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A Norm Convergence Result on Random Products of Relaxed Projections in Hilbert Space

Bauschke, Heinz H. (1994) A Norm Convergence Result on Random Products of Relaxed Projections in Hilbert Space. [Preprint]

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      Abstract

      Suppose $X$ is a Hilbert space and $C_1,\ldots,C_N$ are closed convex intersecting subsets with projections $P_1,\ldots,P_N$. Suppose further $r$ is a mapping from ${\Bbb N}$ onto $\{1,\ldots,N\}$ that assumes every value infinitely often. We prove (a more general version of) the following result: \begin{quote} If the $N$-tuple $(C_1,\ldots,C_N)$ is ``innately boundedly regular'', then the sequence $(x_n)$, defined by $$x_0 \in X ~\mbox{arbitrary}, ~~~ x_{n+1} := P_{r(n)}x_n, ~~\mbox{for all $n \geq 0$},$$ converges in norm to some point in $\bigcap_{i=1}^{N} C_i$. \end{quote} Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: Banach contraction, computerized tomography, convex feasibility problem, convex programming, convex set, Fejer monotone sequence, Hilbert space, image reconstruction, image recovery, innate bounded regularity, Kaczmarz's method, nonexpansive mapping, orthogonal projection, projection algorithm, projective mapping, random product, relaxation method, relaxed projection, signal processing, unrestricted iteration, unrestricted product
      Subjects: 92-xx Biology and other natural sciences, behavioral sciences > 92Cxx Physiological, cellular and medical topics
      90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
      65-xx Numerical analysis > 65Jxx Numerical analysis in abstract spaces
      47-xx Operator theory > 47Hxx Nonlinear operators and their properties
      47-xx Operator theory > 47Nxx Miscellaneous applications of operator theory
      65-xx Numerical analysis > 65Kxx Mathematical programming, optimization and variational techniques
      65-xx Numerical analysis > 65Fxx Numerical linear algebra
      46-xx Functional analysis > 46Cxx Inner product spaces and their generalizations, Hilbert spaces
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 16 Nov 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/76

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