Math 273B, Section 1, Winter 2025

Calculus of Variations

Lecture Meeting Time: Mon, Wed, and Fri 4.00pm - 4.50pm.
Lecture Location: Online over Zoom (lectures recorded and lecture notes posted).
Instructor: Luminita A. Vese
Office hours: After lectures or by appointment. General Course Description: Application of abstract mathematical theory to optimization problems of calculus of variations.

References:
  • I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, 1999 (new edition)
    Online access restricted to UC campuses (2nd edition)
  • J. Nocedal and S.J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer 1999 (1st or 2nd edition).
    Online access restricted to UC campuses (2nd edition)
  • E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Variational Methods and Optimization , Springer-Verlag 1984.
  • P.E. Gill, W. Murray, and M.H. Wright, Practical Optimization, Academic Press 1981.
  • R.T. Rockafellar, Convex Analysis, Princeton University Press 1970.
  • J.-B. Hiriart-Urruty, C. Lemarechal, Fundamentals of Convex Analysis, Springer 2001.
  • S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004 (especially Chapters 9, 10 and 11).
  • M. Giaquinta, S. Hildebrandt, Calculus of variations, Springer, 1996 (two volumes).
  • D. Luenberger, Optimization by Vector Space Methods , John Wiley & Sons, 1969.
  • Dimitri P. Bertsekas, with Angelia Nedic and Asuman E. Ozdaglar, Convex Analysis and Optimization.
  • L.C. Evans, Partial Differential Equations , Chapter 8.
  • H. Attouch, G. Buttazzo, and G. Michaille, Variational Analysis in Sobolev and BV Spaces: applications to PDE's and optimization, MPS-SIAM 2006.

    Specific topics:

    Abstract formulations in calculus of variations and applications to minimization problems on Sobolev spaces. Several sections from Ekeland-Temam will be presented.
  • Abstract minimization problems, existence of minimizers, applications, duality techniques in the continuous case (Ekeland-Temam), polar functions, Lagrangians, saddle points.
  • Duality applied to a particular case on finite dimensional optimization.
  • Several notions of differentiability; characterization of minimizers; computation of Euler-Lagrange equation; associated gradient descent method for a general problem "Min F(u)" for u in V that decreases the objective function (associated time-dependent Euler-Lagrange equation).
  • Algorithms based on duality (from Ekeland - Temam)
  • Applications to abstract minimization problems and to minimization problems on Sobolev spaces; computation of the dual problem.
  • Chambolle's projection algorithm (application of duality)
  • Sobolev gradients
  • Shape optimization and applications to image processing.
  • Functions of bounded variation, minimization of the total variation, and applications to image processing.

    Links:
  • Matlab Optimization Toolbox
  • Optimization Online
  • Optimization Center at Northwestern University
  • SIAM Activity Group on Optimization
  • Numerical Recipies
  • NEOS Guide
  • Convex Analysis and Optimization by Dimitri P. Bertsekas
  • Computational Convex Analysis - CCA numerical library by Yves Lucet

    Assignments Policy: There will be several homework assignments with theoretical and computational questions. Class presentations and working on projects, individually or in group, are strongly encouraged.

    Notes:
  • Summary of optimality conditions
  • Notes on Stable and Normal Problems (following Ekeland-Temam)
  • Connections with the finite dimensional case
  • Duality Examples
  • Notations for Sobolev Spaces

    Homework Assignments, Projects & Practice Problems: