## Math 273b, Section 1, Spring 2016

## Calculus of Variations

**Lecture Meeting Time:** Mon, Wed, and Fri 2.00pm - 2.50pm.

** Lecture Location:** MS 5138.

**Instructor:** Luminita A. Vese

**Office:** MS 7620 D

**Office hours:** TBA

**E-mail:** lvese@math.ucla.edu

** 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).
Applications to abstract minimization problems and to minimization problems on Sobolev spaces; computation of the dual problem.
Sobolev gradients
Shape optimization and applications to image processing.
Other topics

** 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.

**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:**
* Homework #1 (due on Wednesday, April 20)*

* Latex file *
*
Homework #2 (due on Monday, May 2nd)*

* Latex file *
*
Homework #3 (due on Friday, June 3rd)*

* Latex file
*