Algorithms for approximating continuum processes on surfaces

连续过程在整个应用科学和自然科学中无处不在。由于解析解的可能性很小，因此对于相应的偏微分方程(PDE)模型，精确有效的算法的实际重要性怎么强调都不为过。对于许多重要的偏微分方程类，人们已经做出了很大的努力来发展数值方法。这些努力的重点是寻找一、二或三个空间维度的解决方案。但在二维曲面或一维弯曲细丝上也会出现涉及一般微分方程式的问题。例如，表面上的PDE可用于在计算机生成的表面上放置纹理，或增强或恢复扫描表面上的损坏图案。在物体的机器识别中，表面偏微分方程组的解可以用来表征物体的形状。在材料科学中，这样的方程被用来研究材料在曲面上的相变，而在理论生物学中，表面上的偏微分方程作为模拟骨质疏松等骨病理的一部分而出现。尽管曲面上的偏微分方程组广泛存在，但仍然需要一种系统的方法来有效地计算一般曲面上的一般偏微分方程组的解。 我们的研究将继续开发和分析求解曲面上偏微分方程组的算法。重点是(1)适用于一般曲面的方法，(2)适用于一般偏微分方程组的方法，(3)准确有效的方法，以及(4)简单的方法，即利用标准的、现有的技术和软件以统一的方式处理不同的偏微分方程组。预计在这项研究拨款下获得的方法将使研究人员和最终用户能够非常准确地对一般表面过程的数值现实模型进行研究。

Continuous processes are ubiquitous throughout the applied and natural sciences. Because analytical solutions are rarely possible, the practical importance of accurate and efficient algorithms for the corresponding partial differential equation (PDE) models cannot be overemphasized. There has been a great effort made to develop numerical methods for many important classes of PDEs. These efforts focus on finding solutions in one, two or three spatial dimensions. But problems involving general differential equations also arise on two-dimensional curved surfaces or one-dimensional curved filaments. For example, PDEs on surfaces can be used to place a texture on a computer generated surface, or to enhance or restore a damaged pattern on a scanned surface. In machine recognition of objects, the solution of surface PDEs can be used to characterize the shapes of objects. In material science such equations have been used to examine phase change of a material on a curved surface, while in theoretical biology, PDEs on surfaces arise as part of the modeling bone pathologies such as osteoporosis. Despite the widespread occurrence of PDEs on curved surfaces there is still a need for a systematic approach for effectively computing the solution of general PDEs on general surfaces. Our research will continue the development and analysis of algorithms for solving PDEs on surfaces. The focus is on methods which (1) apply to general surfaces, (2) apply to general PDEs, (3) are accurate and efficient, and (4) are simple in the sense that they treat different PDEs in uniform fashion by making use of standard, existing techniques and software. It is anticipated that the methods obtained under this research grant will enable researchers and end-users to numerically investigate realistic models of general surface processes with great accuracy.