Research Group Numerical Mathematics
(Partial Differential Equations)
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Optimal Control of Partial Differential Equations (4V, 2Ü)
Contents
Many processes in industry and natural sciences can be modeled by partial differential equations.
Some examples are
- heat conduction
- cooling of steel profiles
- the behavior of quantum physical particles
- chemical reactions
- crystal growth
- the propagation of sound and electromagnetic waves
- the motion of fluids.
Besides the numerical simulation of such processes, one is often interested in their optimization, e.g., for economical reasons, or in order to improve material qualities.
In this class you will
- get to know some basic examples of optimal control problems with PDEs,
- learn about necessary and sufficient optimality conditions (as a starting point for numerical solution schemes),
- learn to use numerical methods for the solution of optimal control problems.
You may also consider the list of all classes for additional information.
Dates
Exam
There will be oral examinations for participants who would like to acquire a „Fachprüfung“ (subject examination)
or a „Schein mit Note“ (certificate with mark). For participants who would like a
„Schein ohne Note“ (certificate without mark) it is sufficient to hand in
at least 10 successfully processed homework problems (out of a total of 14 problem sheets).
Additional lecture material
- §1 Einführung und Motivation (lecture notes of the first class meeting on October 15, 2008, in German)
- Beiblatt 1 (material concerning §11.5: projection formula for an optimal boundary control problem, for the class meeting on December 01, 2008, in German)
- Beiblatt 2 (material concerning §12: construction of test problems, for the class meeting on December 01, 2008, in German)
- Kapitel 3: Einführung in numerische Verfahren (lecture notes for the class meeting on December 08, 2008, in German)
- Beiblatt 3 (material concerning §14.3: discretization by the finite element method, for the class meeting on December 11, 2008, in German)
- Beiblatt 4 (material concerning §14.3: discretization by the finite element method, for the class meeting on December 11, 2008, in German)
- Beiblatt 5 (material concerning §14.3: discretization by the finite element method, for the class meeting on December 15, 2008, in German)
- Beiblatt 6 (material concerning §14.4 and §14.5: gradient method and method of conjugate gradients, for the class meeting on December 18, 2008, in German)
- Beiblatt 7 (material concerning §18: Sobolev Embedding Theorem and mapping properties of the trace operator, for the class meeting on January 19, 2009, in German)
- Beiblatt 8 (material concerning §20: Solutions in L∞, for the class meeting on January 22, 2009, in German)
- Beiblatt 9 (material concerning §21: Nemyzki-operators, for the class meeting on January 26, 2009, in German)
- Beiblatt 10 (material concerning §22: Existence of a optimal control, for the class meeting on January 29, 2009, in German)
- Beiblatt 11 (material concerning §24: First Order Necessary Optimality Conditions, for the class meeting on February 05, 2009, in German)
Tutorials
Additional tutorial material
- Matlab-program for demonstration of the 1D instationary heat equation
- Matlab-program for demonstration of the influence of the parameter lambda
- Plot of the function log | log (r/2) |
- Uebung10_Aufgabe2.m, Robin_1_0.m, slope_triangle.m
- Ue11Auf2.m, Ue11Auf2_A.m, Ue11Auf2_S.m, Ue11Auf2_Sstar.m, Ue11Auf2_scalar_product.m, setup_Ue11Auf2.m, steepest_descent.m, conjugate_gradients.m
- Ue12Auf2.m, Ue12Auf2_Phi.m, Ue12Auf2_derivative_of_projection.m, Ue12Auf2_projection.m, Ue12Auf3_determine_active_sets.m, Ue12Auf3_solve_equality_constrained_subproblem.m, Armijo.m, primal_dual_active_set.m, project_to_discontinuous.m, projected_gradients.m, setup_Ue12Auf2.m
- Solution of exercise 12, problem 2c
- Solution of exercise 14, problem 3
References
The lecture follows the book (in German)
In addition, you may want to consider a book about functional analysis, e.g.,,
For auxiliary results we will sometimes refer to the following books:
- Adams, Fournier: Sobolev Spaces, Academic Press, 2003
- Kinderlehrer, Stampacchia: An Introduction to Variational Inequalities and their Applications, SIAM, 2000
- Luenberger: Optimization by Vector Space Methods, Wiley, 1998
- Zeidler: Nonlinear Functional Analysis and its Applications, Volume II/B (Nonlinear monotone operators), Springer, 1990
