Abstract
It is well known, that pseudodifferential equations of negative order considered in Sobolev spaces with small smoothness indices are ill–posed. On the other hand, it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations. The main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data. We argue that the combination of this principle with wavelet–based matrix compression techniques leads to algorithms which are order–optimal in the sense of complexity.
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Harbrecht, H., Pereverzev, S. & Schneider, R. Self–regularization by projection for noisy pseudodifferential equations of negative order. Numer. Math. 95, 123–143 (2003). https://doi.org/10.1007/s00211-002-0417-x
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DOI: https://doi.org/10.1007/s00211-002-0417-x