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Unformatted text preview: 1 Lecture 14 1.1 The SVE, SVD and Regularization Reference: Rank-Deficient and Discrete ill-posed problems Christian Hansen, SIAM A Fredholm integral equation of the first kind Find f ( t ) such that Z 1 K ( s, t ) f ( t ) dt = g ( s ) (1) with 0 ≤ s ≤ 1 and k K k 2 ≡ R 1 R 1 K ( s, t ) 2 dsdt ≤ C . Moreover g is the data and K ( s, t ) is the kernel arising from the mathematical model. Often, g is only known at discrete points s 1 , . . . , s m . This lead to Z 1 k i ( t ) f ( t ) dt = b i (2) where K i ( t ) = K ( s i , t ) and b i = g ( s i ). The problem of determining f ( t ) in equation (1) and (2) is ill-posed. Example: Suppose f ( t ) = sin(2 πpt ) , p = 1 , 2 . . . . Then g ( s ) = R 1 K ( s, t ) sin(2 πpt ) dt. By Riemann- Lebesque lemma, lim p →∞ g ( s ) = 0. But as p → ∞ , sin(2 πpt ) becomes highly oscillatory. So small changes in g can correspond to large changes in f ....
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This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.
- Fall '08