Josef Honerkamp

A nonlinear regularization method for the calculation of relaxation spectra

It is well known that the relaxation spectrum characterizing the linear viscoelastic properties of a polymer melt or solution is not directly accessible by an experiment. Therefore, it must be calculated from data for a material function. With Tikhonov regularization the relaxation spectrum in the terminal and plateau region can be calculated from data for a material function in the corresponding region. Serious difficulties arise however, if the spectrum should be determined in a larger range. These difficulties are caused by the considerably different contributions at short and long relaxation times. We show that these difficulties can be avoided by a nonlinear regularization method.

Characteristics of hand tremor time series

Tremor is classified into physiological, essential, and parkinsonian tremor by means of clinical criteria. The aim of our work was to extract quantitative features from the measurements of the acceleration of human postural hand tremor. Different mathematical methods were adopted and modified in order to separate these three types of tremor. Best discrimination between physiological and pathological tremors has been achieved by methods distinguishing nonlinear from linear behavior. On the other hand, methods separating different forms of nonlinear behavior have been found to be superior in discriminating parkinsonian and essential tremor. By these methods physiological and pathological tremors can be separated with an error rate below 20% and essential and parkinsonian tremor with an error rate below 10%. This may help to classify tremor time series by objective mathematical criteria and may increase the understanding of the pathophysiological differences underlying these kinds of tremor.

Ill-posed problems in rheology

Experimental data are always noisy and often incomplete. This leads to ambiguities if one wants to infer from the data some functions, which are related to the measured quantity through an integral equation of the first kind. In rheology many of such so-called ill-posed problems appear. Two techniques to treat such problems, the regularization method and the maximum entropy method, are applied to the determination of the relaxation spectrum from data of small oscillatory shear flow. With simulated data from a reference spectrum it is discussed how the inferred spectrum depends on the region, in which data are available. It turns out that information about the asymptotic behavior of the measured quantity can be of great help in determining the full spectrum also from incomplete data.

Analyzing the dynamics of hand tremor time series

We investigate physiological, essential and parkinsonian hand tremor measured by the acceleration of the streched hand. Methods from the theory of dynamical systems and from stochastics are used. It turns out that the physiological tremor can be described as a linear stochastic process, and that the parkinsonian tremor is nonlinear and deterministic, even chaotic. The essential tremor adopts a middle position, it is nonlinear and stochastic.

A generalized regularization method for nonlinear ill-posed problems enhanced for nonlinear regularization terms☆

In many fields of science one is interested in functions which are not directly accessible by experiment but have to inferred from an experimentally measurable quantity by solving an inverse problem. In general, this constitutes an ill-posed problem. Therefore so-called regularization methods are necessary: Besides the constraint from the experimental data these methods impose additional information on the solution, denoted as prior information and modeled by the so-called regularization term. For example, the Tikhonov regularization respectively its generalization to nonlinear inverse problems, denoted as nonlinear regularization method and implemented in the program NLREG (J. Weese, Comput. Phys. Commun. 77 (1993) 429), are based on the prior information that the solution is smooth. Thus, one is restricted to a specific linear regularization term. However, there exist some regularization methods which make use of more elaborate prior information. Accordingly, there is a need for a program that can handle more general, in particular nonlinear regularization terms. Hence, the nonlinear regularization method is generalized in order to comply with this need. This generalized nonlinear regularization method is implemented in the program GENEREG.

A note on estimating mastercurves

There are several quantities in theology which show a scaling behavior. One well known example is the time-temperature superposition principle of material functions characterizing the linear viscoelastic properties of polymer melts. We propose a mathematical shift procedure for the calculation of mastercurves and the corresponding scaling factors from experimental data which show such a scaling behavior. In order to demonstrate the applicability of the shift procedure mastercurves and scaling factors are determined for material functions of several polystyrene melts and for the specific viscosity of polyisobutylene in cyclohexane.

Using regularization methods for the determination of relaxation and retardation spectra of polymeric liquids

The application of Tikhonov regularization to the determination of relaxation and retardation spectra of viscoelastic fluids is discussed using simulated and real experimental data. It is thereby shown that with this method consistent results for relaxation and retardation spectra can be obtained from experimental data for different material functions. Furthermore, the differences between the most frequently used variants of Tikhonov regularization and the maximum entropy method are discussed. For most calculations the program FTIKREG has been used, which is an especially fast and reliable implementation of Tikhonovs regularization method.

An analytical relation between relaxation time spectrum and molecular weight distribution

In this article it is shown that the relaxation time spectrum can be analytically related to the molecular weight distribution regarding a recently derived generalized mixing rule. This analytical relation greatly reduces the computational effort to determine the molecular weight distribution from the relaxation time spectrum. In this mixing rule a generalized mixing parameter β has been introduced. This parameter has been controversially discussed in the literature. The value of β has been determined theoretically by [The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)] as β=1 and [Europhys. Lett. 5, 437–442 (1988); 6, 475 (1988)] and [Polym. Prepr. (Am. Chem. Soc. Div. Polym. Chem.) 28, 185–186 (1987)] (β=2) and experimentally by [J. Rheol. 42, 1153–1173 (1998)] (β=3.84). In this article the influence of β on shape and position of peaks in bimodal molecular weight distributions is emphasized.In this article it is shown that the relaxation time spectrum can be analytically related to the molecular weight distribution regarding a recently derived generalized mixing rule. This analytical relation greatly reduces the computational effort to determine the molecular weight distribution from the relaxation time spectrum. In this mixing rule a generalized mixing parameter β has been introduced. This parameter has been controversially discussed in the literature. The value of β has been determined theoretically by [The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)] as β=1 and [Europhys. Lett. 5, 437–442 (1988); 6, 475 (1988)] and [Polym. Prepr. (Am. Chem. Soc. Div. Polym. Chem.) 28, 185–186 (1987)] (β=2) and experimentally by [J. Rheol. 42, 1153–1173 (1998)] (β=3.84). In this article the influence of β on shape and position of peaks in bimodal molecular weight distributions is emphasized.

A unifying view on some experimental effects in tapping-mode atomic force microscopy

Several experimental effects that occur in tapping-mode atomic force microscopy are examined: apparent hysteresis effects in force probes and frequency sweeps and distortions in imaging. It is found through examinations of the phase space that they can all be reduced to one common cause: the existence of more than one stable state of the tip vibration for certain parameters. It is shown that the experimental effects can be explained considering only the phase space under the assumption that measurement noise exists. Numerical simulations confirm the theoretical and experimental findings.

A software for recording and analysis of human tremor

For many diseases various methods for the diagnosis and treatment monitoring are available. Presently, such methods are not established for an investigation of tremor diseases, although the different forms of tremor are common neurological symptoms and occur frequently in various neurological diseases and also other conditions. We developed an easy-to-use application for tremor-analysis and recording, running under MS-Windows, that allows us to investigate different forms of tremor by advanced mathematical methods of time series analysis. The application is also applicable for users who are not familiar with these kind of advanced data analysis methods. It provides tools for the diagnosis and treatment monitoring under laboratory conditions, based on previously developed and established methods of spectral and cross spectral analysis of tremor and electromyographic time series.