VARIATIONAL ANALYSIS FOR THE MAXIMAL TIME FUNCTION IN NORMED SPACES

For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.

Computational uncertainty principle in nonlinear ordinary differential equations——Ⅱ.Theoretical analysis

The error propagation for general numerical method in ordinarydifferential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multistep methods are essentially improved, and joining probabilistic theory the “normal” growth of accumulated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numerical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Furthermore, we give the explicitly mathematical expression of the computational uncertainty principle and expound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.

Image reconstruction of a neutron scatter camera

Recently, Sandia Laboratories developed a neutron scatter camera to detect special nuclear materials. This camera exhibits the following advantages: high efficiency, direction discrimination, neutron-gamma discrimination ability, and wide field of view. However, using the direct projection method, the angular resolution of this camera is limited by uncertainties in the energies estimated from pulse height and time of flight measurements. In this study, we established an eight-element neutron scatter camera and conducted the experiment with a ^(252)Cf neutron source. The results show that it has an angular resolution better than 8°(1s) and a detection efficiency of approximately 2.6′10-4. Using maximum likelihood expectation maximization method, the image artifact was eliminated, and the angular resolution was improved. We proposed an average scattering angle method to estimate the scattering energy of neutrons and Compton gamma rays. As such, we can obtain a recognizable image and energy spectrum of the source with some degradation of energy and image resolutions. Finally, a newly measured light response function based on the MPD^(-4) device was used for image reconstruction. Although we did not obtain a better result than that of the standard light response function, we have observed the effects of light response function on image reconstruction.