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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 55
Adaptive Uncertainty Quantification H.G. Matthies and A. Keese
Institute of Scientific Computing, Technische Universität Braunschweig, Germany H.G. Matthies, A. Keese, "Adaptive Uncertainty Quantification", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 55, 2006. doi:10.4203/ccp.83.55
Keywords: uncertainty quantification, adaptive estimation, stochastic systems, polynomial chaos, stochastic Galerkin methods.Summary
Many models in science and engineering involve some element of uncertainty.
If this uncertainty is modelled with probabilistic methods, one has to
consider systems with uncertain or random parameters. The main interest here
is with
random fields, which model the spatial variability of random
parameters.
A simple stationary model of groundwater flow may illustrate this [4]:
where is the hydraulic head, the random conductivity,
f and random sinks and sources, and
the spatial
domain, and a probability space with probability measure
.
Discretisation in space is performed by the finite element method, and various
techniques exist to take account of the random nature of the governing
equation [3]. The stochastic variability may also be treated
with Galerkin methods [2,4]. The solution
is a Following the "Galerkin recipe", one takes finite dimensional subspaces of , using the Karhunen-Loève expansion (KLE) of
the random field
and Wiener's polynomial chaos
expansion (PCE) as a stochastic ansatz, the discrete form of
the problem then looks like [4]:
multi-index to designate the PCE functions.
Often a functional of the solution
is the main point of
interest; in this case the solution process can be used to estimate the error
of the approximated value
, where is the
continuous representation of the discrete solution given by
, and
this estimate can be used to adaptively steer the computation
[1]. For simplicity assume that the functional
is linear, as well as its approximation
, then the error in the functional can be estimated simply by
The error in the functional is
, where is the
References
- 1
- A. Keese, H.G. Matthies, "Adaptivity and sensitivity for stochastic problems." In P. Spanos, G. Deodatis (eds.), "Computational Stochastic Mechanics 4", pp. 311-316, Millpress, Rotterdam, 2003.
- 2
- A. Keese,
*Numerical Solution of Systems with Stochastic Uncertainties : A General Purpose Framework for Stochastic Finite Elements*. Doctoral thesis, Technische Universität Braunschweig, Brunswick, 2004. URL - 3
- H.G. Matthies, C.E. Brenner, C.G. Bucher and C. Guedes Soares,
*Uncertainties in Probabilistic Numerical Analysis of Structures and Solids--Stochastic Finite Elements*, Structural Safety, 19, 283-336, 1997. doi:10.1016/S0167-4730(97)00013-1 - 4
- H.G. Matthies, A. Keese,
*Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations*, Comp. Meth. Appl. Mech. Engrng. 194, 1295-1331, 2005. doi:10.1016/j.cma.2004.05.027
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