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Two results on ill-posed problems

By A. G. Ramm


Let A = A ∗ be a linear operator in a Hilbert space H. Assume that equation Au = f (1) is solvable, not necessarily uniquely, and y is its minimal-norm solution. Assume that problem (1) is ill-posed. Let fδ, ||f − fδ| | ≤ δ, be noisy data, which are given, while f is not known. Variational regularization of problem (1) leads to an equation A ∗ Au + αu = A ∗ fδ. Operation count for solving this equation is much higher, than for solving the equation (A + ia)u = fδ (2). The first result is the theorem which says that if a = a(δ), limδ→0 a(δ) = 0 and limδ→0 δ a(δ) = 0, then the unique solution uδ to equation (2), with a = a(δ), has the property limδ→0 ||uδ −y| | = 0. The second result is an iterative method for stable calculation of the values of unbounded operator on elements given with an error

Year: 2007
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