package,” http://www.amath.washington.edu/∼claw/.

38.

O’Rourke

, P. J.

, and Sahota

, M. S.

, 1998

, “A variable explicit/implicit numerical method for calculating advection on unstructured meshes

,” J. Comput. Phys.

, 143

, p. 312

312

.39.

DeVolder, B. G., Sahota, M. S., and Cline, M. C., 2000, “Verification of the CHAD hydrodynamics code,” Report No. LA-UR-00-6039, Los Alamos National Laboratory, Los Alamos, NM.

40.

Grove

, J. W.

, 1994

, “Applications of front tracking to the simulation of shock refractions and unstable mixing

,” J. Appl. Num. Math.

, 14

, pp. 213

–237

.41.

Glimm

, J.

, Grove

, J. W.

, Li

, X. L.

, Oh

, W.

, and Sharp

, D. H.

, 2001

, “A critical analysis of Rayleigh-Taylor growth rates

,” J. Comput. Phys.

, 169

, pp. 652

–677

, LANL report No. LA-UR-99-5582.42.

The MathWorks. MATLAB. http://www.mathworks.com/.

43.

M. P. Wand and M. C. Jones, 1995,

*Kernel Smoothing*. Chapman & Hall, 2–6 Boundary Row, London SE1 8HN, UK.Copyright © 2002

by ASME

You do not currently have access to this content.