PhD Course in Applied Nonsmooth Analysis and Multilevel Methods – University of Copenhagen

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PhD Course in Applied Nonsmooth Analysis and Multilevel Methods

Location and Date: Copenhagen, Denmark, January 30th-February 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, sub-differentials, 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
    two-body 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 Euler-Lagrange 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


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 physics-based 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 finite-dimensional 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 non-smooth 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 non-differentiability 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 two-body contact problem with comments on the multi scale nature of friction as motivation for multi-scale problems. A short overview on multiscale coupling methods and on our Mortar-related 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: Kernel-based 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 Gauss-Seidel 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

To obtain course credit, students must give a short presentation of their PhD project and participate in all programming exercises


Registration via this page is closed. For inquiries regarding late registration, please send a mail to

The course fees are:

  • Free of charge for all PhD students in academia
  • 300€ for others


Preliminary program.

 Hour  Monday  Tuesday
 09-10am  Registration  
 10-11am  Kenny Erleben
 Michael Ulbrich
 Francois Lauze
 Christian Igel
 Rolf Krause
 12-13pm Lunch
 13-14pm  Michael Ulbrich
 Michael Ulbrich  Rolf Krause
 Presentations  Rolf Krause
 15-16pm  Exercise 1
 Exercise 2  Exercise 3
 Exercise 4
 Evening    Network event