Wednesday,
March 28, 2:00 p.m. GWC 604
W. Steven Rosenthal
Dept. Math. & Stats
Discrete Polynomial Least Squares Approximation for Image Reconstruction and Solving Partial Differential Equations
Abstract
Fourier spectral methods accurately approximate smooth and periodic
functions from equally-spaced data sets. However, the Gibbs
phenomenon destroys the Fourier reconstruction of non-periodic and/or
piecewise-smooth functions. High-order polynomial interpolations of
smooth, non-periodic functions given on equally-spaced points provoke
Runge effects near the boundaries. A natural alternative is to use a
Chebyshev distribution of nodes; although, in applications to
numerical partial differential equations (PDEs), stable time
integration is relatively slow. Moreover, the approximation method is
incompatible with imaging applications, as data is usually restricted
to an equally-spaced grid. Stretching the distribution to be more
equally-spaced can accelerate time integration, but with notable loss
in accuracy. We present a suitable alternative using a least-squares
approach on equally-spaced points. The method employs a polynomial
basis mutually orthogonal with respect to the Freud weight function to
obtain a spectrally-accurate approximation of a smooth function. We
show how to choose least-squares parameters to minimize the pointwise
error in the majority of the domain [-1,1]. Consequently, the Runge
effects are partially alleviated. This least-squares method forms the
foundation for a multi-domain method to integrate PDEs which yields a
less restrictive stability criterion and recovers exponential
convergence in [-1,1].
For further information please contact:
mittelmann@asu.edu