Research project "Nonlinear Dynamics in Process Design"

(Konstruktive Methoden der Nichtlinearen Dynamik zum Entwurf verfahrenstechnischer Prozesse)

Researchers

Thomas Beelitz
Bruno Lang
Paul Willems

Duration and funding

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

Description

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

determine small boxes that are guaranteed to contain all solutions of a nonlinear system (such as the augmented system mentioned above),
prove that these boxes do indeed contain solutions of the system, and
solve unconstrained and constrained optimization problems, again with the guarantee that the global optimum is not missed.

Our nonlinear solver is composed of three components:

a symbolic preprocessing module, which sets up the augmented system and its derivatives from the process description and simplifies the terms,
an exclusion module, which determines and eliminates those parts of the search area that cannot contain a solution; this module is based on a "branch-and-bound" bisection scheme, accelerated with Newton-type operators and the resolution of equations for single variables, and
a verification module providing several algorithms for computationally proving the existence of a solution within a certain box.

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. [ DOI \| 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. [ DOI \| 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. [ DOI \| 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. [ DOI \| 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. [ DOI \| 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. [ DOI \| 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. [ DOI \| Abstract ]
[9]	Andreas Frommer and Bruno Lang. On preconditioners for the Borsuk existence test. Proc. Appl. Math. Mech., 4:638--639, 2004. [ DOI \| 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. [ DOI \| 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 ]

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