Randomized Kaczmarz solver for noisy linear systems

The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax = b . Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomiz...

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Vydáno v:	BIT (Nordisk Tidskrift for Informationsbehandling) Ročník 50; číslo 2; s. 395 - 403
Hlavní autor:	Needell, Deanna
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Dordrecht Springer Netherlands 01.06.2010 Springer
Témata:	Acceleration of convergence Algebra Algebraic geometry Computational Mathematics and Numerical Analysis Exact sciences and technology Mathematics Mathematics and Statistics Nonlinear algebraic and transcendental equations Numeric Computing Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Sciences and techniques of general use Randomized algorithms 65F10 65F22 Algebraic reconstruction technique Kaczmarz method 65F20 Transcendental equation Error estimation Convergence acceleration Numerical linear algebra Pseudoinverse Iterative method Algorithm Equation system Non linear equation Direct method Numerical analysis Linear system Matrix inversion overdetermined system Algebraic equation Convergence rate
ISSN:	0006-3835, 1572-9125
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Popis
Shrnutí:	The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax = b . Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method converges with expected exponential rate, independent of the number of equations in the system. Here we analyze the case where the system Ax = b is corrupted by noise, so we consider the system Ax ≈ b + r where r is an arbitrary error vector. We prove that in this noisy version, the randomized method reaches an error threshold dependent on the matrix A with the same rate as in the error-free case. We provide examples showing our results are sharp in the general context.
ISSN:	0006-3835 1572-9125
DOI:	10.1007/s10543-010-0265-5