We will cover some of the following topics:-
- Optimisation in Banach spaces.
- Optimisation in Hilbert spaces with and without constraints.
- Optimality conditions and Lagrange multipliers.
- Lower semi-continuity.
- Convex functionals.
- Variational inequalities
- Gradient descent and iterative methods.
- Banach, Brouwer Schauder fixed point theorems.
- Monotone mappings.
- Applications in differential equations, inverse problems, optimal control, obstacle problems, imaging.
The module will form a fourth year option on the MMath Degree.It builds upon modules in the second and third year like Metric Spaces, Functional Analysis I and Theory of PDEs to present some fundamental ideas in nonlinear functional analysis with a view to important applications, primarily in optimisation and differential equations. The aims are: introduce the concept of unconstrianed and constrained optimisation in Banach and Hilbert spaces; existence theorems for nonlinear equations; importance in applications to calculus of variations, PDEs, optimal control and inverse problems.
By the end of the module the student should be able to:-
- Recognise situations where existence questions can be formulated in terms of fixed point problems or optimisation problems.
- Recognise where the Banach fixed point approach can be used.
- Apply Brouwers and Schauders fixed point theorems.
- Apply the direct method in the calculus of variations.
- Apply elementary iterative methods for fixed point equations and optimisation.
The instructor has own printed lecture notes which will provide the primary source. The printed lecture notes will also have a bibliography.
List A (These books contain material directly relevant to the module):-
- G. Allaire, Numerical analysis and optimisation, Oxford Science Publications 2009
- P.G. Ciarlet, Linear and nonlinear functional analysis with applications. SIAM 2013
- P. G. Ciarlet, Introduction to numerical linear algebra and optimisation, Cambridge 1989
- L.C. Evans, Partial Differential Equations , Graduate Studies in Mathematics 19, AMS, 1998.
- F. Troltzsch, Optimal control of partial differential equations AMS Grad Stud Math Vol 112 (2010)
List B (The following texts contain relevant and more advanced material):-
- G. Aubert and P. Kornprobst. Mathematical problems in Image Processing, Applied Mathematical Sciences (147). Springer Verlag 2006.
- M. Chipot. Elements of nonlinear analysis . Birkhauser, Basel-Boston-Berlin, 2000.
- D. Kinderleher and G. Stampacchia, An introduction to variational inequalities and their applications Academic Press 1980
- E. Zeidler, Nonlinear functional analysis and its applications I, Fixed Point theorems , Springer New York, 1986