On Convergence Rates Equivalency and Sampling Strategies in Functional Deconvolution Models

Using the asymptotical minimax framework, we examine convergence rates equivalency between a continuous functional deconvolution model and its real-life discrete counterpart, over a wide range of Besov balls and for the $L^2$-risk. For this purpose, all possible models are divided into three groups: {\it uniform}, {\it regular} and {\it irregular}. We formulate the conditions when each of these situations takes place. In the regular case, not only we point out the number and the selection of sampling points which deliver the fastest convergence rates in the discrete model but also investigate when, in the case of an arbitrary sampling scheme, the convergence rates in the continuous model coincide or do not coincide with the convergence rates in the discrete model. We also study what happens if one chooses a uniform, or a more general pseudo-uniform, sampling scheme which can be viewed as an intuitive replacement of the continuous model. Finally, as a representative of the irregular case, we study functional deconvolution with a box-car like kernel since this model has a number of important applications. All theoretical results presented are illustrated by numerous examples many of which are motivated directly by a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. We conclude that in both regular and irregular cases one should be extremely careful when replacing a discrete functional deconvolution model by its continuous counterpart.