Maka Karalashvilia, Sven Großb, Adel Mhamdia, Arnold Reuskenb, Wolfgang Marquardta

a Process Systems Engineering, b Institute of Geometry and Practical MathematicsRWTH-Aachen University, Aachen, Germany

Aachener Verfahrenstechnik – RWTH Aachen University – Germany

www.avt.rwth-aachen.de

Acknowledgements. The authors gratefully acknowledge the finantial support of Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center (SFB) 540 „Model-based experimental analysis of kinetic phenomena in fluid multiphase reactive systems“.

A decomposition approach for the solution of inverse convection-diffusion problems

[1] H. Akaike, A new look at the statistical model identification, 19 (1974), pp. 716–723.[2] O. M. Alifanov, Inverse Heat Transfer Problems, Springer, Berlin, 1994. [3] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer

Academic Publishers, Dordrecht, 1996.[4] M. Hanke, Regularization properties of a truncated Newton-CG algorithm for nonlinear

inverse problems, Numer. Funct. Anal. Optim., 18 (1998), pp. 971-993.[5] C. Hansen, Rank-Deficient and Discrete Ill-posed Problems, SIAM, Philadelphia, 1998.[6] M. Karalashvili, S. Groß, A. Mhamdi, A. Reusken, and W. Marquardt, Incremental

identification of transport coefficients in convection-diffusion systems, SIAM J. Sci. Comput.,30 (2008), pp. 3249 –3269.

Motivation and problem statement

The model problem

A decomposition approach

Treatment of inverse problems

References

model structure and parameters

B balances

flux laws

BFT transportmodelsflux laws

measurement data

balances

source

transport coefficient

parameters

structure ofbalance equation

model structure fortransport coefficient

balancesBFmodel structure for flux

Incremental model identification

Model correction

transport coefficient transport models

coefficient correction

measurement data

incremental model identification

parameter correction model selection

optimal initial values

B:

affine-linear; CG method [2].

BFT:

coefficient correction:

nonlinear; standard least-squares.

parameter correction:nonlinear; standard least-squares.

Model selection Akaike‘s minimum information theoretic criterion [1]:

the model with the best fit of the data and a minimal no. of parameters.

Energy transport in laminar wavy film flow, the “flat-film“ model problem:

simulation settings:

discretization:

Numerical and software realization

incremental model identification [6]:-no a priori knowledge on transport is necessary

model correction:- high confidence in

parameters• transport coefficient • best suited transport model • corrected transport model

nonlinear; truncated Newton-CGNE method [4].

nonlinear; truncated Newton-CGNE method [4].

BF:

Information on physical parameters, i.e. transport coefficients, is lacking.

There exist no transport model: design of technical systems challenging.

consider:

transport coefficient:

Forward simulation and solution of arising direct problems:

DROPS - multilevel FE-method and one step θ-scheme

CG and truncated Newton-CGNE solution strategies:

MATLAB MEX DROPS Regularization of inverse problems:

fixed spatial and temporal discretization.

intoroduction of the weighted-minimal norm solution [3].

early termination of iterations: the discrepancy principle [3]; L-curve method [4].