# Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels

## TL;DR

This paper evaluates various iterative linear solvers and preconditioners for cardiovascular simulations involving rigid and deformable vessels, highlighting the superior performance of incomplete factorization preconditioners and the BIPN method in specific scenarios.

## Contribution

It provides a comprehensive comparison of solver and preconditioner performance in realistic cardiovascular models, introducing insights into their scalability and effectiveness.

## Key findings

- Incomplete factorization preconditioners offer best scalability.
- BIPN outperforms other methods in rigid wall models.
- BICG with diagonal and LU preconditioners excels in deformable wall models.

## Abstract

Computing the solution of linear systems of equations is invariably the most time consuming task in the numerical solutions of PDEs in many fields of computational science. In this study, we focus on the numerical simulation of cardiovascular hemodynamics with rigid and deformable walls, discretized in space and time through the variational multi-scale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual (GMRES) and stabilized bi-conjugate gradient (BICGS) methods, and a recently proposed, problem specific, bi-partitioned (BIPN) method. We also perform a comparative analysis of several preconditioners, including diagonal, block-diagonal, incomplete factorization, multi-grid, and resistance based methods. Solver performance and matrix characteristics (diagonal dominance, symmetry, sparsity, bandwidth and spectral properties) are first examined for an idealized cylindrical geometry with physiologic boundary conditions and then successively tested on several patient-specific anatomies representative of realistic cardiovascular simulation problems. Incomplete factorization pre-conditioners provide the best performance and results in terms of both strong and weak scalability. The BIPN method was found to outperform other methods in patient-specific models with rigid walls. In models with deformable walls, BIPN was outperformed by BICG with diagonal and Incomplete LU preconditioners.