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

本研究项目旨在为两个应用领域开发改进的高效数值方法：纳米尺度材料的电和磁特性的高精度模拟和电阻抗层析成像。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。本项目旨在开发基于所研究的偏微分方程解的概率表示的新计算方法。该项目的成果有望具有广泛的适用性，从太阳能电池的开发到癌症的检测。该项目涉及开发高精度和高效的数值方法来模拟纳米线、量子点和DNA等复杂形状纳米粒子的电和磁极化张量，以及电阻抗断层扫描（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，如果不是不可能的话。为了应对这些挑战，在本项目中，将研究基于偏微分方程解的费曼-卡茨概率表示的路径积分蒙特卡罗（PIMC）方法，用于材料科学应用以及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出现的高度不规则几何物体的能力，另一方面提供了EIT正演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.