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

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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 multiscale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual 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, multigrid, 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 preconditioners 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.

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This work was supported by NIH grant (NIH R01-EB018302), NSF SSI grants 1663671 and 1339824, and NSF CDSE CBET 1508794. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [35], which is supported by National Science Foundation grant number ACI-1548562. We thank Mahidhar Tatineni for assisting on building Trilinos on Comet cluster, which was made possible through the XSEDE Extended Collaborative Support Service (ECSS) program [1]. The authors also thank Michael Saunders, Michael Heroux, Mahdi Esmaily, Ju Liu, and Vijay Vedula, for fruitful discussions that helped in the preparation of this paper. The authors would like to thank the two anonymous reviewers whose comments greatly contributed to improve the completeness of the present study. We also acknowledge support from the open source SimVascular project at

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A GMRES restart

We tested different GMRES restart numbers in the pipe benchmark problems. In Fig. 20, we plot compute times for preconditioned GMRES using a pipe model with rigid wall and a pipe with deformable wall. Our test shows that decreasing restart number increases the compute time of linear solver in the rigid pipe model. The FSI model does not show a notable difference.

Fig. 20
figure 20

Compute times for GMRES and preconditioners using (top) a rigid pipe model, and (bottom) a pipe model with deformable wall using different the GMRES restart numbers

B Choice of smoother and subsmoother for ML

In the ML package, multiple options are available for the smoother and the subsmoother. As shown in Fig. 21, the Gauss–Seidel smoother works best among Chebyshev, symmetric Gauss–Seidel, and ILUT. For the subsmoother, the symmetric Gauss–Seidel is the best among Chebyshev and MLS (Fig. 22).

Fig. 21
figure 21

Compute times for linear solvers preconditioned with ML using a a rigid and b an FSI pipe model with tolerance \(\epsilon =10^{-3}\), with different smoothers. The symmteric Gauss–Seidel subsmoother is used. For the rigid wall model, 38 cores are used. For the FSI model, 48 cores are used

Fig. 22
figure 22

Compute times for linear solvers preconditioned with ML using a a rigid and b FSI pipe model with tolerance \(\epsilon =10^{-3}\), with different subsmoothers. The Gauss–Seidel smoother is used. For the rigid wall model, 38 cores are used. For the FSI model, 48 cores are used

C Effect of reordering in ILU

We evaluated and compared compute times of linear solvers with different reordering methods. RCM and METIS reordering for ILUT via the Trilinos IFPACK are implemented. We use 2 level fill-in and \(10^{-2}\) dropping tolerance for this test. Figure 23 shows performance differences between ILUT with different reordering schemes. From the testing, we confirm that the RCM is the fastest method against METIS and no reordering. The superior performance of RCM is notable when GMRES is used with ILUT.

Fig. 23
figure 23

Compute times for linear solvers preconditioned with ILUT using a a rigid and b FSI pipe model with tolerance \(\epsilon =10^{-3}\), with different reorderings. For the rigid wall, 38 cores are used. For the FSI, 48 cores are used

D Eigenvalue spectrums of the local and global matrix.

In this section we compare the spectrum of eigenvalues in the local and global matrices and investigate how our analysis on local eigenvalues can be generalized to the global matrix. We use a pipe model in the same dimension shown in Fig. 1 meshed with 24,450 elements with \(N_{nd}=5462\). We use one core to extract the global matrix, and four cores to examine local matrices. As shown in Fig. 24, the eigenvalue distributions of the global and local matrices are similar. Although the eigenvalues from the global and local matrices are not exactly the same, the distribution of eigenvalues of local matrices is a good approximations to the distribution of eigenvalues in the global matrix.

Fig. 24
figure 24

Spectrums of eigenvalues for a rigid pipe model with Neumann BC at the outlet. (top) eigenvalues obtained from four local matrices. Different colors are used to represent eigenvalues from different local matrices. (bottom) eigenvalues obtained from the global matrix

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Seo, J., Schiavazzi, D.E. & Marsden, A.L. Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels. Comput Mech 64, 717–739 (2019).

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