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PhD Course in Applied Nonsmooth Analysis and Multilevel Methods
Location and Date: Copenhagen, Denmark, January 30thFebruary 3rd 2012
This course aims to provide the participants with a basic working understanding of nonsmooth analysis and understanding of multiscale methods. The intent is that upon completion of the course the students will be able to apply nonsmooth numerical methods on own research problems. This knowledge is gained through theory lectures on the elements of nonsmooth analysis and multiscale methods. The theory is supported by study group exercises.
Specific application examples will be given in areas of Machine learning, Variational Methods for Video Sequences Processing, Computational Contact Mechanics, Elastodynamics and more.
Learning Goals
At completion of the course the student will be able to
 explain basic concepts and definitions in nonsmooth analysis, such as generalized
Jacobians, subdifferentials, and give examples of simple problems illustrating the
definitions and concepts  identify various nonsmooth mathematical problems, VI, NCP, LCP, PROX,
constrained PDEs etc.  derive and implement nonsmooth and semismooth Newton methods
 explain basic concepts and definitions of multiscale methods
 explain how to apply multiscale to nonsmooth problems in frictional
twobody contact problems  elaborate on where nonsmooth modeling appears in Machine Learning, Video
Compression, Contact Mechanics etc.  apply the numerical methods taught during the course to own problems
Recommended Prerequisites
It is expected that students
 can apply introductory level mathematical analysis (apply limits, differentiate and integrate vector functions etc.), equivalent to first year undergraduate university level course
 are well versed in linear algebra (vectors and matrices, vector spaces, norms etc. vector products, convexity), equivalent to first year undergraduate university level course
 are familiar with calculus of variation (functional and EulerLagrange equations)
 understand data structures and algorithms in a computer science sense and use high level programming languages, equivalent to undergraduate computer science university level
 have an understanding of classical mechanics on at least high school level (Newtonian
mechanics)
Lecturers
 Professor Michael Ulbrich, Technical University of Munich, Germany.
Homepage: http://wwwm1.ma.tum.de/bin/view/Lehrstuhl/MichaelUlbrich  Professor Rolf Krause, Università della Svizzera Italiana, Lugano.
Homepage: http://www.ics.inf.usi.ch/people/profrolfkrause.html  Professor Christian Igel, Image Group, University of Copenhagen.
Homepage: http://www.diku.dk/english/staff/beskrivelse/?id=400547
 Associate Professor Francois Lauze, Image Group, University of Copenhagen.
Homepage: http://www.diku.dk/english/staff/beskrivelse/?id=200294  Associate Professor Kenny Erleben, Image Group, University of Copenhagen.
Homepage: http://www.diku.dk/english/staff/beskrivelse/?id=110537
Detailed Course Content
As an appetizer, Prof. Erleben will introduce linear complementarity problems (LCPs) and show a few problem examples of nonsmooth modeling from the area of physicsbased animation. Following this Erleben will present a few selected numerical methods for this particular class of problems. An open source Matlab library of LCP solvers will be provided for the participants to play and experiment with.
Prof Ulbrich will address important aspects of nonsmooth analysis and nonsmooth equations and possibly nonsmooth minimization. Topics intended to be covered are: complementarity problems (CP) and variational inequalities (VI), nonsmooth reformulation of CPs, VIs, and other problems, elements of finitedimensional nonsmooth analysis, semismoothness and semismooth Newton methods, constrained optimization with partial differential equations (PDEs) and much more.
Prof. Lauze will talk on variational methods for recovery of optical flow in vector valued sequences and estimation of missing data in video sequences. The variational formulation involves L1 norms and benefit greatly of nonsmooth optimization techniques, projections onto convex sets (POCS), proximal operators etc.
Prof. Igel will give an introduction to machine learning (ML) and address the problem of solving support vector machines (SVMs) using sequential minimal optimization (SMO). The main message will be that by considering the dual one can sometimes circumvent nondifferentiability issues.
Prof. Krause will give a short introduction to continuum mechanics and frictional contact problems and discuss nonsmooth multiscale methods and possibly nonlinear domain decomposition methods for solving contact problems. On the discretization side, Mortar methods will be introduced for treating twobody contact problem with comments on the multi scale nature of friction as motivation for multiscale problems. A short overview on multiscale coupling methods and on our Mortarrelated approach in multiscale coupling will be given and if time permits comment on the discretization in time of inequality constrained problems in mechanics will close this session.
Course material
Lecture notes
Prof. Kenny Erleben's lecture notes  Linear Complementarity Problems (.pdf file)
Prof. Christian Igel's lecture notes  Machine Learning: Kernelbased Method (.pdf file)
Prof. Francois Lauze's lecture notes  Convex Methods in Image Analysis (.pdf file)
MatLab source code
Prof. Kenny Erleben  Linear Complementarity Problems (.zip file)
Exercise text
Prof. Rolf Krause  Exercises on the Projected GaussSeidel method (.pdf file)
Recommended literature
A.M. Sändig: Variational Methods for nonlinear boundary value problems in elasticity
Glowinski et al: Numerical analysis of variational inequalities
Course Credit
5 ECTS
To obtain course credit, students must give a short presentation of their
PhD project and participate in all programming exercises
Registration
Registration via this page is closed. For inquiries regarding late registration, please send a mail to niebe@diku.dk.
The course fees are:
 Free of charge for all PhD students in academia
 300€ for others
Program
Preliminary program.
Hour  Monday  Tuesday 
Wednesday 
Thursday 
Friday 
0910am  Registration  
1011am  Kenny Erleben 
Michael Ulbrich 
Francois Lauze 
Christian Igel 
Rolf Krause 
1112pm  
1213pm  Lunch 

1314pm  Michael Ulbrich 
Michael Ulbrich  Rolf Krause 
Presentations  Rolf Krause 
1415pm  
1516pm  Exercise 1 
Exercise 2  Exercise 3 
Exercise 4 

1617pm  
Evening  Network event 