A comparison of different choices for the regularization parameter in inverse electrocardiography models

Calculating the potentials on the heart’s epicardial surface from the body surface potentials constitutes one form of inverse problems in electrocardiography (ECG). Since these problems are ill-posed, one approach is to use zero-order Tikhonov regularization, where the squared norms of both the residual and the solution are minimized, with a relative weight determined by the regularization parameter. In this paper, we used three different methods to choose the regularization parameter in the inverse solutions of ECG. The three methods include the L-curve, the generalized cross validation (GCV) and the discrepancy principle (DP). Among them, the GCV method has received less attention in solutions to ECG inverse problems than the other methods. Since the DP approach needs knowledge of norm of noises, we used a model function to estimate the noise. The performance of various methods was compared using a concentric sphere model and a real geometry heart-torso model with a distribution of current dipoles placed inside the heart model as the source. Gaussian measurement noises were added to the body surface potentials. The results show that the three methods all produce good inverse solutions with little noise; but, as the noise increases, the DP approach produces better results than the L-curve and GCV methods, particularly in the real geometry model. Both the GCV and L-curve methods perform well in low to medium noise situations.

Comparisons of the structural and catalytic properties of Ti-HMS synthesized using the hydrothermal and molecular designed dispersion methods

Titanium containing wormhole-like mesoporous silicas, denoted Ti-HMS, synthesized both via the hydrothermal synthesis route and the post synthesis grafting technique, known as molecular designed dispersion, have been successfully applied in the gas phase oxidation of Toluene to CO and CO2. Selectivity towards CO2 for all catalysts, at temperatures between 400-600degreesC, was above 80%. Benzene and benzaldehyde were observed at temperatures above 450degreesC, but in very low concentrations. The conversion of toluene was shown to increase significantly when the V-TEX/N-MESO ratios were increased from 0.07 to 0.84. No significant difference in catalytic activity was observed for catalysts prepared via the different synthesis techniques. The catalytic activity also depends on the concentration of tetrahedrally coordinated titanium atoms and not on the total concentration of titanium in the catalyst.