On convergence rates equivalency and sampling strategies in functional deconvolution models

Abstract

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 L²-risk. For this purpose, all possible models are divided into three groups. For the models in the first group, which we call uniform, the convergence rates in the discrete and the continuous models coincide no matter what the sampling scheme is chosen, and hence the replacement of the discrete model by its continuous counterpart is legitimate. For the models in the second group, to which we refer as regular, one can point out the best sampling strategy in the discrete model, but not every sampling scheme leads to the same convergence rates; there are at least two sampling schemes which deliver different convergence rates in the discrete model (i.e., at least one of the discrete models leads to convergence rates that are different from the convergence rates in the continuous model). The third group consists of models for which, in general, it is impossible to devise the best sampling strategy; we call these models irregular.

We formulate the conditions when each of these situations takes place. In the regular case, we not only 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 boxcar-like blurring function since this model has a number of important applications. All theoretical results presented in the paper 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. The theoretical performance of the suggested estimator in the multichannel deconvolution model with a boxcar-like blurring function is also supplemented by a limited simulation study and compared to an estimator available in the current literature. The paper concludes that in both regular and irregular cases one should be extremely careful when replacing a discrete functional deconvolution model by its continuous counterpart.