Fractal-based methods in analysis

基于近年来的工作，“基于分形的方法”不仅指分形本身的理论和工具，而且还指这种数学背后的驱动哲学。例如，在我最近关于常微分方程和偏微分方程逆问题的合作研究中，没有出现分形，但数学方法显然受到了分形成像中成功方法的启发。本提案中的工作从这个基于分形的核心扩展到包括分析，动力系统，数学建模，数值分析和其他数学主题和特定应用领域研究的广泛元素。******提议的研究计划的目标是开发基于分形的框架，用于分析*******(A)物理和工程中的直接和反向问题，包括*******-多孔介质和穿孔域，建立在我在过去一年中发表的一些初步合作工作的基础上。通常，微分方程上的穿孔域是处理均质理论，其中非均质材料是由一个虚构的均质介质；我最近的研究有力地证明，这种尺度转换也可以用于相关的逆问题。例如，人们可以通过使用观测数据来恢复多孔介质的可变热扩散率，以解决无孔介质上相应的逆问题*******-分形边界数据的PDE问题*******(B)环境科学，重点是可持续性，包括*******-污染驱动的种群动态；可能具有经济影响*******-生态系统建模*******-具有延迟的反应-扩散方程耦合系统的反问题解决框架*******在这两个问题中，感兴趣的关键点是可持续性，特别是在存在不良因素（一些可容忍的污染水平）或可能的大扰动（收获）的情况下。******(C)生物医学科学，包括继续在肿瘤检测、建模和分析方面的工作，以及图像驱动的逆问题解决框架的发展*******(D)基于分形分析的最新思想的新成像框架，包括*******-星形集反演系统*******-迭代多功能系统*******提议的研究计划结合了严格的理论元素(分析，分形几何，微分和积分方程)与应用驱动的问题（建模，信号/图像处理）和实际问题（编程，算法设计，数值分析，近似）。在该项目中工作的学生将在这一领域接受深入而广泛的培训。研究计划提案的一些小部分已经由当前/最近的学生开发，随后的部分将由未来的学生建立。该提案的其他部分反映了当前合作中最近工作的下一步。**

Based on the work of recent years, "fractal-based methods" refers not just to the theory and tools of fractals themselves, but also to the driving philosophy behind this mathematics. For example, in my recent collaborative work in inverse problems for ordinary and partial differential equations, no fractal appears, yet the mathematical approach is clearly inspired by successful approaches in fractal imaging. The work in this proposal extends from this fractal-based core to include broad elements from analysis, dynamical systems, mathematical modeling, numerical analysis, and other topics in mathematics and the specific application domain research.******The goals of the proposed research program are to develop fractal-based frameworks for analyzing direct and inverse problems in*******(A) physics and engineering, including*******- porous media and perforated domains, building upon some preliminary collaborative work that I have published in the past year. Typically, differential equations on perforated domains are treated with homogenization theory in which heterogeneous material is replaced by a fictitious homogenous medium; my recent work rigorously justifies that this sort of scale shift can also be done for related inverse problems. For example, one can recover the variable thermal diffusivity of a porous medium by using observational data to solve the corresponding inverse problem on the medium with no holes*******- PDE problems with fractal-like boundary data*******(B) environmental science, with a focus on sustainability, including*******- pollution-driven population dynamics, perhaps with economic impact*******- ecosystem modeling*******- an inverse problem solution framework for coupled systems of reaction-diffusion equations with delay(s)*******In both problems, a key point of interest is sustainability, particularly in the presence of undesirable elements (some tolerable pollution level) or possibly large perturbations (harvest). ******(C) biomedical science, including continuing work on tumor detection, modeling, and analysis, and the development of an image-driven inverse problem solution framework*******(D) new imaging frameworks based on very recent ideas in fractal-based analysis, including*******- star-shaped set inversion systems*******- iterated multifunction systems*******The proposed research program combines rigorous theoretical elements (analysis, fractal geometry, differential and integral equations) with application-driven matters (modeling, signal/image processing) and practical issues (programming, algorithm design, numerical analysis, approximations). Students working in the program will receive both deep and broad training across this spectrum. Some initial small pieces of the research program proposal have been developed by current/recent students, with subsequent pieces to be established by future students. Other parts of the proposal reflect the next step of very recent work in current collaborations. **