2019/08/16 by Peter Mathé, Mathé, Peter, M. Thamban Nair +3
Engineering · Mathematics · #47A52 #62G05 #65J20 #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Statistics Theory (math.ST) #Ultrasonics and Acoustic Wave Propagation
paper · pdf · doi:10.48550/arxiv.1908.05871
openalex publication_date 2019/08/16 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We study regularization of ill-posed equations involving multiplication\noperators when the multiplier function is positive almost everywhere and zero\nis an accumulation point of the range of this function. Such equations\nnaturally arise from equations based on non-compact self-adjoint operators in\nHilbert space, after applying unitary transformations arising out of the\nspectral theorem. For classical regularization theory, when noisy observations\nare given and the noise is deterministic and bounded, then non-compactness of\nthe ill-posed equations is a minor issue. However, for statistical ill-posed\nequations with non-compact operators less is known if the data are blurred by\nwhite noise. We develop a regularization theory with emphasis on this case. In\nthis context, we highlight several aspects, in particular we discuss the\nintrinsic degree of ill-posedness in terms of rearrangements of the multiplier\nfunction. Moreover, we address the required modifications of classical\nregularization schemes in order to be used for non-compact statistical\nproblems, and we also introduce the concept of the effective ill-posedness of\nthe operator equation under white noise. This study is concluded with\nprototypical examples for such equations, as these are deconvolution equations\nand certain final value problems in evolution equations.\n