Path Integral Monte Carlo Methods for Computing Polarizability Tensors of Nano-materials and Electrical Impedance Tomography

该研究项目旨在为两个应用领域开发改进的高效数值方法：纳米级材料的电磁特性的高精度模拟和电阻抗断层成像。 在这两个领域，用传统方法进行数值计算即使不是不可能，也是具有挑战性的。 该项目旨在开发新的计算方法的基础上的概率表示的解决方案的偏微分方程的研究。 本项目的研究成果，从太阳能电池的开发到癌症的检测，都具有广泛的应用价值。本项目的研究内容是开发用于模拟纳米线、量子点、DNA等形状复杂的纳米粒子的电、磁极化率张量的高精度、高效率的数值方法，以及电阻抗断层成像（EIT）的快速算法。由于纳米颗粒的几何复杂性，传统的基于网格的离散化方法，如有限元和边界元方法的数值计算面临着巨大的挑战，如果不是不可能的。为了应对这些挑战，在这个项目中，路径积分蒙特卡罗（PIMC）方法，基于Feynman-Kac概率表示的偏微分方程的解决方案，将研究材料科学应用以及EIT问题。与传统的基于网格的数值方法相比，PIMC方法一方面提供了处理材料科学应用中高度不规则几何形状物体的能力，另一方面提供了EIT正问题中电极上偏微分方程的局部解。

This research project aims to develop improved efficient numerical methods for two application areas: highly accurate simulation of the electric and magnetic properties of nanometer-scale materials, and electrical impedance tomography. In both areas, numerical computations with traditional methods are challenging, if not impossible. This project aims to develop novel computational methods based on probabilistic representations of solutions to the partial differential equations under study. Results of the project are expected to have wide applicability, from the development of solar cells to the detection of cancer.This project concerns the development of highly accurate and efficient numerical methods to simulate the electric and magnetic polarizability tensors of nanoparticles of complex shapes as in nanowires, quantum dots, and DNA, and fast algorithms for electrical impedance tomography (EIT). Due to the geometric complexities of nanoparticles, numerical computations with traditional mesh-based discretization methods such as finite element and boundary element methods face great challenges, if not impossibility. To meet these challenges, in this project, path integral Monte Carlo (PIMC) methods, based on Feynman-Kac probabilistic representations of solutions to partial differential equations, will be studied for material science applications as well as EIT problems. Compared with traditional grid-based numerical methods, the PIMC methods offer the capability of handling objects with highly irregular geometries arising from materials science applications on the one hand, and provide local solutions of partial differential equations over electrodes in forward problems in EIT on the other hand.