Arnoldi preconditioning for solving large linear biomedical systems.

Area of Science:

• Computational science
• Numerical analysis
• Biomedical engineering

Background:

• Solving large, sparse, linear systems (Ax = b) is crucial for biomedical simulations.
• Iterative solvers, like preconditioned conjugate gradient, minimize memory but can be slow.
• Effective preconditioners are key to improving computational efficiency and remain an active research area.

Purpose of the Study:

• To introduce a novel preconditioner for large, sparse, linear systems based on Arnoldi method order reduction.
• To evaluate the performance of this new preconditioner against existing methods for biomedical systems.

Main Methods:

• Developed a novel preconditioner utilizing system order reduction via the Arnoldi method.
• Applied the preconditioner to large-scale systems (up to 1 million orders) from finite element method formulations of bidomain equations.
• Compared performance against other preconditioners, including incomplete LU decomposition (ILU).

Main Results:

• The Arnoldi-based preconditioner demonstrated considerably faster convergence, often within a single iteration.
• The new method exhibited lower memory usage compared to incomplete LU decomposition.
• While requiring recomputation for repeated solves, the Arnoldi transformation is quick to update.

Conclusions:

• The Arnoldi preconditioner offers significantly reduced solution times for single system solves.
• For repeated system solves, the Arnoldi preconditioner remains faster than ILU.
• This method presents a promising advancement for accelerating biomedical simulations requiring iterative solvers.