This course will provide graduate students with a panorama of functional analysis and approximation theory in multiple dimensions, adopting a systematic dual point of view (functions defined through a collection of measurements, weak formulations). The emphasis will be laid on the simplest, albeit modern mathematical concepts and mechanisms, with a view to avoid extraneous formalism and more abstract (e.g., topological) considerations. This knowledge will be used to devise methods for solving exactly or approximately various inverse problems; e.g., resulting from partial differential equations.
This knowledge will be used to model engineering problems (e.g., data acquisition, sampling), to devise methods for solving exactly or approximately the inverse problems that are related (e.g., resulting from partial differential equations), and to analyse the error resulting from the approximations.
Course contents: elements of
- measure theory, Lebesgue integral, Hilbert spaces
- generalized functions/distributions (delta, etc.), duality
- Fourier, convolutions
- functionals, Lagrangians, variational calculus
- partial differential equations, integral equations, Green functions
- approximations in partial differential equations
- approximation of functions, Ritz-Galerkin approximations
- splines, finite elements, wavelets, Gabor functions
- interpolation of uniform and scattered data, radial basis functions
- examples in electromagnetism, continuum mechanics, image processing etc.
The goal of this course is, for the students, to learn the most important mathematical ideas that are needed to be able to solve mathematically and numerically a number of multivariate problems that arise in engineering fields (including physics, signal processing). It is not meant to provide very detailed, complete mathematical coverage of each theory encountered, though.
At the end of this course, the students are expected to be able to
- Understand the working principle of the theory of measure/integration (Lebesgue)
- Understand and apply generalized functions in an engineering/physics context
- Understand and apply the Fourier transformation in multiple dimensions
- Understand and apply variational calculus
- Understand and apply exact approaches for solving partial differential equations
- Understand and apply numerical approaches for solving partial differential equations (approximation theory)
- Apply a part of the knowledge learnt in a Matlab project (computer implementation)