An adaptive regularization by projection for noisy pseudodifferential equations

It is well known that pseudodifferential equations of negative order considered in the Sobolev space with a small smoothness index are ill-posed. On the other hand, it is known that effective discretization schemes with properly chosen discretization parameter allow to obtain a regularization effect for such equations. The main accomplishment of the present talk is the principle for the adaptive choice of the discretization parameter directly from noisy discrete data. We argue that the combination of this principle with wavelet-based matrix compression technique leads to algorithms which are order-optimal in the sense of complexity.