TY - JOUR A1 - Wulkow, Niklas A1 - Telgmann, Regina A1 - Hungenberg, Klaus-Dieter A1 - Schütte, Christof A1 - Wulkow, Michael T1 - Deterministic and Stochastic Parameter Estimation for Polymer Reaction Kinetics I: Theory and Simple Examples JF - Macromolecular Theory and Simulations N2 - Two different approaches to parameter estimation (PE) in the context of polymerization are introduced, refined, combined, and applied. The first is classical PE where one is interested in finding parameters which minimize the distance between the output of a chemical model and experimental data. The second is Bayesian PE allowing for quantifying parameter uncertainty caused by experimental measurement error and model imperfection. Based on detailed descriptions of motivation, theoretical background, and methodological aspects for both approaches, their relation are outlined. The main aim of this article is to show how the two approaches complement each other and can be used together to generate strong information gain regarding the model and its parameters. Both approaches and their interplay in application to polymerization reaction systems are illustrated. This is the first part in a two-article series on parameter estimation for polymer reaction kinetics with a focus on theory and methodology while in the second part a more complex example will be considered. Y1 - 2021 U6 - https://doi.org/10.1002/mats.202100017 VL - 30 ER - TY - JOUR A1 - Hungenberg, Klaus-Dieter A1 - Schwede, Christian A1 - Wulkow, Michael A1 - Wulkow, Niklas T1 - Determination of reactivity ratios for acrylic acid and its dimer from classical parameter estimation and Bayesian approach JF - Canadian Journal of Chemical Engineering N2 - The reactivity ratios of acrylic acid (AA, M1) and its dimer beta-acroyloxypropionic acid (diAA, M2) are determined from cumulative copolymerization data by two different methods: classical parameter estimation (PE) by minimizing the objective function and a Bayesian analysis. Classical PE gives r1 =0.74 and r2 = 1.23 at the minimum of the residual. From the Bayesian analysis, the probability distribution of the parameter sets is obtained, revealing the existence of parameter sets with rather the same probability. The influence of the number of data and the size of the measurement error are discussed. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1002/cjce.24330 SP - 1 EP - 10 ER - TY - GEN A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation N2 - \begin{abstract} In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equat ion, also known as the chemical master equation (CME). Up to now, the CME has almost exclusively been treated by Monte-Carlo techniques, the most prominent of which is the simulation algorithm suggest ed by Gillespie in 1976. Since this algorithm requires an update for each single reaction event, realizations can be computationally very costly. As an alternative, we here propose a novel approach, which focuses on the discrete partial differential equation (PDE) structure of the CME and thus allows to adopt ideas from adaptive discrete Galerkin methods (as designed by two of the present authors in 1989), which have proven to be highly efficient in the mathematical modelling of polyreaction kinetics. Among the two different options of discretizing the CME as a discrete PDE, the method of lines approach (first space, then time) and the Rothe method (first time, then space), we select the latter one for clear theoretical and algorithmic reasons. First numeric al experiments at a challenging model problem illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further research. \end{abstract} T3 - ZIB-Report - 07-04 KW - adaptive discrete Galerkin methods KW - adaptive Rothe method KW - discrete Chebyshev polynomials KW - stochastic reactions kinetics KW - chemical master equation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9471 ER - TY - GEN A1 - Ackermann, Jörg A1 - Wulkow, Michael T1 - MACRON - A Program Package for Macromalecular Reaction Kinetics. N2 - This paper presents the new program package MACRON for the simulation of macromolecular kinetics including standard chemical reactions. Such problems lead to countable (possibly) infinite systems of ordinary differential equations (CODE's), which are numerically treated by the so-called discrete Galerkin method here. By a chemical compiler the required analytical preprocessing is performed, such that the complete reaction system, standard kinetics as well as macromolecular reactions, can be entered in the chemical formalism. Typical macromolecular reaction steps are chain addition, termination, chain transfer and degradation (cracking). In order to ensure efficiency and reliability, high sophisticated numerical routines are built within the package. MACRON can be used without a detailed knowledge of the used numerical methods. As an illustration the application of MACRON to some realistic problems is presented. T3 - ZIB-Report - SC-90-14 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-423 ER - TY - GEN A1 - Wulkow, Michael A1 - Ackermann, Jörg T1 - Numerical Treatment of Polyreactions - Recent Developments. N2 - The mathematical modeling of macromolecular reactions leads to countable (possibly infinite) systems of ordinary differential equations (CODE's). This paper reviews two recent developments of the so-called discrete Galerkin method, which has been developed for the numerical treatment of countable systems, which arise e.g. in polymer chemistry. The first approach can be considered as a method of lines with moving basis functions and has been implemented recently in the program package MACRON. The second type of the Galerkin method is characterized by a so-called outer time discretization of the complete problem and an appropriate and efficient solution of the arising subproblems. This method is realized in the research code CODEX. T3 - ZIB-Report - SC-90-15 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-439 ER - TY - GEN A1 - Wulkow, Michael T1 - Numerical Treatment of Countable Systems of Ordinary Differential Equations. N2 - Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and medicine. They can be considered as ordinary differential equations in sequence spaces. In this work, a full adaptive algorithm for the computational treatment of such systems is developed. The method combines time discretization with extrapolation in Hilbert spaces with a discrete Galerkin approach as discretization of the stationary subproblems. The Galerkin method is based on orthogonal functions of a discrete variable , which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations can be assembled either by use of analytical properties of the orthogonal functions or numerically by a multilevel summation algorithm. The resulting algorithm CODEX is applied to many examples of technological interest, in particular from polymer chemistry. T3 - ZIB-Report - TR-90-08 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4728 ER - TY - GEN A1 - Wulkow, Michael T1 - Adaptive Treatment of Polyreactions in Weighted Sequence Spaces. N2 - Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and statistics. They can be considered as ordinary differential equations in sequence spaces. In this work, a fully adaptive algorithm for the computational treatment of such systems is developed. The method is based on a time discretization of an abstract Cauchy problem in Hilbert space and a discrete Galerkin approach for the discretization of the arising stationary subproblems. The Galerkin method uses orthogonal functions of a discrete variable, which are generated by certain weight functions. A theory of countable systems in the associated weighted sequence spaces is developed as well as a theory of the Galerkin method. The Galerkin equations are solved adaptively either by use of analytical properties of the orthogonal functions or by an appropriate numerical summation. The resulting algorithm CODEX is applied to examples of technological interest, in particular from polymer chemistry. T3 - ZIB-Report - SC-91-17 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-673 ER - TY - GEN A1 - Ackermann, Jörg A1 - Wulkow, Michael T1 - The Treatment of Macromolecular Processes with Chain-Length-Dependent Reaction Coefficients - An Examplefrom Soot Formation. N2 - The description of chain length distributions in macromolecular reaction kinetics leads to so-called countable systems of differential equations. In particular, when the appearing reaction rate coefficients depend on the chain length of the reacting macromolecules itself, an efficient numerical treatment of these systems is very difficult. Then even the evaluation of the right-hand side of the system can become prohibitively expensive with respect to computing time. In this paper we show how the discrete Galerkin method can be applied to such problems. The existing algorithm CODEX is improved by use of a multiplicative error correction scheme for time discretization and a new type of numerical preprocessing by means of a Gauss summation. Both ideas are exemplary for a wide class of approximation types and are described very briefly here. The new numerical techniques are tested on an example from soot formation, where the coagulation of molecules is modeled in terms of reaction coefficients depending on the surface of the particles and their collision frequency. T3 - ZIB-Report - SC-91-18 Y1 - 1991 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-682 ER - TY - GEN A1 - Deuflhard, Peter A1 - Wulkow, Michael T1 - Computational Treatment of Polyreaction Kinetics by Orthogonal Polynomials of a Discrete Variable. N2 - The paper presents a new approach to the computational treatment of polyreaction kinetics. This approach is characterized by a Galerkin method based on orthogonal polynomials of a discrete variable, the polymer degree (or chain length). In comparison with the known competing approaches (statistical moment treatment, Galerkin methods for continuous polymer models), the suggested method is shown to avoid the disadvantages and preserve the adventages of either of them. The basic idea of the method is the construction of a discrete inner product associated with a reasonably chosen probability density function. For the so-called Schulz-Flory distribution one thus obtains the discrete Laguerre polynomials, whereas the Poisson distribution leads to the Charlier polynomials. Numerical experiments for selected polyreaction mechanisms illustrate the efficiency of the proposed method. T3 - ZIB-Report - SC-88-06 Y1 - 1988 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-140 ER - TY - GEN A1 - Schütte, Christof A1 - Wulkow, Michael T1 - Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of RamanSpectroscopy. N2 - Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown. T3 - ZIB-Report - SC-92-07 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-778 ER - TY - GEN A1 - Budde, Uwe A1 - Wulkow, Michael T1 - Computation of Molecular Weight Distributions for Free Radical Polymerization Systems. N2 - Modeling of free radical polymerization leads to very large and usually stiff systems of ordinary differential equations which cannot be solved directly in an efficient way. This paper presents the application of a new approach called discrete Galerkin method to a realistic example - the polymerization of methyl methacrylate(MMA). The method is characterized by a Galerkin approximation on the basis of orthogonal polynomials of a discrete variable which represents the polymer degree. It allows the efficient computation of solutions of complete kinetic schemes with time- or moment-dependent reaction coefficients by reducing the complexity to a few differential equations. The approximation error can be controlled by an error estimation. In the case of MMA polymerization a reduction of computational effort by a factor of about 25 compared to a standard method can be obtained for the quasi-steady-state approximation of the model. In addition solutions of the instationary kinetic scheme can be easily computed. T3 - ZIB-Report - SC-89-07 Y1 - 1989 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-253 ER - TY - GEN A1 - Wulkow, Michael A1 - Deuflhard, Peter T1 - Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions. N2 - The discrete Galerkin method developed by the authors has turned out to be an efficient tool for the computational treatment of very large scale ODE systems arising in polyreaction kinetics. Up to now, this approach has been worked out in detail for homogeneous polymer reactions. The present paper deals with one line of possible extensions of the method to the case of so-called heterogeneous processes, which may appear e. g. in smog reactions. The associated mathematical models involve reaction coefficients depending on the chain length of the reacting polymer. The herein suggested extension is worked out in some detail on the basis of the earlier paper. In addition, a numerical example describing polymer degradation is included. T3 - ZIB-Report - SC-90-01 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-292 ER - TY - GEN A1 - Deuflhard, Peter A1 - Nowak, Ulrich A1 - Wulkow, Michael T1 - Recent Developments in Chemical Computing. N2 - The paper surveys three aspects of chemical computing, which seem to play a role in recent developments. First, extrapolation methods for the numerical treatment of differential- algebraic equations are introduced. The associated extrapolation code LIMEX has reached a certain level of sophistication, which makes it a real competitor to the elsewhere widely used multi-step code DASSL of Petzold. Second, adaptive methods of lines for partial differential equations such as those arising in combustion problems are treated. Both static and dynamic regridding techniques are discussed in some detail. Finally, some new ideas about the treatment of the kinetic equations arising from polymer reactions are presented. The new feature of the suggested approach is the application of a Galerkin procedure using sets of orthogonal polynomials over a discrete variable (which, of course, in the case of polymer reactions is the polymer degree). The new approach may open the door to a new reliable low dimensional treatment of complex polymer reactions. T3 - ZIB-Report - SC-90-07 Y1 - 1990 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-359 ER - TY - GEN A1 - Deuflhard, Peter A1 - Wulkow, Michael T1 - Simulationsverfahren für die Polymerchemie N2 - Die enorme algorithmische Beschleunigung durch diskrete Galerkin-Methoden f{ü}r abz{ä}hlbare Differentialgleichungssysteme hat der Simulation von Polymerisationsprozessen neue, industriell relevante Problembereiche er{ö}ffnet, die mit den bis dahin verf{ü}gbaren Methoden nicht zug{ä}nglich waren. T3 - ZIB-Report - SC-94-22 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1502 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Nowak, Ulrich A1 - Wulkow, Michael T1 - Recent Developments in Chemical Computing JF - Computers in Chemical Engineering Y1 - 1990 VL - 14 SP - 1249 EP - 1258 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Wulkow, Michael T1 - Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable JF - IMPACT Comput. Sci. Eng. Y1 - 1989 VL - 1 IS - 3 SP - 269 EP - 301 ER - TY - CHAP A1 - Deuflhard, Peter A1 - Wulkow, Michael ED - et al. Bachem, Achim T1 - Simulationsverfahren für die Polymerchemie T2 - Mathematik in der Praxis. Fallstudien aus Industrie, Wirtschaft, Naturwissenschaften und Medizin Y1 - 1995 SP - 117 EP - 136 PB - Berlin: Springer-Verlag ER - TY - JOUR A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation JF - SIAM J. Sci. Comput Y1 - 2008 VL - 30 IS - 6 SP - 2990 EP - 3011 ER - TY - JOUR A1 - Nikitin, B. A. A1 - Wulkow, Michael A1 - Schütte, Christof T1 - Modeling of Free Radical Styrene/Divinylbenzene Copolymerization with the Numerical Fractionation Technique JF - Macromolecular Theory and Simulation Y1 - 2013 U6 - https://doi.org/10.1002/mats.201300125 VL - 22 IS - 9 SP - 475 EP - 489 ER - TY - CHAP A1 - Wulkow, Michael A1 - Deuflhard, Peter ED - Fatunla, S. T1 - Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions T2 - Computational Ordinary Differential Equations Y1 - 1992 SP - 287 EP - 306 PB - University Press, Nigeria ER - TY - JOUR A1 - Appel, Jörg A1 - Bockhorn, Henning A1 - Wulkow, Michael T1 - Numerical Simulation of Soot Particle Size Distributions with a Discrete Galerkin Method JF - Proceedings of the Workshop Scientific Computing in Chemical Engineering II Y1 - 1999 VL - 2 ER - TY - JOUR A1 - Bockhorn, Henning A1 - Hornung, Andreas A1 - Hornung, Ursel A1 - Jakobströer, Petra A1 - Wulkow, Michael T1 - Modeling of Thermal Degradation of Polymers JF - Proceedings of the Workshop Scientific Computing in Chemical Engineering II Y1 - 1999 VL - 1 ER - TY - JOUR A1 - Teichmann, L. A1 - Liebelt, U. A1 - Reuschenbach, P. A1 - Wulkow, Michael A1 - Horn, H. T1 - Two dimensional simulation of mass transport and nitrification in the river Rhine JF - Eng. Life Sci. Y1 - 2001 VL - 1 IS - 4 SP - 145 EP - 149 ER - TY - JOUR A1 - Wulkow, Michael T1 - Feature article - Computer Aided Modeling of Polymer Reaction Engineering - The Status of Predici, 1 - Simulation JF - Macromol. React. Eng. Y1 - 2008 VL - 2 SP - 461 EP - 494 ER - TY - JOUR A1 - Wulkow, Michael T1 - Feature article - The Simulation of Molecular Weight Distributions in Polyreaction Kinetics by Discrete Galerkin Methods JF - Macromol. Theory Simul. Y1 - 1996 VL - 5 SP - 393 EP - 416 ER - TY - JOUR A1 - Wulkow, Michael A1 - Busch, Markus A1 - Davis, Thomas P. A1 - Barner-Kowollik, Christopher T1 - Implementing the Reversible Addition Fragmentation Chain Transfer (RAFT) Process in Predici. JF - Journal of Polymer Science Y1 - 2004 VL - 42 SP - 1441 ER - TY - JOUR A1 - Wulkow, Michael A1 - Gerstlauer, A. A1 - Nieken, U. T1 - Modeling of Crystallization Processes using PARSIVAL JF - Chem. Eng. Sci. Y1 - 2001 VL - 56 SP - 2575 EP - 2588 ER - TY - JOUR A1 - Iedema, P. A1 - Wulkow, Michael A1 - Hoefsloot, H. T1 - Conditional Monte Carlo sampling to find branching architectures of polymers from radical polymerizations with transfer to polymer and recombination termination JF - Polymer Y1 - 2007 VL - 48 IS - 6 SP - 1770 EP - 1784 ER - TY - JOUR A1 - Iedema, P. A1 - Wulkow, Michael A1 - Hoefsloot:, H. T1 - Modeling Molecular Weight and Degree of Branching Distribution of low density Polyethylene JF - Macromolecules Y1 - 2000 VL - 33 SP - 7173 EP - 7184 ER - TY - JOUR A1 - Katzer, J. A1 - Pauer, W. A1 - Moritz, H.-U. A1 - Wulkow, Michael T1 - Thermal polymerization of styrene Part B JF - Macromol. React. Eng Y1 - 2012 VL - 6 SP - 225 EP - 238 ER - TY - JOUR A1 - Lin, P.-J. A1 - Grimm, L. A1 - Wulkow, Michael A1 - Hempel, D. A1 - Krull, R. T1 - Population Balance Modeling of the Conidial Aggregation of Aspergillus niger JF - Biotechnology and Bioengineering Y1 - 2008 VL - 99 IS - 2 SP - 341 EP - 350 ER - TY - JOUR A1 - Müller, A. A1 - Yan, D. A1 - Wulkow, Michael T1 - Molecular Parameters of Hyperbranched Polymers Made by Self-condensing Vinyl Polymerization, 1. Molecular weight distribution JF - Macromolecules Y1 - 1997 VL - 30 SP - 7015 ER - TY - JOUR A1 - Ochoa, J.G. A1 - Wulkow, Michael T1 - DNA damages as a Depolymerization Process JF - International Journal of Modern Physics C Y1 - 2012 VL - 23 IS - 03 SP - 1250018 EP - 1 ER - TY - JOUR A1 - Reuter, H. A1 - Krause, G. A1 - Wulkow, Michael A1 - Mönig, A. A1 - Horn, H. T1 - A water quality management tool for river basins JF - Wat. Sci. Techn. Y1 - 2003 VL - 48 IS - 10 SP - 47 EP - 53 ER - TY - JOUR A1 - Reynhout, X. A1 - Meuldijk, J. A1 - Drinkenburg, B. A1 - Iedema, P. A1 - Wulkow, Michael T1 - A Novel Method to Model Emulsion Polymerization Kinetics JF - Polymer-Plastics Tech. Eng. Y1 - 2005 VL - 44 IS - 4 SP - 707 EP - 740 ER - TY - JOUR A1 - Schmidt, Christian-Ulrich A1 - Busch, Markus A1 - Lilge, Dieter A1 - Wulkow, Michael T1 - Detailed Molecular Structure Modeling – A Path Forward to Designing Application Properties of ldPE JF - Macromolecular Materials and Engineering Y1 - 2005 VL - 290 SP - 404 ER - TY - JOUR A1 - Schütte, Christof A1 - Wulkow, Michael T1 - A hybrid Galerkin-Monte-Carlo approach to higher-dimensional population balances in polymerization kinetics JF - Macromol. React. Eng. Y1 - 2010 VL - 4 SP - 562 EP - 577 ER - TY - JOUR A1 - Brüggemann, Stefan A1 - Bauer, Nanette A1 - Fuchs, Eberhard A1 - Polt, Axel A1 - Wagner, Bernhard A1 - Wulkow, Michael T1 - Support of Strategic Business Decisions at BASF’s Largest Integrated Production Site Based on Site-Wide Verbund Simulation JF - Computer Aided Chem. Eng. Y1 - 2008 VL - 25 SP - 925 EP - 930 ER - TY - JOUR A1 - Busch, Markus A1 - Müller, M. A1 - Wulkow, Michael T1 - The use of Simulation Techniques in Developing Kinetic Models for Polymerizations JF - Chem. Eng. Technol. Y1 - 2003 VL - 26 SP - 1031 ER - TY - JOUR A1 - Busch, Markus A1 - Wulkow, Michael T1 - Modellierung differenzierter mikrostruktureller Eigenschaften JF - Chem. Ing. Tech. Y1 - 2003 VL - 75 SP - 1027 ER - TY - JOUR A1 - Grosch, R. A1 - Briesen, Heiko A1 - Wulkow, Michael A1 - Marquardt, Wolfgang T1 - Generalization and numerical investigation of QMOM for crystallization processes JF - AIChE J. Y1 - 2007 VL - 53 SP - 207 EP - 227 ER - TY - JOUR A1 - Horn, H. A1 - Neu, T. A1 - Wulkow, Michael T1 - Modelling the Structure and Function of Extracellular Polymeric Substances in Biofilms with new Numerical Techniques JF - Wat. Sci. Techn. Y1 - 2001 VL - 43 IS - ´6 SP - 121 EP - 127 ER - TY - JOUR A1 - Horn, H. A1 - Wulkow, Michael T1 - Simulation von Wachstum und Abtrag der Biomasse – Eine exemplarische Betrachtung für eine 2D-Modellierung JF - Chem. Ing. Technik Y1 - 2005 VL - 77 IS - 4 SP - 418 EP - 424 ER - TY - JOUR A1 - Horn, H. A1 - Wulkow, Michael T1 - Modellierung von Einleitungen in kleine Fließgewässer JF - gwf Wasser Abwasser Y1 - 1996 VL - 137 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Telgmann, R. A1 - Wulkow, Michael T1 - The virtual lab approach to pharmacokinetic JF - Drug Discov Today Y1 - 2006 VL - 11 SP - 800 EP - 805 ER - TY - JOUR A1 - Hungenberg, K.-D. A1 - Chen, C.-C. A1 - Zhang, F. A1 - Wulkow, Michael A1 - Stubbe, G. A1 - Nieken, U. T1 - Design of Polymer Processes Using the Coupling of Commercial Simulation Packages Polymers Plus and PREDICI JF - DECHEMA Monographs Y1 - 2001 VL - 137 SP - 237 EP - 245 ER - TY - JOUR A1 - Hungenberg, K.-D. A1 - Knoll, K. A1 - Wulkow, Michael T1 - Absolute Propagation Rate Coefficients in Radical Polymerization from Gel Permeation Chromatography of Polymers Produced by Intermittent Initiation JF - Macromol. Theory Simul. Y1 - 1997 VL - 6 ER - TY - JOUR A1 - Schütte, Christof A1 - Wulkow, Michael T1 - Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of Raman Spectroscopy. JF - preprint Y1 - 1992 UR - http://publications.imp.fu-berlin.de/109/ ER - TY - JOUR A1 - Schütte, Christof A1 - Wulkow, Michael T1 - A Hybrid Galerkin-Monte-Carlo Approach to Higher-Dimensional Population Balances in Polymerization Kinetics JF - Macromol. React. Eng. Y1 - 2010 UR - http://publications.imp.fu-berlin.de/956/ U6 - https://doi.org/10.1002/mren.200900073 VL - 4 IS - 9-10 SP - 562 EP - 577 ER -