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- W1964577368 abstract "Abstract Consider the prototype ill-posed problem of a first kind integral equation ℛ with discrete noisy data d i , = f(x i ) + ε i , i = 1, …, n . Let u 0 be the true solution and u n α a regularised solution with regularisation parameter α. Under certain assumptions, it is known that if α → 0 but not too quickly as n → ∞, then u n α converges to u 0 . We examine the dependence of the optimal sequence of α and resulting optimal convergence rate on the smoothness of f or u 0 , the kernel K , the order of regularisation m and the error norm used. Some important implications are made, including the fact that m must be sufficiently high relative to the smoothness of u 0 in order to ensure optimal convergence. An optimal filtering criterion is used to determine the order where is the maximum smoothness of u 0 . Two practical methods for estimating the optimal α, the unbiased risk estimate and generalised cross validation, are also discussed." @default.
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- W1964577368 date "1988-07-01" @default.
- W1964577368 modified "2023-09-23" @default.
- W1964577368 title "Assessing regularised solutions" @default.
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- W1964577368 doi "https://doi.org/10.1017/s0334270000006019" @default.
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