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In the present paper we consider the problem of estimating a periodic$(r+1)$-dimensional function $f$ based on observations from its noisyconvolution. We construct a wavelet estimator of $f$, derive minimax lowerbounds for the $L^2$-risk when $f$ belongs to a Besov ball of mixed smoothnessand demonstrate that the wavelet estimator is adaptive and asymptoticallynear-optimal within a logarithmic factor, in a wide range of Besov balls. Weprove in particular that choosing this type of mixed smoothness leads to ratesof convergence which are free of the "curse of dimensionality" and, hence, arehigher than usual convergence rates when $r$ is large. The problem studied inthe paper is motivated by seismic inversion which can be reduced to solution ofnoisy two-dimensional convolution equations that allow to draw inference onunderground layer structures along the chosen profiles. The common practice inseismology is to recover layer structures separately for each profile and thento combine the derived estimates into a two-dimensional function. By studyingthe two-dimensional version of the model, we demonstrate that this strategyusually leads to estimators which are less accurate than the ones obtained astwo-dimensional functional deconvolutions. Indeed, we show that unless thefunction $f$ is very smooth in the direction of the profiles, very spatiallyinhomogeneous along the other direction and the number of profiles is verylimited, the functional deconvolution solution has a much better precisioncompared to a combination of $M$ solutions of separate convolution equations. Alimited simulation study in the case of $r=1$ confirms theoretical claims ofthe paper.

Anisotropic Denoising in Functional Deconvolution Model with Dimension-free Convergence Rates