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Iterative methods for solving integral equations
Author(s) -
Kleinman R. E.,
den Berg P. M.
Publication year - 1991
Publication title -
radio science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.371
H-Index - 84
eISSN - 1944-799X
pISSN - 0048-6604
DOI - 10.1029/90rs00934
Subject(s) - krylov subspace , conjugate gradient method , mathematics , conjugate residual method , derivation of the conjugate gradient method , iterative method , integral equation , operator (biology) , self adjoint operator , convergence (economics) , neumann series , mathematical analysis , mathematical optimization , computer science , hilbert space , gradient descent , biochemistry , chemistry , repressor , machine learning , artificial neural network , transcription factor , economics , gene , economic growth
A number of iterative algorithms to solve integral equations arising in field problems are discussed. We describe the essential features of the Neumann Series, overrelaxation methods, Krylov subspace methods, and the conjugate gradient technique. Proofs of convergence of the conjugate gradient method are directly available when the underlying integral operator is self‐adjoint, and in this case the method is equivalent to the Krylov method. However, for non‐self‐adjoint operators the conjugate gradient method requires an implicit symmetrization which results in poorer convergence than that obtained using the Krylov method. Some convergence results are also available for overrelaxation methods for both self‐adjoint and non‐self‐adjoint operators. Relations between all of the methods will be described and numerical performance will be contrasted using a uniform square error criterion. All the methods are treated in the continuous operator form which is especially useful in using the physical setting to arrive at effective preconditioners.