Two results on ill-posed problems

2005/11/14 by А. Г. Рамм, Ramm, A. G.
Computer Science · Mathematics · #35R30 #45A50 #47A05 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.math/0511354

openalex publication_date 2005/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 f_\d, ||f-f_d||≤ \d, be noisy data, which are given, while f is not known. Variational regularization of problem (1) leads to an equation A^*Au+\a u=A^*f_\d. Operation count for solving this equation is much higher, than for solving the equation (A+ia)u=f_\d (2). The first result is the theorem which says that if a=a(\d), lim_{\d → 0}a(\d)=0 and lim_{\d → 0}\frac \d a(\d)=0, then the unique solution u_\d to equation (2), with a=a(\d), has the property lim_{\d → 0}||u_\d-y||=0. The second result is an iterative method for stable calculation of the values of unbounded operator on elements given with an error.

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