Bergische Universität Wuppertal
Fakultät für Mathematik und Naturwissenschaften
Angewandte Informatik - Algorithmik


Research project "Nonlinear Dynamics in Process Design"

(Konstruktive Methoden der Nichtlinearen Dynamik zum Entwurf verfahrenstechnischer Prozesse)


Thomas Beelitz
Bruno Lang
Paul Willems

Duration and funding

July 2001 to September 2004, funded by (Geschäftszeichen I/79288) .


Chemical processes are typically controlled via adjustable parameters, such as the concentrations of the educts in the inflow, the inflow heating, etc. Depending on the values of these parameters, the process may reach a single steady state (described, e.g., by the concentrations of educts and products in the outflow, the energy released in the course of the reaction, etc.), or may potentially jump between different steady states, which may reduce the quality of the product of may even necessitate shutting down the process. Therefore it is essential to know the singularities of the process, which mark the transitions from unique to multiple steady states, in order to choose parameters that guarantee safe operation.

One way to obtain guaranteed bounds for the singularities is to work with "augmented systems" (describing the process and the type of singularity). In a collaborative effort with Process Systems Engineering (Lehrstuhl für Prozesstechnik) and the Institute for Scientific Computing (Lehrstuhl für Hochleistungsrechnen), both at Aachen University, we have developed the SONIC (Solver and Optimizer for Nonlinear problems based on Interval Computations) framework, which allows to

Our nonlinear solver is composed of three components:

The optimizer contains additional components.

In addition to the one-processor version of SONIC we have parallelized versions for shared memory and distributed memory architectures (based on OpenMP and MPI, respectively).

For more details see the project-related publications.

Project-related publications

[1] Andreas Frommer, Fatmir Hoxha, and Bruno Lang. Proving the existence of zeros using the topological degree and interval arithmetic. J. Comput. Appl. Math., 199(2):397--402, February 2007. [ Abstract ]
[2] Martin Mönnigmann, Wolfgang Marquardt, Christian H. Bischof, Thomas Beelitz, Bruno Lang, and Paul Willems. A hybrid approach for efficient robust design of dynamic systems. SIAM Rev., 49(2):236--254, 2007. [ Abstract ]
[3] Thomas Beelitz, Bruno Lang, and Christian H. Bischof. Efficient task scheduling in the parallel result-verifying solution of nonlinear systems. Reliab. Comput., 12(2):141--151, April 2006. [ Abstract ]
[4] Wolfgang Marquardt, Martin Mönnigmann, Thomas Beelitz, Bruno Lang, Paul Willems, and Christan H. Bischof. Robust design of dynamic systems. ECMI Newsletter, 39:15, March 2006. [ Abstract ]
[5] Thomas Beelitz, Andreas Frommer, Bruno Lang, and Paul Willems. Symbolic--numeric techniques for solving nonlinear systems. Proc. Appl. Math. Mech., 5(1):705--708, December 2005. [ Abstract ]
[6] Andreas Frommer and Bruno Lang. Existence tests for solutions of nonlinear equations using Borsuk's theorem. SIAM J. Numer. Anal., 43(3):1348--1361, 2005. [ Abstract ]
[7] A. Frommer, B. Lang, and M. Schnurr. A comparison of the Moore and Miranda existence tests. Computing, 72(3--4):349--354, June 2004. [ Abstract ]
[8] Thomas Beelitz, Christian H. Bischof, and Bruno Lang. A hybrid subdivision strategy for result-verifying nonlinear solvers. Proc. Appl. Math. Mech., 4:632--633, 2004. [ Abstract ]
[9] Andreas Frommer and Bruno Lang. On preconditioners for the Borsuk existence test. Proc. Appl. Math. Mech., 4:638--639, 2004. [ Abstract ]
[10] Thomas Beelitz, Christian Bischof, Bruno Lang, and Klaus Schulte Althoff. Result-verifying solution of nonlinear systems in the analysis of chemical processes. In René Alt, Andreas Frommer, R. Baker Kearfott, and Wolfram Luther, editors, Numerical Software with Result Verification --- Proc. Intl. Dagstuhl Seminar, January 19--24, 2003, Dagstuhl Castle, Germany, number 2991 in LNCS, pages 198--205. Springer, Berlin, 2004. [ Abstract ]
[11] Thomas Beelitz, Christian H. Bischof, Bruno Lang, and Paul Willems. SONIC---A framework for the rigorous solution of nonlinear problems. Report BUW-SC 04/7, Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 2004. [ Preprint | Abstract ]
[12] Thomas Beelitz, Christian H. Bischof, and Bruno Lang. Intervals and OpenMP: Towards an efficient parallel result-verifying nonlinear solver. In Dieter an Mey, editor, Proc. EWOMP '03, Fifth European Workshop on OpenMP, September 22-26, 2003, Aachen, Germany, pages 119--125, Aachen, Germany, 2003. Center for Computing and Communication, Aachen University. [ Abstract ]

Project-related theses

[1] Paul Willems. Symbolisch-numerische Techniken zur verifizierten Lösung nichtlinearer Gleichungssysteme. Diploma thesis, RWTH Aachen, Germany, May 2004.
[2] André Kienitz. Untersuchungen zum Einsatz von Taylormodellen bei der verifizierten Lösung von Gleichungssystemen. Diploma thesis, RWTH Aachen, Germany, April 2003. [ Abstract ]
[3] Thomas Beelitz. Methoden zum Einschluss von Funktions- und Ableitungswerten. Diploma thesis, RWTH Aachen, Germany, April 2002. [ Abstract ]
[4] Klaus Schulte Althoff. Algorithmen zum verifizierten Lösen nichtlinearer Gleichungssysteme. Diploma thesis, RWTH Aachen, Germany, May 2002.

See also

the pages of our partners at Aachen University:

The work on the SONIC framework is continued; see the SONIC project page.

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics and Computer Science
Applied Computer Science Group

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