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。

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.