MTH 628: Variational Methods for PDEs


LECTURE: MWF 1400 - 1450 Kidder 356 CRN 35348
Instructor: R.E. Showalter Kidder 286 show@math.oregonstate.edu
We begin with the minimization of quadratic functions over closed convex subsets and applications to boundary-value problems for elliptic partial differential equations, variational inequalities and optimal control problems. These are prototypes of penalty or augmented Lagrangian methods and mixed formulations for problems of minimization with constraints, the characterization of their solutions, and algorithms for their approximation. These problems will be placed in the more general context of convex analysis, non-smooth optimization with constraints, saddle points and Lagrange multipliers, primal and dual problems. These concepts will be illustrated with classic 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. (MTH 627 is NOT prerequisite.)
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.

Material will be drawn from various advanced textbooks and NOTES will be developed and made available during the term.

BOOKS
Boyd: Convex Optimization
Luenberger: Optimization
NOTES
Optimization Topics
Variational Method
Constrained Optimization
Boundary-Value Problems
Stokes_3
Evolution Eqns
Problems

Schedule:
1. Introduction & Dirichlet's Principle, pp. 169 - 170 in Optimization Topics. (01/06)
Send email with recent related courses and objectives (tools, topics) to show@math.oregonstate.edu.
2.Minimization of Convex Functions, pp. 171 - 172.
3. derivative pp. 172 - 174.
4. monotone, coercive pp. 174 - 177. (01/13)
5. projection & more examples pp. 177-178.
6. Sobolev space; see pp. 9-10 of Variational Method.
7. No class. Read Sections 1 & 2 of Variational Method. (01/20)
8. M = VI = WVI = CC (cone) = VE (linear); Boundary Value Problem.
9. Examples: VE & CC on boundary, pp. 178-180 in Optimization Topics.
10. CC on interior of domain (01/27)
11. Optimal control of BVPs, pp. 181-184.
12. Boundary control; interior control, pp. 184-186.
13. Inf-sup condition, pp. 1-3 in Constrained Optimization. (02/03)
14. Closed range theorem, pp. 3-5.
15. Snow Day.
16. Mixed systems, Saddle-point, pp. 5-8. (02/10)
17. MinMax=MaxMin, primal problem, pp. 9-10.
18. dual problem, pp. 11-12.
19. Neumann problem, pp. 1-2 in Boundary-Value Problems. (02/17)
20. Trace and normal trace, pp. 3-4.
21. Gradient mixed formulation of Dirichlet-Neumann problem, p. 5.
22. Divergence mixed formulations of Dirichlet-Neumann problem, pp. 6-7. (02/24)
23. Interface problems: direct formulation, pp. 8 - 10.
24. Interface problems: a mixed-mixed formulation, pp. 11-12.
25. Stokes system, pp. 1-2 in Stokes. (03/03)
26. stationary equation, pp. 3-4.
27. mixed system, 4-5.
28. non-homogeneous system, mixed boundary conditions, 6-8. (03/10)
29. evolution equations, p. 1 in Evolution Eqns.
30. evolution systems, pp. 2-5.