Cascadic multilevel methods for ill-posed problems

Authors:
Lothar Reichel;Andriy Shyshkov
Affiliations:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA;Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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Abstract

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error. This paper shows that cascadic multilevel methods with a conjugate gradient-type method as basic iterative scheme are regularization methods. The iterations are terminated by a stopping rule based on the discrepancy principle.