Randomized progressive iterative approximation for B-spline curve and surface fittings

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA

On the extended randomized multiple row method for solving linear least-squares problems

The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple row method to solve a given overdetermined and inconsistent linear system and analyze its computational complexities at each iteration. We prove that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm. Several numerical studies are presented to corroborate our theoretical findings. The real-world applications, such as image reconstruction and large noisy data fitting in computer-aided geometric design, are also presented for illustration purposes

Life fingerprints of nuclear reactions in the body of animals

Nuclear reactions are a very important natural phenomenon in the universe. On the earth, cosmic rays constantly cause nuclear reactions. High energy beams created by medical devices also induce nuclear reactions in the human body. The biological role of these nuclear reactions is unknown. Here we show that the in vivo biological systems are exquisite and sophisticated by nature in influence on nuclear reactions and in resistance to radical damage in the body of live animals. In this study, photonuclear reactions in the body of live or dead animals were induced with 50-MeV irradiation. Tissue nuclear reactions were detected by positron emission tomography (PET) imaging of the induced beta+ activity. We found the unique tissue "fingerprints" of beta+ (the tremendous difference in beta+ activities and tissue distribution patterns among the individuals) are imprinted in all live animals. Within any individual, the tissue "fingerprints" of 15O and 11C are also very different. When the animal dies, the tissue "fingerprints" are lost. The biochemical, rather than physical, mechanisms could play a critical role in the phenomenon of tissue "fingerprints". Radiolytic radical attack caused millions-fold increases in 15O and 11C activities via different biochemical mechanisms, i.e. radical-mediated hydroxylation and peroxidation respectively, and more importantly the bio-molecular functions (such as the chemical reactivity and the solvent accessibility to radicals). In practice biologically for example, radical attack can therefore be imaged in vivo in live animals and humans using PET for life science research, disease prevention, and personalized radiation therapy based on an individual's bio-molecular response to ionizing radiation