Solving 3-D PDEs by Tensor B-Spline Methodology: A High Performance Approach Applied to Optical Diffusion Tomography

Solutions of 3-D elliptic PDEs form the basis of many mathematical models in medicine and engineering. Solving elliptic PDEs numerically in 3-D with fine discretization and high precision is challenging for several reasons, including the cost of 3-D meshing, the massive increase in operation count, and memory consumption when a high-order basis is used, and the need to overcome the "curse of dimensionality." This paper describes how these challenges can be either overcome or relaxed by a Tensor B-spline methodology with the following key properties: 1) the tensor structure of the variational formulation leads to regularity, separability, and sparsity, 2) a method for integration over the complex domain boundaries eliminates meshing, and 3) the formulation induces high-performance and memory-efficient computational algorithms. The methodology was evaluated by application to the forward problem of Optical Diffusion Tomography (ODT), comparing it with the solver from a state-of-the-art Finite-Element Method (FEM)-based ODT reconstruction framework. We found that the Tensor B-spline methodology allows one to solve the 3-D elliptic PDEs accurately and efficiently. It does not require 3-D meshing even on complex and non-convex boundary geometries. The Tensor B-spline approach outperforms and is more accurate than the FEM when the order of the basis function is > 1, requiring fewer operations and lower memory consumption. Thus, the Tensor B-spline methodology is feasible and attractive for solving large elliptic 3-D PDEs encountered in real-world problems.