Regularization Methods: new theory, analysis and applications

正则化方法是基于偏微分方程的精确解的数值技术，当Dirac delta被光滑近似（如高斯近似）取代时，以与基本解相同的方式导出偏微分方程。 涡滴法就是一个例子。 所得到的偏微分方程的解是基本解的有界逼近。 本项目将推进这些方法的理论、计算实现和在Stokes流问题中的应用。正则化的高阶元素，如偶极子和四极杆，系统地推导出所需的性质的斑点提出和家庭的斑点与这些属性的开发。正则化的流体速度表达式形成了新的数值方法的基础，其收敛性和效率特性进行了分析。提供了选择数值参数的基于性能的指南。 最后，这些方法的几个应用包括微生物种群的动态和它们产生的大规模流动模式，纤毛的运动，因为它们跳动以产生有组织的净流，流过可渗透的壁，等等。研究人员开发必要的数学工具，以设计新的可靠的方法来生成涉及液体环境中微生物的生物现象的计算机模拟。 这些包括细菌在人体内的运动，肺部纤毛的同步运动，血液运输的化学物质的运输和过滤，以及其他生物学上重要的情况。 这些现象的计算机模拟补充了在医学实验室进行的实验研究，并且非常有用，特别是对于理解人体在无法在体内测试的情况下的行为。通常情况下，计算机模拟可以检测到当参数超出实验室通常使用的范围时发生的反应，从而指出应该进行的新实验。 这项工作中开发的方法增加了我们对E。大肠杆菌可能在身体周围移动，或者肺纤毛的缺陷如何导致功能下降和可能的疾病。

Regularization methods are numerical techniques based on exact solutions of partial differential equations that are derived in the same way as fundamental solutions when the Dirac delta is replaced by a smooth approximation, such as a Gaussian. The vortex blob method is an example. The resulting solutions of the PDEs are bounded approximations to the fundamental solutions. This project advances the theory, the computational implementation and the application of these methods to problems of Stokes flows. Regularized high-order elements, such as doublets and quadrupoles, are systematically derived, the required properties of blobs are proposed and families of blobs with such properties are developed. The regularized fluid velocity expression forms the basis of new numerical methods, whose convergence and efficiency properties are analyzed. Performance-based guidelines for choosing the numerical parameters are provided. Finally, several applications of the methods include the dynamics of microorganism populations and the large-scale flow patterns they generate, the motion of cilia as they beat to generate an organized net flow, flows across permeable walls, and more. The investigator develops mathematical tools necessary to design new reliable ways to generate computer simulations of biological phenomena involving microorganisms in a liquid environment. These include the motion of bacteria in the human body, the synchronized motion of cilia in the lungs, the transport and filtration of chemicals transported by blood, and other biologically important situations. The computer simulation of these phenomena complement experimental studies done in medical laboratories and are extremely useful, particularly for understanding the behavior of the human body under circumstances that cannot be tested in vivo. Often, computer simulations can detect responses that occur when parameters are outside the range typically used in the laboratory, and thus point to new experiments that should be pursued. The methods developed in this work increase our understanding of how E. coli may move around the body or how defects in the lung cilia can translate into decreased functionality and possibly disease.