Minimization of errors of calculating wavelet coefficients while solving inverse problems

Statistical inverse problems arise in many applied fields, including medicine, astronomy, biology, plasma physics, chemistry, etc. At the same time, there are always errors in the observed data due to imperfect equipment, background noise, data discretization, and other reasons. To reduce these errors, it is necessary to apply special regularization methods that allow constructing approximate stable solutions of inverse problems. The classical regularization methods are based on the use of windowed singular value decomposition. However, this approach takes into account only the type of operator involved in the formation of observable data and does not take into account the properties of the object of observation. For linear homogeneous operators, this problem is solved with the help of special methods of wavelet analysis, which allow adapting simultaneously to the form of the operator and local features of the function describing the object. In this paper, the authors consider the problem of inverting a linear homogeneous operator in the presence of noise in the observational data by thresholding the wavelet expansion coefficients of the observed function. The asymptotically optimal thresholds and orders of the loss function are calculated when minimizing the averaged probability of error of wavelet coefficient calculation.