(523a) Strategies for Dealing With Ill-Posed Problems in On-Line Optimal Experimental Design

The idea to exploit measurement information as soon as it is generated, in order to improve the design of a running experiment has been initially developed in the 70ies (Mehra, 1974). However, applications of corresponding methodologies to mechanistic chemical and biological models have only been presented in the last decade. These are the adaptive input design (see e.g. Stigter, Vries, and Keesman, 2006) and the online model-based redesign of experiment (se e.g. Galvanin, Barolo, and Bezzo, 2009).

The common assumption in these contributions is that all parameters are identifiable at any time instant. Thus, it is assumed that well-posed parameter estimation (PE) and optimal experiment design (OED) problems are solved during on-line calculations. However, in a real case study this is not always true, especially at the beginning of an experiment, when the experimental data is scarce. Accordingly, adequate measures to ensure the robustness and reliability of the online algorithm have to be introduced.

In this work, different approaches for handling ill-posed problems in on-line OED are presented and critically discussed. We compare singular value decomposition (SVD) for selecting identifiable parameter subsets and regularization techniques such as Tikhonov regularization. For doing so, results presented by Haber, Horesh, and Tenorio (2010) regarding off-line experimental design for nonlinear ill-posed problems will be considered. Moreover, the (ordinary) SVD and generalized SVD (GSVD) will be the numerical tools for analyzing the rank-deficient and discrete ill-posed problems (Hansen, 1998). The application of the different approaches is presented for the parameter determination of a fermentation reactor.

Galvanin, F.; Barolo, M. and Bezzo, F. (2009): Online Model-Based Redesign of Experiments for Parameter Estimation in Dynamic Systems, Industrial & Engineering Chemistry Research 48, pp. 4415–4427.

Haber, E.; Horesh, L. and Tenorio, L. (2008): Numerical methods for experimental design of large-scale linear ill-posed inverse problems, Inverse Problems 24 [5], p. 055012.

Hansen, P. C. (1998): Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, ISBN: 0898714036.

Mehra, R. (1974): Optimal input signals for parameter estimation in dynamic systems--Survey and new results, Automatic Control, IEEE Transactions on 19 [6], pp. 753-768.

Stigter, J. D.; Vries, D. and Keesman, K. J. (2006): On adaptive optimal input design: a bioreactor case study, AIChE journal 52 [9], pp. 3290-3296.