We address the issue of epistemic uncertainty quantification in the context of constrained differential systems. To illustrate our approach we have chosen a certain chromatographic adsorption model which is a coupled system of partial differential, ordinary differential and algebraic equations. The difficulty in solving this type of a system is that typically certain unknowns lack a natural time evolution equation. The standard approach in such cases is to devise specific numerical schemes which somehow try to take into account the implicit structure of the system. In our approach, we complete the system by finding the appropriate missing evolution equations. This makes the system overdetermined and more complicated in some way but, on the other hand, the completed system provides extra information useful for error estimation and uncertainty quantification. We will also show that reducing the epistemic uncertainties also leads to better estimations of aleatory uncertainties.
DOI : 10.1051/m2an/2015002
Mots-clés : Overdetermined PDE, uncertainty quantification, sensitivity analysis, chromatographic adsorption, constrained coupled systems
@article{M2AN_2015__49_4_1047_0,
author = {Mohammadi, Bijan and Tuomela, Jukka},
title = {Uncertainty quantification in the numerical solution of coupled systems by involutive completion},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1047--1062},
publisher = {EDP-Sciences},
volume = {49},
number = {4},
year = {2015},
doi = {10.1051/m2an/2015002},
mrnumber = {3371903},
zbl = {1322.65064},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015002/}
}
TY - JOUR AU - Mohammadi, Bijan AU - Tuomela, Jukka TI - Uncertainty quantification in the numerical solution of coupled systems by involutive completion JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1047 EP - 1062 VL - 49 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015002/ DO - 10.1051/m2an/2015002 LA - en ID - M2AN_2015__49_4_1047_0 ER -
%0 Journal Article %A Mohammadi, Bijan %A Tuomela, Jukka %T Uncertainty quantification in the numerical solution of coupled systems by involutive completion %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1047-1062 %V 49 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015002/ %R 10.1051/m2an/2015002 %G en %F M2AN_2015__49_4_1047_0
Mohammadi, Bijan; Tuomela, Jukka. Uncertainty quantification in the numerical solution of coupled systems by involutive completion. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1047-1062. doi : 10.1051/m2an/2015002. http://www.numdam.org/articles/10.1051/m2an/2015002/
AIAA Guide for the verification and validation of computational fluid dynamics simulations. AIAA (1998) G-077. | Zbl
P.I. Dudnikov and S.N. Samborski, Linear overdetermined systems of partial differential equations, in Partial Differential Equations VIII, edited by M.A. Shubin. Vol. 65 of Encycl. Math. Sci. Springer (1996) 1–86. | MR | Zbl
A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Appl. Math. Sci. Springer-Verlag, New York (2004). | MR | Zbl
and , On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data. J. Comput. Phys. 217 (2006) 63–81. | DOI | MR | Zbl
R. Ghanem and P. Spanos, Stochastic finite elements: A spectral approach. Springer Verlag, New York (1991). | MR | Zbl
G. Iaccarino, Quantification of uncertainty in flow simulations using probabilistic methods. VKI Lect. Series (2008).
, and , Overdetermined elliptic systems. Found. Comp. Math. 6 (2006) 309–351. | DOI | MR | Zbl
and , Completion of overdetermined parabolic PDEs. J. Symb. Comput. 43 (2008) 153–167. | DOI | MR | Zbl
O.P. Le Maître and O.M. Knio, Spectral methods for uncertainty quantification. Scientific Computation. Springer, New York (2010). | MR | Zbl
, and , A novel partial differential algebraic equation (PDAE) solver: iterative spacetime conservation element/solution element (CE/SE) method. Comput. Chem. Eng. 28 (2004) 1309–1324. | DOI
, and , Computer-aided model analysis for ionic strength-dependent effective charge of protein in ion-exchange chromatography. Biochem. Eng. J. 25 (2005) 125–140. | DOI
and , Simplifying numerical solution of constrained PDE systems through involutive completion. ESAIM: M2AN 39 (2005) 909–929. | DOI | Numdam | MR | Zbl
and , Involutive upgrade of Navier–Stokes solvers. IJCFD 23-6 (2009) 439–447. | MR | Zbl
and , Involutive formulation and simulation for electroneutral microfluids. ESAIM: M2AN 45 (2011) 901–913. | DOI | Numdam | MR | Zbl
G. Obinata and B. Anderson, Model reduction for control system design. Springer, Berlin (2000). | Zbl
J. F. Pommaret, Systems of partial differential equations and Lie pseudogroups. Vol. 14 of Math. Appl. Gordon and Breach Science Publishers (1978). | MR | Zbl
Z. Qu, Model order reduction techniques with applications in finite element analysis. Springer, Berlin (2004). | MR | Zbl
W. Schilders, H. Van der Vorst and J. Rommes, Model order reduction: Theory, research aspects and applications. Vol. 13 of Springer Math Indus. Series. Berlin (2008). | MR | Zbl
W.M. Seiler, Involution – the formal theory of differential equations and its applications in computer algebra. Vol. 24 of Algorithms Comput. Math. Springer (2010). | MR | Zbl
R. Smith, Uncertainty quantification. Vol. 12 of Comput. Sci. Eng. SIAM, Philadelphia, PA (2014). | MR | Zbl
, Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc. 75 (1969) 179–239. | DOI | MR | Zbl
D. Xiu, Numerical methods for stochastic computations: A spectral method approach. Princeton University Press (2010). | MR | Zbl
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