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MA4K2 Content

Content:
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.

Aims:
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.

Objectives:
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.

Books:
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