Our aim is to provide new analytic insight to the relationship between the continuous and practical inversion models corrupted by white Gaussian noise. Let us consider an indirect noisy measurement M of a physical quantity u
M = Au + d*N
where A is linear smoothing operator and d > 0 is noise magnitude.
If N was an L2-function we could use the classical Tikhonov regularization to achieve an estimate. However, realizations of white Gaussian noise are almost never in L2. That is why we present a modification of Tikhonov regularization theory covering the case of white Gaussian measurement noise. We will also consider the question in which space does the estimate convergence to a correct solution when the noise amplitude tends to zero and what is the speed of the convergence.
This is joint work with Matti Lassas and Samuli Siltanen (University of Helsinki).