MTH 619: Convex Analysis and PDEs

LECTURE: MWF 1100 - 1150 BEXL 412 CRN 39581
Instructor: R.E. Showalter Kidder 286
Office Hours: Mondays 1500, Wednesdays 1600, and by appointment.
The minimization of quadratic functions over closed convex subsets has classical applications to boundary-value problems for elliptic partial differential equations, variational inequalities and optimal control problems, penalty or augmented Lagrangian methods and mixed formulations. We shall develop these ideas and applications to boundary-value problems and algorithms for their approximation. These topics will be placed in the more general context of convex analysis, saddle points and Lagrange multipliers, primal and dual problems in Hilbert space. These concepts will be illustrated with examples from mechanics (internal or boundary obstacles, friction), fluid flow (Stokes or Bingham flow) and diffusion (heat conduction, porous media flow).
The choice of Topics covered and Examples developed will depend in part on preferences of the class.
Prerequisite: MTH 582 or MTH 512 or MTH 622.
Special topics from linear analysis will be developed or described as needed. Contact instructor for questions about prerequisites or content, especially for general science or engineering students.
Accommodations for students with disabilities are determined and approved by Disability Access Services (DAS). If you, as a student, believe you are eligible for accommodations but have not obtained approval please contact DAS immediately at 541-737-4098 or at their website. DAS notifies students and faculty members of approved academic accommodations and coordinates implementation of those accommodations. While not required, students and faculty members are encouraged to discuss details of the implementation of individual accommodations.
NOTES will be developed and made available during the term.
Variational Method in Hilbert Space (VM) (Background material as needed.)
Optimization Topics (OT) (Chapter VII of Hilbert Space Methods for PDEs (HSM)).
Constrained Optimization (CO)

1. Snow Day (01/09)
2. Preview, pp. 1 - 4 in VM.
3. pp. 5 - 8 in VM. Read/review pp. 8 - 12.
4. MLK Holiday (01/16)
5. Examples of BVPs.
6. Divergence Theorem
7. minimization and pp. 12 - 13 in VM.(01/23)
8. Projection: pp. 13 - 15 in VM. Exercise 1 due on Day 12.
9. minimization again!, pp. 169 - 172 in OT.
10. derivative pp. 172 - 174. (01/30)
11. monotone, coercive pp. 174 - 177.
12. Examples: boundary constraints, pp. 178-180 in OT.
13. Examples: interior constraints, pp. 15 - 18 in VM (02/06).
14. Examples: continued; start reading Section II.2: Sobolev spaces in HSM.
15. Section II.3: Boundary Trace
16. Inf-sup condition, pp. 1-2 in CO. (02/13)
17. Equivalent conditions, p.3.
18. Mixed fornulation and Lagrange multiplier, pp, 3-5. Exercise 2 due on Day 21.
19. Saddle point & Lagrangian, pp, 6-7. (02/20)
20. p. 7-8.
21. p. 8-9.
22. No class. 02/27)
23. No class.
24. Neumann problem, pp. 9-11 in Constrained Optimization.
25. Trace & Normal Trace, p. 12. (03/06)
26. Dirichlet-Neumann Problem, pp. 13-14. Exercise 3, a,b,c due on March 24.
27. Mechanics 101.
28. Stokes Equation, pp. 20-21 in CO. (03/13)
29. pp. 22-23.