A perturbation basis for calculating NMR Diffusometry

TL;DR

This paper introduces an approximate method using a perturbation basis to solve the Bloch-Torrey equation for NMR diffusometry in porous media, enabling efficient and accurate analysis of spin magnetization.

Contribution

It presents a novel perturbation basis approach that simplifies solving the eigenvalue problem in porous media for NMR diffusometry, with scalable computational efficiency.

Findings

Approximate solutions are orthogonal in low-frequency Fourier space.

Error scales with N^{-2} relative to eigenvalues.

Computations scale quadratically with basis size using fast multipole methods.

Abstract

An approximative method for solving the Bloch-Torrey equation in general porous media is presented. The method expand the boundaries defining the porous media using electrostatic charges. As a result the eigenvalue problem of the Laplace operator in a confined geometry can approximately solved. Importantly the approximative solution is orthogonal in the low-frequent region of Fourier space. This gives a natural approach for studying spin magnetization in presence of magnetic fields. The error in the approximation scales with N^{-2} times the magnitude of each eigenvalue, where N is the size of the expansion matrix. From a computational point of view, the calculations scale quadratically with the number of basis functions using fast multipole methods.