Blockwise SVD with error in the operator and application to blind deconvolution

We consider linear inverse problems in a nonparametric statistical framework. Both the signal and the operator are unknown and subject to error measurements. We establish minimax rates of convergence under squared error loss when the operator admits a blockwise singular value decomposition (blockwise SVD) and the smoothness of the signal is measured in a Sobolev sense. We construct a nonlinear procedure adapting simultaneously to the unknown smoothness of both the signal and the operator and achieving the optimal rate of convergence to within logarithmic terms. When the noise level in the operator is dominant, by taking full advantage of the blockwise SVD property, we demonstrate that the block SVD procedure overperforms classical methods based on Galerkin projection or nonlinear wavelet thresholding. We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere

Sorption of Perfluorochemicals to Granular Activated Carbon in the Presence of Ultrasound

Perfluorochemicals (PFCs) are emerging pollutants of increasing public health and environmental concern due to their worldwide distribution, environmental persistence, and bioaccumulation potential. Activated carbon adsorption is an effective method to remove PFCs from water. Herein, we report on the sorption of four PFCs: perfluorooctane sulfonate (PFOS), perfluorooctanoate (PFOA), perfluorobutane sulfonate (PFBS), and perfluorobutanoate (PFBA), from deionized water (MQ) and landfill groundwater (GW) by granular activated carbon (GAC) in the absence and presence of 20 kHz ultrasound. In all cases, the adsorption kinetics were found to be well-represented by a pseudosecond-order model, with maximum monolayer sorption capacity and initial sorption rate values following the orders q_(e)^(PFOS) > q_(e)^(PFOA) > q_(e)^(PFBS) > q_(e)^(PFBA) and v_(0)^(PFOS) > v_(0)^(PFBS) > v_(0)^(PFOA) > v_(0)^(PFBA), respectively. The equilibrium adsorption was quantified by the BET multilayer absorption isotherm, and the monolayer sorption capacity increased with increasing PFC chain length: q_(m)^(PFOS) > q_(m)^(PFOA) > q_(m)^(PFBS) > q_(m)^(PFBA). The equilibrium PFC sorption constants, q_e and q_m, and the sorption kinetic constants, v_0 and k_2, were greater in Milli-Q water than in landfill groundwater with or without pretreatment, indicating competition for sorption sites by natural and cocontaminant groundwater organics. Ultrasonic irradiation significantly increased the PFC−GAC sorption kinetics, 250−900%, and slightly increased the extent of PFC equilibrium adsorption, 5−50%. The ultrasonic PFC−GAC sorption kinetics enhancement increased with increasing PFC chain length, suggesting ultrasound acts to increase the PFC diffusion rate into GAC nanopores