Math 273b, Section 1, Fall 2016
Calculus of Variations
Lecture Meeting Time: Mon, Wed, and Fri 12.00pm - 12.50pm.
Lecture Location: MS 6221.
Instructor: Luminita A. Vese
Office: MS 7620 D
Office hours: MWF 1pm - 1.30pm (after the class).
General Course Description: Application of abstract mathematical theory to optimization problems of calculus of variations.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, 1999 (new edition)
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).
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,
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.
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)
Shape optimization and applications to image processing.
Functions of bounded variation, minimization of the total variation, and applications to image processing.
Matlab Optimization Toolbox
Optimization Center at Northwestern University
SIAM Activity Group on Optimization
Convex Analysis and Optimization by Dimitri P. Bertsekas
Computational Convex Analysis - CCA numerical library by Yves Lucet
There will be several homework assignments with theoretical and computational questions.
Summary of optimality conditions
Notes on Stable and Normal Problems (following Ekeland-Temam)
Connections with the finite dimensional case
Notations for Sobolev Spaces
Homework Assignments, Projects & Practice Problems:
Homework #1 (pdf) (due on Friday, October 21)
Homework #1 (Latex)
Homework #2 (pdf) (due on Wednesday, November 23rd)
Homework #2 (Latex)
Homework #3 (pdf) (due on Wednesday, November 30)
Homework #3 (Latex)