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Advanced Scientific Computing
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Uncertainty Quantification
J. Glimm and Y. Yu
Quantification of Uncertainty
and Computer Assisted Decision Making
The need for computer assisted decision making is driven by two
related factors. The first is the importance of complex
scientific/technical decisions, such as those related to global warming,
for which controlled experiments are not feasible. The second is the
need for rapid or timely decisions, using incomplete information, such
as in shortening the time to market of a product design cycle, mandating
a reduction of the role of the human in the loop.
A key issue, and the central one considered here, is an accurate
assessment of errors in numerical simulations. Uncertainty
quantification (UQ) can be viewed as the process of adding error bars to
a simulation prediction. The error bars refer to all sources of
uncertainty in the prediction, including data, physics and numerical
modeling error. Our approach to uncertainty quantification uses a
Bayesian framework. Specifically the Bayesian likelihood is (up to
normalization) a probability, which specifies the probability of
occurrence of an error of any given size. Our approach is to use
solution error models as defining one contribution to this likelihood.
We provide a scientific basis for the probabilities associated with
numerical solution errors.
We have studied UQ for shock physics simulations [1,2,3], with a focus
on statistical analysis of errors in numerical solutions. We decomposed
the total simulation error into components attributed to various
subproblems, using wave filters to locate significant shock and contact
waves. In 1D, a simple error model described the errors introduced at
each wave interaction, and (for a spherical geometry) the power law
growth or decay of the errors propagated between interactions. A
scattering type formula was derived to allow summation of errors
propagated through individual interactions to a final error after many
interactions [1,2].
The 2D shock interaction problem leads to chaotic interfacial mixing;
the central UQ problem here is to define a methodology to describe
solution errors for chaotic flow regimes. The challenge is to establish
a reliable error analysis for chaotic simulations that do not converge
in a pointwise sense, but rather add new complexity with each new level
of mesh refinement (see Figure 1). The solution to this conundrum is to
look for convergence in averaged quantities, i.e., the statistical
moments, and the spatial, temporal, and ensemble averages that will
define them. 2D wave filters are used to obtain flow regions having a
comparable flow history. Within a single homogeneous flow region, we
find that a modest amount of averaging leads to convergent flow
quantities under mesh refinement [3].
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Figure 1. Density
plot for a spherical implosion simulation with a perturbed
interface. The outer orange-blue boundary is the edge of the
computational domain. The red-orange circular boundary is an
outgoing reflected shock, and the chaotic inner interface is the
object of study. The grid size is 800 x 1600. |
References
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[1] Glimm, J., Grove, J.W., Kang, Y., Lee, T.W., Li, X., Sharp, D.H.,
Yu, Y., Ye, K., and Zhao, M. Statistical Riemann problems and a
composition law for errors in numerical solutions of shock physics
problems. SISC 26: 666-697 (2004). University at Stony Brook preprint
number SB-AMS-03-11 and LANL report number LA-UR-03-2921.
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[2] Glimm, J., Grove, J.W., Kang, Y., Lee, T., Li, X., Sharp, D.H., Yu,
Y., and Zhao, M. Errors in numerical solutions of spherically symmetric
shock physics problems. Contemporary Mathematics 371: 173-179 (2005).
University at Stony Brook preprint number SB-AMS-04-03 and LANL report
number LA-UR-04-0713.
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[3] Yu, Y., Zhao, M., Lee, T., Pestieau, M.N., Bo, W., Glimm, J., and
Grove, J.W. Uncertainty quantification for chaotic computational fluid
dynamics. J. Comp. Phys. 217: 200-216, 2006. Stony Brook Preprint number
SB-AMS-05-16 and LANL preprint number LA-UR-05-6212.
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