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

In the present paper we consider the problem of estimating a periodic $(r+1)$-dimensional function $f$ based on observations from its noisy convolution. We construct a wavelet estimator of $f$, derive minimax lower bounds for the $L^2$-risk when $f$ belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the "curse of dimensionality" and, hence, are higher than usual convergence rates when $r$ is large. The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function $f$ is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of $M$ solutions of separate convolution equations. A limited simulation study in the case of $r=1$ confirms theoretical claims of the paper.Comment: 29 pages, 1 figure, 1 tabl

Dynamical mass generation: from elementary fields to bound states

We investigate the dynamical generation of fermion mass in Quantum Electrodynamics (QED) and in Quantum Chromodynamics (QCD). This non-perturbative study is performed using a truncated set of Schwinger-Dyson equations for the fermion and photon propagator and the quark propagator. First, we study dynamical fermion mass generation in QED using a cancellation mechanism for the full photon-electron vertex that respects multiplicative renor- malisability and reproduces perturbation theory and determine the critical coupling in different approximations. We then study the quark equation using a model for the strong coupling with two parameters and compare this study with previous ones. Finally, we show how bound states masses derived by lattice calculations can be extrapolated to low quark masses using the Nambu Jona-Lasinio model (NJL) and demonstrate the limitation of the NJL model. As an outlook, we present a functional method to control the quantum fluctuations of a given theory. We derive an exact equation for the effective action T and using a gradient expansion for T we derive evolution equations for different couplings