DocumentCode :
592210
Title :
Optimization-based estimation of random distributed parameters in elliptic partial differential equations
Author :
Borggaard, Jeff ; van Wyk, Hans-Werner
Author_Institution :
Interdiscipl. Center for Appl. Math., Virginia Tech, Blacksburg, VA, USA
fYear :
2012
fDate :
10-13 Dec. 2012
Firstpage :
2926
Lastpage :
2933
Abstract :
As simulation continues to replace experimentation in the design cycle, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. For distributed parameter models, current approaches assume the mean and variance of parameters are known, then use recently developed efficient numerical methods for approximating stochastic partial differential equations. However, the statistical descriptions of the model parameters are rarely known. A number of recent works have investigated adapting existing variational methods for parameter estimation to account for parametric uncertainty. In this paper, we formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers and the first order necessary conditions. A spectral approximation of the uncertain observations (via a truncated Karhunen-Loève expansion) allows an approximation of the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called `finite noise´ problem, in the space of functions with bounded mixed derivatives. We prove convergence of `finite noise´ minimizers to the corresponding infinite dimensional solutions, and devise a gradient based strategy for locating these numerically. Lastly, we illustrate our method with a numerical example.
Keywords :
approximation theory; control system synthesis; distributed parameter systems; gradient methods; optimisation; parameter estimation; partial differential equations; statistical analysis; bounded mixed derivatives; design cycle; deterministic optimization problem; elliptic partial differential equations; finite noise problem; gradient based strategy; infinite dimensional constrained optimization problem; model outputs; optimization-based estimation; parameter identification problem; parametric uncertainty; random distributed parameter model; spectral approximation; statistical descriptions; stochastic partial differential equations; uncertain observations; variational methods; Helium; Least squares approximation; Noise; Numerical models; Piecewise linear approximation; Uncertainty;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
Decision and Control (CDC), 2012 IEEE 51st Annual Conference on
Conference_Location :
Maui, HI
ISSN :
0743-1546
Print_ISBN :
978-1-4673-2065-8
Electronic_ISBN :
0743-1546
Type :
conf
DOI :
10.1109/CDC.2012.6425896
Filename :
6425896
Link To Document :
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