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A modified discrepancy principle to attain optimal convergence rates under unknown noise. (English) Zbl 1472.62066

Summary: We consider a linear ill-posed equation in the Hilbert space setting. Multiple independent unbiased measurements of the right-hand side are available. A natural approach is to take the average of the measurements as an approximation of the right-hand side and to estimate the data error as the inverse of the square root of the number of measurements. We calculate the optimal convergence rate (as the number of measurements tends to infinity) under classical source conditions and introduce a modified discrepancy principle, which asymptotically attains this rate.

MSC:

62G20 Asymptotic properties of nonparametric inference
65J22 Numerical solution to inverse problems in abstract spaces
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