Fractal-based methods in analysis

基于近几年的工作，“基于分形学的方法”不仅指的是分形学本身的理论和工具，而且还指的是这门数学背后的驱动哲学。例如，在我最近合作的常微分方程和偏微分方程反问题的工作中，没有出现分形，但这种数学方法显然受到了成功的分形成像方法的启发。这项建议中的工作从这个基于分形的核心扩展到包括来自分析、动力系统、数学建模、数值分析和其他数学主题和具体应用领域研究的广泛元素。 拟议的研究计划的目标是开发基于分形学的框架，用于分析 (A)物理学和工程学，包括 -多孔介质和穿孔领域，基于我在过去一年发表的一些初步合作工作。通常，多孔区域上的微分方程是用齐化理论处理的，在齐化理论中，非均匀材料被虚拟的均匀介质所代替；我最近的工作严格地证明了这种尺度变换也可以用于相关的反问题。例如，人们可以通过使用观测数据来求解无孔介质上的相应逆问题来恢复多孔介质的可变热扩散率 -具有类分形边界数据的偏微分方程组问题 (B)环境科学，重点是可持续性，包括 -污染驱动的人口动态，可能会对经济产生影响 -生态系统建模 -时滞反应扩散方程组的反问题解框架(S) 在这两个问题中，一个关键的关注点是可持续性，特别是在存在不良因素(一些可容忍的污染水平)或可能存在较大干扰(收获)的情况下。 (C)生物医学，包括继续开展肿瘤检测、建模和分析方面的工作，以及开发图像驱动的逆问题解决框架 (D)基于最新的基于分形学的分析思想的新成像框架，包括 -星形集合反演系统 -迭代多功能系统 拟议的研究计划将严格的理论元素(分析、分形几何、微分方程组和积分方程式)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)结合在一起。参与该项目的学生将在这一领域接受深入和广泛的培训。一些最初的研究项目提案的小部分已经由当前/最近的学生开发，后续的部分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开展的工作的下一步。

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.