No abstract

By enabling the estimation of difficult-to-measure target variables using available indirect measurements, mechanistic soft sensors have become important tools for various bioprocess monitoring and control scenarios. Despite promising higher process efficiencies and increased process understanding, widespread application of soft sensors has been stalled by uncertainty about the feasibility and reliability of their estimations given present process analytical constraints. Observability analysis can provide an indication of the possibility and reliability of soft sensor estimations by analyzing the structural properties of first-principle (mechanistic) models. In addition, it can provide a criteria for selection of suitable measurement methods with respect to their information content; thereby leading to successful implementation of soft sensors in bioprocess development and manufacturing environments. We demonstrate the utility of observability analysis for two classes of upstream bioprocesses: the processes involving growth and ethanol formation by Saccharomyces cerevisiae and the process of penicillin production by Penicillium chrysogenum. Results obtained from laboratory-scale cultivations in addition to in-silico experiments enable a comparison of theoretical aspects of observability analysis and the real-life performance of soft sensors. By taking the expected error of measurements provided to the soft sensor into account, an innovative scaling approach facilitates a higher degree of comparability of observability results among various measurement configurations and process conditions.

In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear ill-posed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landweber's method is numerically very intensive. Therefore we propose an adaptive variant of Landweber's iteration that significantly may reduce the computational expense, i.e. leading to a compressed version of Landweber's iteration. We lend the concept of adaptivity that was primarily developed for well-posed operator equations (in particular, for elliptic PDE's) essentially exploiting the concept of wavelets (frames), Besov regularity, best N-term approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landweber's iteration. In combination with an adequate refinement/stopping rule (a-priori as well as a-posteriori principles) we prove that the proposed procedure is an regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging.

We present a discrepancy-based parameter choice and stopping rule for iterative algorithms performing approximate Tikhonov-functional minimization which adapts the regularization parameter value during the optimization procedure. The suggested parameter choice and stopping rule can be applied to a wide class of penalty terms and iterative algorithms which aim at Tikhonov regularization with a fixed parameter value. It leads, in particular, to computable guaranteed estimates for the regularized exact discrepancy in terms of numerical approximations. Based on these estimates, convergence to a solution is shown. As an example, the developed theory and the algorithm is applied to the case of sparse regularization. We prove order optimal convergence rates in case of sparse regularization, i.e., weighted p norms, which turn out to be the same as for the a-priori parameter choice rule already obtained in the literature as well as for Morozov's principle applied to exact regularized solutions. Finally, numerical results for two different minimization techniques, iterative soft thresholding (ISTA) and monotone fast iterative soft thresholding (MFISTA), are presented, confirming, in particular, the results from the theory.

SUMMARYWe develop a regularization technique for Perona-Malik diffusion equations that relies on multiresolution techniques. The main result of this paper is to show that the chosen discretization overcomes the ill posedness of the nonlinear Perona-Malik model. The resulting algorithm is tested and the results are compared with pixel-based methods.