The Geometric Minimum Action Method: A Least Action Principle on the Space of Curves

173 Pages·2007·4.85 MB·English

by  Matthias Heymann

The Geometric Minimum Action Method:
A Least Action Principle on the Space of Curves
by
Matthias Heymann
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
New York University
September 2007
Eric Vanden-Eijnden—Advisor
c Matthias Heymann
(cid:13)
All Rights Reserved, 2007
To my parents.
iii
Acknowledgements
How miraculous an invention the science of mathematics is: it enables us to
express some of the most beautiful and ingenious ideas of mankind by placing
only a few carefully selected symbols into the right order. Even more
miraculous it seems however that at the same time these symbols entirely fail to
express the blood, sweat and tears that oftentimes accompany the creation of
such pieces of work. Instead, the last evidence for those can be found in the
only part that is written in plain text: the acknowledgements.
Little to none of this particular work would exist without the help or influence
of several people who crossed my way in the decisive moments.
An important role during the first two years of my PhD was played by
Prof. Jonathan Goodman and Mr. Robert Fechter who invested both their time
and money into my scientific and non-scientific ideas, thus being the first people
at NYU to take me serious as a collaborator. Tamar Arnon gave me both logistic
support and much needed encouragement when late in my third year I had to
switch my advisor for the second time, and the fellowship committee gave me
the opportunity to use my then remaining time more effectively by generously
lowering my teaching load.
iv
Especially during that time I was very grateful that I could always rely on
the moral support of my parents and my girlfriend Sarah all of whom stubbornly
believed in me without truly understanding a single page of my work, and that
I could always count on my friend Giuseppe when I needed someone to explore
New York City with.
ThekeyplayerforthisthesishoweverwascertainlyProf.EricVanden-Eijnden
who has been more than only a scientific advisor in the past two years. Always
tryingtogetthebestforandoutofme, hespentlotsoftimewithmeinnumerous
one-on-one sessions (in person and via skype) to introduce me to large deviations
theory, and to fill many of my gaps in numerical analysis. When necessary, he
himself was working feverishly to push our project forward, throwing in many
decisive tricks to make my algorithms work, while at other times he gave me all
the freedom to explore more analytical questions by using my own set of tools. In
the end it was just this combination of our very distinct backgrounds that made
this collaboration so fruitful.
Finally, I want to thank Prof. Oliver Bu¨hler, Prof. Weinan E, Prof. Robert
Kohn, Prof. Maria Reznikoff and Prof. Marco Avellaneda for their useful com-
ments and suggestions.
“La la ......la la!”
little girl on swing, Washington Square Park
v
Abstract
Dynamicalsystemswithsmallnoise(e.g.SDEsorcontinuous-timeMarkovchains)
allow for rare events that would not be possible without the presence of noise,
e.g. for transitions from one stable state into another. Large deviations theory
provides the means to analyze both the frequency of these transitions and the
maximum likelihood transition path. The key object for the determination of
both is the quasipotential,
V(x ,x ) = inf S (ψ),
1 2 T
T,ψ
where S (ψ) is the action functional associated to the system, and where the
T
infimum is taken over all T > 0 and all paths ψ : [0,T] Rn leading from x to
1
→
x . The numerical evaluation of V(x ,x ) however is made difficult by the fact
2 1 2
that in most cases of interest no minimizer exists.
ˆ
Here, this issue is resolved by introducing the action S(ϕ) on the space of
ˆ
curves (i.e. S is independent of the parametrization of ϕ) and proving the alter-
native geometric formulation of the quasipotential
ˆ
V(x ,x ) = infS(ϕ),
1 2
ϕ
vi
where the infimum is taken over all curves ϕ : [0,1] Rn leading from x to
1
→
x . In this formulation a minimizer exists, and we use this formulation to build
2
a flexible algorithm (the geometric minimum action method, gMAM) for finding
the maximum likelihood transition curve.
We demonstrate on several examples that the gMAM performs well for SDEs,
SPDEs and continuous-time Markov chains, and we show how the gMAM can
be adjusted to solve also minimization problems with endpoint constraints or
endpoint penalties.
Finally, we apply the gMAM to problems from mathematical finance (the
valuation of European options) and synthetic biology (the design of reliable stan-
dard genetic parts). For the latter, we develop a new tool to identify sources of
instability in (genetic) networks that are modelled by continuous-time Markov
chains.
vii
Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 Introduction and main results 1
1.1 Freidlin-Wentzell theory of large deviations . . . . . . . . . . . . . 2
1.2 The role of the quasipotential . . . . . . . . . . . . . . . . . . . . 5
1.3 Geometric reformulation . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Numerical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Organization, notations and assumptions . . . . . . . . . . . . . . 12
2 Theoretical background 17
2.1 A large deviations action on the space of curves . . . . . . . . . . 17
2.2 Probabilistic interpretation . . . . . . . . . . . . . . . . . . . . . . 22
ˆ
2.3 Lower semi-continuity of S, proof of Proposition 1 . . . . . . . . . 25
viii
2.4 Recovering the time parametrization . . . . . . . . . . . . . . . . 32
3 Numerical algorithms 37
3.1 The Euler-Lagrange equation and the steepest descent flow . . . . 38
3.2 The outer loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Accuracy and efficiency of the outer loop . . . . . . . . . . . . . . 46
3.4 The inner loop (computing ϑˆ(ϕ,ϕ0)) . . . . . . . . . . . . . . . . . 49
4 Examples 54
4.1 SDE: The Maier-Stein model . . . . . . . . . . . . . . . . . . . . . 54
4.2 SPDE: An SPDE generalization of the Maier-Stein model . . . . . 62
4.2.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Continuous-time Markov chain: The genetic switch . . . . . . . . 77
5 Minimization with variable endpoints 82
5.1 Endpoint constraints . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Endpoint penalties . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Example: SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Finance application: Valuation of equity index options . . . . . . 93
6 Application: Synthetic Biology 106
6.1 Introduction to Synthetic Biology . . . . . . . . . . . . . . . . . . 106
6.2 The Genetic Switch . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 A tool to identify sources for instability in networks . . . . . . . . 115
6.4 Proof of Lemma 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
ix
7 Conclusions 125
Appendices 129
Bibliography 155
x