Fractal-based methods in analysis

根据近年来的工作，“基于分形的方法”不仅指分形本身的理论和工具，还指这种数学背后的驱动哲学。例如，在我最近的合作工作在反问题的常微分方程和偏微分方程，没有分形出现，但数学方法显然是灵感来自成功的方法在分形成像。本提案中的工作从基于分形的核心扩展到包括分析，动力系统，数学建模，数值分析以及数学和特定应用领域研究中的其他主题的广泛元素。 拟议的研究计划的目标是开发基于分形的框架，用于分析正问题和逆问题， (A)物理学和工程学，包括 - 多孔介质和穿孔域，建立在一些初步的合作工作，我已经发表在过去的一年。通常情况下，微分方程穿孔域处理均匀化理论，其中非均匀材料被替换为一个虚构的均匀介质;我最近的工作严格证明，这种规模的转变也可以做相关的反问题。例如，在无孔介质上，利用观测数据求解相应的反问题，可以恢复多孔介质的变热扩散率 - 具有分形边界数据的偏微分方程问题 (B)环境科学，重点是可持续性，包括 - 污染驱动的人口动态，可能具有经济影响 - 生态系统模拟 - 具有时滞的耦合反应扩散方程组反问题求解框架 在这两个问题中，关注的一个关键点是可持续性，特别是在存在不良因素（某种可容忍的污染水平）或可能的大扰动（收获）的情况下。 (C)生物医学科学，包括肿瘤检测，建模和分析的持续工作，以及图像驱动的逆问题解决方案框架的开发 (D)基于分形分析的最新思想的新成像框架，包括 - 星型集反演系统 - 迭代多功能系统 拟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）与应用驱动的事项（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）。在该计划中工作的学生将在这一范围内接受深入和广泛的培训。研究计划提案的一些最初的小片段已经由当前/最近的学生开发，随后的片段将由未来的学生建立。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.