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

Area of Science:

• Computational mathematics
• Numerical analysis
• Scientific computing

Background:

• 3-D elliptic PDEs are crucial for modeling in medicine and engineering.
• Numerical solutions face challenges: 3-D meshing costs, high operation counts, memory demands, and the curse of dimensionality.
• Existing methods like Finite Element Method (FEM) struggle with complex geometries and high-order bases.

Purpose of the Study:

• To introduce and evaluate a Tensor B-spline methodology for solving 3-D elliptic PDEs.
• To demonstrate how this method overcomes meshing requirements and computational challenges.
• To compare its performance against FEM in a relevant application.

Main Methods:

• Utilizing a Tensor B-spline formulation with inherent regularity, separability, and sparsity.
• Employing integration over complex domain boundaries to eliminate the need for 3-D meshing.
• Developing high-performance, memory-efficient computational algorithms based on the formulation.
• Applying the methodology to the forward problem of Optical Diffusion Tomography (ODT).

Main Results:

• The Tensor B-spline method accurately and efficiently solves 3-D elliptic PDEs.
• It eliminates the need for 3-D meshing, even for complex, non-convex boundaries.
• Outperforms FEM in accuracy and efficiency for basis function orders > 1.
• Requires fewer operations and less memory compared to FEM.

Conclusions:

• The Tensor B-spline methodology is a feasible and attractive alternative for solving large-scale 3-D elliptic PDEs.
• It offers significant advantages over FEM in terms of accuracy, efficiency, and memory usage.
• Enables advanced modeling in fields like medical imaging (ODT) and engineering.