A unifying theory for nonlinear additively and multiplicatively preconditioned globalization strategies [Elektronische Ressource] : convergence results and examples from the field of nonlinear elastostatics and elastodynamics / vorgelegt von Christian Groß

Christian Groß

Sujets

• Mathematics
• Physik

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2009
• Nombre de lectures: 8
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

A Unifying Theory for Nonlinear Additively and
Multiplicatively Preconditioned Globalization Strategies:
Convergence Results and Examples From the Field of
Nonlinear Elastostatics and Elastodynamics
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakulta¨t
der
Rheinischen Friedrich-Wilhelms-Universita¨t Bonn
Vorgelegt von
Christian Groß
aus
Remagen
Bonn, Juli 2009I
Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakulta¨t der Rheinischen
Friedrich-Wilhelms Univerversita¨t Bonn
1. Gutachter: Prof. Dr. Rolf Krause
2. Gutachter: Prof. Dr. Helmut Harbrecht
Tag der Promotion: 11.09.2009
Diese Arbeit ist mit Unterstu¨tzung der von der Deutschen Forschungsgemeinschaft getragenen
Bonn International Graduate School (BIGS) und des SFB 611 entstanden.II
Mano Brangutei
Fu¨r meine LiebsteAbstract
The solution of nonlinear programming problems is of paramount interest for various applications,
such as for problems arising from the field of elasticity. Here, the objective function is a smooth,
but nonlinear and possibly nonconvex functional describing the stress-strain relationship for material
classes. Often, additional constraints are added to model, for instance, contact. The discretization
of the resulting partial differential equations, for example with Finite Elements, gives rise to a finite
dimensional minimization problem of the kind
nu∈B⊂R :J(u) = min! (M)
nwheren ∈N, andJ :R →R, sufficiently smooth. The set of admissible solutionsB is given by
n nB ={u∈R |φ ≤u ≤φ for alli = 1,...,n} whereφ,φ∈R .i ii
The solution of such a minimization problem can be carried out with various numerical methods.
From an analytical point of view it is of interest under which assumptions a numerical solution
strategy computes a (local) solution of the minimization problem. Here, basically two classes of
globalization strategies, Linesearch and Trust-Region methods, exist which are able to solve (M)
even if J is nonconvex. Though, the interest of a user lies in the efficiency and robustness of the
employed tool. In fact, it is of great importance that a solution is, independent of the employed
parameters, rapidly carried out.
In particular, a modern nonlinear solution strategy must necessarily be able to be applied for (mas-
sive) parallel computing. The first step would, indeed, be employing parallelized linear algebra for
the Trust-Region and Linesearch strategy. But, to guarantee convergence, traditional solution strate-
gies damp the computed Newton corrections which might slow down the convergence.
Therefore, different extensions for the traditional schemes were developed, such as the two (additive)
schemes PARALLEL VARIABLE DISTRIBUTION (PVD) [FM94], PARALLEL GRADIENT DISTRIBU-
TION (PGD) [Man95] and the (multiplicative) schemes MG/OPT [Nas00], recursive Trust-Region
methods (RMTR) [GST08, GK08b] and recursive Linesearch methods (MLS) [WG08]. Both, the
nonlinear additive and multiplicative scheme, aim at a solution of related but “smaller” minimization
problems to compute corrections or search directions. In particular, the paradigm of the PVD and
PGD schemes is to asynchronously compute solutions of local minimization problems which are
combined to a global correction. The recombination process itself is the solution of another non-
linear programming problem. The multiplicative schemes, in contrast, aim at a solution of coarse
level problems starting from a projection of the current fine level iterate. As numerical examples
+in [GK08b, GMS 09] and [WG08] have shown, combining multiplicative schemes with a “global”
smoothing step yields clearly improved rates of convergence with little computational overhead.
In the present thesis we will show that these additive and multiplicative schemes can be regarded as
a nonlinear right preconditioning of a globalization strategy. Moreover, novel, generalized nonlinear
additive and multiplicative frameworks are introduced which fit into the nonlinear preconditioning
context. In numerous examples, we comment on the relationship to state-of-the-art domain decom-
position frameworks such as hierarchical and vertical decompositions and explain how these decom-
positions fit into the presented context. In a second step, Trust-Region and Linesearch variants of the
preconditioning frameworks are presented and first–order convergence is shown.IV
As it turns out, the presented multiplicative Trust-Region concept is based on the RMTR framework
employed in [GK08b] extending it to more arbitrary domain decompositions. On the other hand,
the multiplicative Linesearch methods are based on the MLS scheme in [WG08]. Here, the original
assumptions are weakend allowing for the solution of non-smooth nonlinear programming prob-
lems. Moreover, we present a novel nonlinear additive preconditioning framework, along with actual
Trust-Region and Linesearch implementations. As it turns out, well-balanced a priori and a posteriori
strategies and a novel subset objective function which allow for straight-forwardly implementing the
presented frameworks and showing first–order convergence. As will be highlighted, these novel ad-
ditive preconditioning strategies are perfectly suited to be employed for massive parallel computing.
Furthermore, remarks on second–order convergence are stated.
To motivate the presented solution strategies, systems of PDEs and equivalent minimization problems
arising from the field of elasto-statics and elasto-dynamics are introduced. Moreover, we will show
that – after discretization – the resulting objective functions satisfy the assumptions stated for show-
ing convergence of the respective globalization strategies. Furthermore, various numerical examples
employing these objective functions are presented showing the efficiency and robustness of the pre-
sented nonlinear preconditioning frameworks. Comments on the computation times, the number of
iterations, the computation of search directions, and the actual implementation of the frameworks are
stated.
Danksagung
An dieser Stelle mo¨chte ich mich bei meinem Erstbetreuer Rolf Krause bedanken, der mir
vorgeschlagen hat, dieses außerordentliche interessante, breite und anspruchsvolle Forschungsthema
zu bearbeiten. Zudem stand er mir oft mit Rat und Tat zur Seite und hat sich immer darum bemu¨ht,
dass ich meine Ergebnisse auch in fru¨hen Stadien meines Forschungsprojektes in (Konferenz-)
¨Vortra¨gen darstelle. Desweiteren danke ich Helmut Harbrecht fu¨r die reibungslose Ubernahme der
Betreuerpflichten an der Universita¨t Bonn, seine Unterstu¨tzung und sein ausfu¨hrliches Feedback.
Auch mo¨chte ich mich sehr bei Andreas Weber fu¨r die Fo¨rderung noch wa¨hrend meines Diplom-
studiums danken.
Besonderer Dank gilt meinen Kollegen Thomas Dickopf und Mirjam Walloth, die immer ein offenes
Ohr fu¨r oftmals technische Fragen hatten. Aufgrund ihrer u¨beraus aufgeschlossenen Einstellung
wurde oft aus einer Idee ein mathematisch korrektes Resultat. Gleichsam danke ich Johannes Steiner
und Britta Joswig fu¨r die Wirbelgeometrie, die sie erstellt haben und die ich im Abschnitt 5.6.8
verwenden durfte. Auch danke ich allen Kollegen am INS und am ICS, insbesondere Dorian Krause
fu¨r das schnelle Bereitstellen des Servers in Lugano.
Ich danke in besonderem Maße der Bonn International Graduate School, die mir nicht nur ein
großzu¨giges Promotionsstipendium gewa¨hrt hat, sondern auch viele Konferenzteilnahmen und einen
Aufenthalt an der Columbia University in the City of New York zu großen Teilen finanziert hat. Fu¨r
¨das Bereitstellen einer hervorragenden Infrastruktur danke ich besonders dem Institut fur Numerische
Simulation der Rheinischen Friedrich-Wilhelms Universita¨t Bonn und dem Institute of Computa-
tional Science der Universita` della Svizzera italiana in Lugano.
Die Begebenheiten, die zu dieser wissenschaftlichen Arbeit gefu¨hrt haben sind vielfa¨ltig. Jedoch
haben gerade die fru¨hen Weichenstellungen in ganz besonderem Maße dazu gefu¨hrt, dass ich studiert
habe und diese Arbeit nun geschrieben habe. Daher mo¨chte ich mich ganz besonders bei meinen
wichtigsten Fo¨rderern und Vorbildern, meinen Eltern Elisabeth und Wolfgang Groß und meinem
Bruder Thomas, bedanken. Zu guter Letzt mo¨chte ich mich ganz herzlich bei meiner Frau Rimante
fu¨r das viele Zuho¨ren, das gute Zureden und das Tolerieren u¨berlanger Arbeitstage bedanken.Contents
1 Introduction 1
1.1 The Nonlinear Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Constitutional Equations and their Discretization . . . . . . . . . . . . . . . . . 5
1.2.1 Kinematics and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 6
11.2.2 Elastodynamic and Elastostatic Model Problems inH . . . . . . . . . . . . 7
1.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 State of the Art Globalization Strategies 15
2.1 The “Traditional” Trust-Region Framework . . . . . . . . . . . . . .