24 research outputs found
Accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes
The von Neumann-Halperin method of alternating projections converges strongly
to the projection of a given point onto the intersection of finitely many
closed affine subspaces. We propose acceleration schemes making use of two
ideas: Firstly, each projection onto an affine subspace identifies a hyperplane
of codimension 1 containing the intersection, and secondly, it is easy to
project onto a finite intersection of such hyperplanes. We give conditions for
which our accelerations converge strongly. Finally, we perform numerical
experiments to show that these accelerations perform well for a matrix model
updating problem.Comment: 16 pages, 3 figures (Corrected minor typos in Remark 2.2, Algorithm
2.5, proof of Theorem 3.12, as well as elaborated on certain proof
Functions with Prescribed Best Linear Approximations
A common problem in applied mathematics is to find a function in a Hilbert
space with prescribed best approximations from a finite number of closed vector
subspaces. In the present paper we study the question of the existence of
solutions to such problems. A finite family of subspaces is said to satisfy the
\emph{Inverse Best Approximation Property (IBAP)} if there exists a point that
admits any selection of points from these subspaces as best approximations. We
provide various characterizations of the IBAP in terms of the geometry of the
subspaces. Connections between the IBAP and the linear convergence rate of the
periodic projection algorithm for solving the underlying affine feasibility
problem are also established. The results are applied to problems in harmonic
analysis, integral equations, signal theory, and wavelet frames
How good are projection methods for convex feasibility problems?
We consider simple projection methods for solving convex feasibility problems. Both successive and sequential methods are considered, and heuristics to improve these are suggested. Unfortunately, particularly given the large literature which might make one think otherwise, numerical tests indicate that in general none of the variants considered are especially effective or competitive with more sophisticated alternatives