Interval analysis, computer graphics algorithms and social networks

区间分析使用实数紧凑的区间进行算法开发，而不是实数。这使得能够在计算中包括计算误差和实验误差，并开发在间隔内提供有保证的结果的算法。在这方面，建议继续利用区间分析技术设计各种函数的有效方法。本文的研究结果将应用于区间牛顿方法，它是牛顿方法在数值分析中求解非线性方程组的区间推广。研究了区间Bezier曲线曲面的升阶和降阶问题。这些曲线曲面是计算机图形学中常用的Bezier曲线曲面的区间推广。它们基于PI最先研究的区间Bernstein多项式。当Bernstein多项式升阶和降阶时，计算中的误差意味着多项式的次数不再正确。在计算机图形学中，建议继续关于渲染效率的工作，特别是在图形处理器上。将考虑一种复杂的索引方案，以避免多余的计算。在计算几何学中，关于三维实体的有效相交的工作将继续下去。这项工作对于涉及几何计算的几个领域具有重要意义，例如计算机图形学、地理信息系统和实体建模。这项工作使用了区间分析技术和精确计算工具。社交网络中的应用问题将被考虑，因为它们与当地行业需求有关。特别是，与当地一家公司就一项慈善捐款计划开展的合作取得了成功。该项目由MITACS提供支持，并将向MITACS申请下一阶段的支持。一家在众包领域处于世界级领先地位的当地公司正在开发一种扩展的系统，他们需要专家的建议和实施协助。目前正在考虑由PI和一名同事提出的支持MITACS的申请。

Interval analysis uses real compact intervals for algorithm development as opposed to real numbers. This enables the inclusion of both computational and experimental error in the computations and to develop algorithms that deliver guaranteed result within intervals. In this area it is proposed to continue the work on devising efficient methods for the range of functions using interval analysis techniques. The result of this investigation will be applied to the interval Newton method, the interval generalization of Newton's method in numerical analysis for the solution of non-linear equations. It is also proposed to investigate the degree elevation and reduction of interval Bezier curves and surfaces. These curves and surfaces are interval generalizations of the commonly used Bezier curves and surfaces in computer graphics. They based on the interval Bernstein polynomials first investigated by the PI. When Bernstein polynomials are degree elevated and reduced the error committed in the computations means that the polynomial degree is no longer correct. In computer graphics, it is proposed to continue the work on rendering efficiency in particular on GPU's. A sophisticated indexing scheme will be considered that avoid superfluous computations. In computational geometry, the work on efficient intersections for 3D solids will be continued. This work is of importance for several areas which involve geometric computations such as for example computer graphics, GIS and solid modeling, This work used techniques from interval analysis combined with exact computation tools. Applied problems in social networks will be considered as they pertain to local industry needs. In particular the work with a local company on a scheme for charitable donations has been successful. Support for this project was provided by MITACS and will be applied for from MITACS for the next phase. A local company that is a world class leader in crowdsourcing is developing an expanded system for which they need expert advise and implementation assistance. An application by the PI and a colleague for MITACS support is currently being considered.