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

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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.