Table Of ContentLecture Notes in Mathematics 1739
Editors:
A. Dold, Heidelberg
E Takens, Groningen
B. Teissier, Paris
Subseries: Fondazione C. I. M. E., Firenze
Adviser: Arrigo Cellina
regnirpS
Berlin
Heidelberg
New kroY
Barcelona
Hong Kong
London
Milan
siraP
eropagniS
oykoT
R. Burkard R Deuflhard A. Jameson
J.-L. Lions G. Strang
lanoitatupmoC
scitamehtaM Driven
by lairtsudnI smelborP
Lectures given at the 1st Session of the
Centro Internazionale Matematico Estivo
(C.I.M.E.) held in Martina Franca, Italy,
June 21-27, 1999
Editor: .V Capasso H. Engl J. Periaux
enoizadl;oF
C.I.M.E.
regnirpS
Authors Editors
Rainer E, Burkard Vincenzo Capasso
Technische Universitat Graz MIRIAM Milan Research Centre for
Institut ftir Mathematik Industrial - and Applied Mathematics
Steyrergasse 30 Department of Mathematics
8010 Graz, Austria University of Milan
E-mail: [email protected] Via C. Saldini, 50
20133 Milan, Italy
Antony Jameson E-mail: vince nzo.capasso @ mat.unimi.it
Dept. of Aeronautics and Astronautics
Stanford University Jacques Periaux
Durand 279 Dassault Aviation
Stanford, CA, 94305-4035, USA 78 quai Marcel Dassault
E-mail: jameson @baboon.stanford.edu 92214 Saint Cloud, France
E-mail: periaux @rascasse.inria.fr
Gilbert Strang
Dept. of Mathematics Heinz W. Engl
Massachusetts Institute of Technology Industrial Mathematics Institute
Room 2-240 Johannes Kepler University
77 Massachusetts Avenue Altenbergerstrasse 69
Cambridge, MA 02139-4307, USA 4040 Linz, Austria
E- mail: gs @ math. mit.edu E-mail: [email protected]
Peter Deuflhard
Konrad-Zuse-Zentrum
Takustrasse 7
14195 Berlin-Dahlem, Germany
E-mail: [email protected]
Jacques-Louis Lions
Coll6ge de France
3 rue d'Ulm
75231 Paris cedex 05, France
Library of Congress Cataloging-in-Publication Data
Computational mathematics driven by industrial problems : lectures given at the 1st
session of the Centro internazionale matematico estivo (C.I.M.E.) held in Mar'tim
Franca, Italy, June 21-27, 1999 / R. Burkard ... [et al.] ; editors V. Capasso, H. Engek J. Periaux.
p. cm. -- (Lecture notes in mathematics, ISSN 0075-8434 ; 1739)
Includes bibliographical references.
ISBN 3540677828 (soffcover : alk. paper)
.1 Mathematical models--Congresses. 2. Mathematics--Industrial
applications--Congresses. I. Burkard, Rainer E. II. Capasso, V. (Vincenzo), 1945- IlL
Engl, Heinz W. IV. Periaux, Jacques. V. Lecture notes in mathematics (Springer-Verlag)
QA3 .L28 no. 1739
[QA401 ]
510 s--de21
[511'.8]
00-063777
Mathematics Stibject Classification (2000): 65-XX, 49-XX, 90CXX, 41-XX, 76-XX, 60D05, 60GXX, 62M30
ISSN 0075-8434
ISBN 3-540-67782-8 Springer-Verlag Berlin Heidelberg New York
This work subject is copyright. to All rights are reserved, whether whole the or part of the material concerned, is specifically
rights lbe "ro translation, repriming, re-use of illustrations, recitation, broadcasting, reproduction microlilms oil or in any other
and storage way. ni data hanks. Duplication of publication this or parts thereof is pmmined only under the provisions of the
German Copyright Law of September 9, in its current 1965, version, permission and use fur must always be obtained from
Springer-Verlag. Violations are liable prosecution for under German the Copyright Law.
Springer-Verlag Heidelberg Berlin New kroY
a member "ro BertelsmannSpringer Science+Business Media GmbH
© Springer-Verlag Berlin Heidelberg 2000
Printed lli Germany
The use of descriptive etc. general names, registered names, trademarks, ni even not does imply, publication this ni absence the of
a specific statement, thai such protective regulations names and laws and are relevant exempt therefore from tile free for general
use.
Typesetting: Camera-ready TEX output authors by the
Printed on paper acid-fi'ee SPIN: 10724313 41/3142-543210
Preface
The Centro Internazionale Matematico Estivo (CIME) organized a summer
course on Computational Mathematics Driven by Industrial Problems from
June 21-27, 1999 at the Ducal Palace in Martina Franca ( a nice baroque
village in Apulia, Italy).
The relevance of methods of advanced mathematics for innovative tech-
nology has become well-recognized ever since the 1984 publication of the
"David Report" by an ad hoc committee chaired by the Vice-President of
EXXON, Dr. Edward E. David, jr.
As a direct consequence of the "revolution" in information technologies,
mathematics has become more and more visible. The truth is that mathe-
matics is not just applied but rather continuously created to respond to the
challenges of technological development and competitiveness.
Today traditional machines are no longer the core of industrial develop-
ment; computer simulation of highly complex mathematical models substi-
tute the traditional mechanical models of real systems. This poses challeng-
ing problems in the development of new mathematics and new computational
methods.
This course was designed to offer young European scientists an opportu-
nity of acquiring knowledge in selected areas of mathematics and advanced
scientific computing which have benefited from the needs of competitive in-
dustry. The application of mathematics for the solution of industrial problems
includes mathematical modelling of the real system; mathematical analysis
of well posedness; computational methods; identification of models and of pa-
rameters (inverse problems); optimization and control. The extensive courses
and seminars included all of these aspects. Furthermore, some case stud-
ies were presented in two-hour seminars on areas of industrial excellence in
Europe, namely polymers and glass, for which there are two Special Inter-
est Groups within the European Consortium for Mathematics in Industry
(ECMI).
In this volume you will find the written account of all the contributions.
We are grateful to all their authors. It may be of interest to the reader that the
President (G.S) of SIAM (Society for Industrial and Applied Mathematics),
three of the recent Presidents (V.C., H.E., and R.M.) of ECMI (European
Consortium for Mathematics in Industry), and the former President (J.P.)
of ECCOMAS (European Council on Computational Methods in Applied
Sciences and Engineering) were among the lecturers in Martina Franca.
It is a pleasure to also thank Professor Jacques Louis Lions, who was
impeded from participating at the last moment due to relevant commitments
with the President of France. He kindly accepted our invitation to include
the lecture notes he had prepared for the course in this monograph.
VI
The Directors are most grateful to CIME for giving us the opportunity
of bringing together excellent lecturers and a surprisingly large number of
brilliant and enthusiastic mathematicians, physicists and engineers from 18
countries. It is a great pleasure to acknowledge the assistance of Dr. Sabrina
Gaito and Dr. Alessandra Micheletti without whom the course could impos-
sibily have run so smoothly. Dr. Daniela Morale's assistance in editing this
volume is also kindly acknowledge. We extend our warmest thanks to them,
to the lecturers and to all participants.
We gratefully acknowledge the financikl support from CIME and the Eu-
ropean Union. Finally, we would like to thank the Mayor, the General Secre-
tariat and the complete staff of the Town Council of Martina Franca, which
offered the XVII century Ducal Palace as site of the course, for the warm
hospitality and continuous assistance.
Milano, Linz, Paris Vincenzo Capasso
March, 2000 Heinz Engl
Jacques Periaux
CONTENTS
Trees and Paths: Graph Optimisation Problems with Industrial Applications
.R .E Burkard
Mathematical Models for Polymer Crystallization Processes
K Capasso 39
DifferentiEaqlu ations in Technology and Medicine:
Computational Concepts, Adaptive Algorithms, and Virtual Labs
.P Deuflhard 69
InversPer oblems and Their Regularization
.H .W Engl 721
Aerodynamic Shape OptimizationT echniques Based on Control Theory
A. Jameson, .L Martinetli 151
Complexity in Industrial Problems. Some remarks.
J.-L. Lions 223
Flow and Heat Transfer in Pressing of Glass Products
.K Laevksy, .B .J van der Linden, . R..MM Mattheij 267
Drag Reduction by Active Control for Flow Cylinders Past
J.-W. ,eH .M Chevalier, .R GlowinskL .R Metcalfe, .A Nordlander, .J Periaux 287
Signal Processing for Everyone
G. Strang 365
List of Participants 413
Trees and paths: graph optimisation problems
with industrial applications *
Rainer E. Burkard
Technische Universit~t Graz, Institut fiir Mathematik, Steyrergasse 30,
A-8010 Graz, Austria. Email: burkard~opt.math.tu-graz.ac.at
Contents
Introduction 2
1.1 Graph optimization problems .................. 2
1.2 Basic properties of graphs .................... 2
1.2.1 Undirected graphs .................... 2
1.2.2 Directed graphs and networks .............. 4
1.3 Complexity ............................ 5
Shortest trees 7
2.1 Shortest connection of points .................. 7
2.2 Minimum spanning tree algorithms ............... 10
2.3 The greedy algorithm and matroids ............... 14
2.4 Shortest spanning trees in the plane .............. 18
2.5 Steiner tree problems ....................... 20
Shortest paths 22
3.1 Shortest paths problems ..................... 22
3.2 Dijkstra's algorithm ....................... 23
3.3 Different objective functions ................... 26
3.4 Bellman-Moore algorithm .................... 27
3.5 The all-pairs problem ...................... 30
3.6 An application: 1-median problem ............... 34
References 37
*This research has been supported by Spezialforschungsbereich F 300 "Optimierung
und Kontrolle", Projektbereich Diskrete Optimierung.
2 R.E. Burkard
1 Introduction
1.1 Graph optimization problems
Graphs offer a simple model for "connecting" objects. Thus they frequently
occur in models from such different fields like telecommunication, traffic,
location and relational analysis. And in many of these problems we want to
connect objects in an "optimal" way. The underlying mathematical theory
deals with optimization strategies for trees and paths in graphs. This will be
the object of these lectures.
After introductory remarks on undirected and directed graphs we out-
line the basic notions of computational complexity. Complexity issues play a
crucial role in the evaluation of the efficiency of algorithms. The second chap-
ter is devoted to tree problems. We distinguish between spanning trees and
Steiner trees which may use additional vertices, so-called Steiner points. We
shall see that minimum spanning trees can be computed in polynomial time,
whereas no polynomial time algorithm is known for the Steiner minimum tree
problem. One of the basic algorithms for finding a minimum spanning tree in
a connected graph will turn out as a basic tool in combinatorial optimization:
Kruskal's algorithm is the prototype of a greedy algorithm.
In the third chapter we investigate different shortest path problems. We
shall see how the positiveness of single links will influence the computational
complexity of solution routines. And we shall see how an analysis of the algo-
rithm leads to solution procedures for path problems stated in ordered alge-
braic structures like ordered semigroups. As an application of path problems
we mention not only classical cases like shortest routes in traffic networks,
but also the critical path problem for project networks and the 1-median
problem in location theory.
Summarizing we see that even so simple structures like trees and paths
lead to interesting mathematical questions in different fields like computa-
tional geometry, matroids, systolic arrays and computational complexity, just
to mention a few ones.
1.2 Basic properties of graphs
1.2.1 Undirected graphs
An undirected graph G = (V, E) is defined by its vertex set V and by its edge
set E, where every edge e C E connects two vertices of set V. Throughout
this lecture we shall always assume that
• the sets V and E are finite,
* there is no loop, i.e., every edge connects two different vertices u and v,
• there are no parallel edges, i.e., every edge is uniquely defined by the
pair of vertices it connects.
Graph optimisation problems 3
A graph G fulfilling the above properties is called a finite, simple graph.
If edge e connects the vertices u and v, we say, edge e is incident with u and
v: e = [u, v]. The number of edges incident with a vertex v is called the
degree d(v) of v. The complete graph Kn on n vertices is a graph where all
vertices are pairwisely connected. Fig. 1.1 depicts the complete graphs K1
to/(5.
w w
K1 K2 /(3 K4
K5
Figure 1.1: The complete graphs K1-K5
Figure 1.2: Two different trees on four vertices
A path from vertex u to vertex v can be described by a sequence of different
vertices u = vo, vl, ..., kV --- v, where [vi-1, vii are edges for all i =- 1, 2, ..., k.
If we add the edge [v~,v0] to this path, we get a cycle. Every cycle in G
can be described in this way. A cycle is called Hamiltonian cycle, if it passes
through all vertices of the graph just once. The question whether a given
graph admits a Hamiltionian cycle or not, is one of the main problems in
graph theory.
A graph is connected, if there exists a path between any two vertices of G.
A graph G = (V, E) is a tree, if G is connected and does not contain a cycle.
Fig. 1.2 depicts two different trees on 4 vertices. Trees play a crucial role
in graph optimisation. The following fundamental theorem exhibits several
