AMCS 325 Hierarchical Basis and Wavelets



We study Hierarchical Basis (HB) and its inherent ability to represent linear operators in a compact efficient representation. Due to decorrelation across scales, the HB representation of certain linear operators is easy to obtain. In particular, these linear operators can be sparsified, pre-conditioned or decoupled. First, we introduce the concept of wavelets and its generalization to Adapted Hierarchical Basis. Wavelets and their Fourier properties are discussed. Applications to the sparsification and preconditioning of linear operators on equidistant grid domains are introduced and their limitations pointed out. However, for non equidistant geometries we can extend wavelets to Hierarchical Basis (HB). We discuss the construction of polynomial orthogonal HB that allows the sparsification and pre-conditioning of Integral Equations and Radial Basis Function (RBF) interpolation for non trivial geometries. Moreover, the HB approach stabilizes and efficiently solves the RBF interpolation and Best Linear Unbiased Estimation problem. HB is further generalized to allow scale decoupling of Partial Differential Equations. Finally, we will introduce the concept behind compressive sensing and explore applications to linear operators. The course will include recitations to work out some homework problems and a project as evaluation.
Course period02/5/11 → …
Course level300