Table 1.

Benchmarking data-driven GMRES against data-free preconditioning techniques.

Method	Training time	Setup time	GMRES time	Total solve time	Number of iterations
GMRES	×	×	23.34 s	23.34 s	115.7
PCA: $N=5$	7.63 s	×	0.87 s	0.87 s	6.1
PCA: $N=10$	7.63 s	×	0.59 s	0.59 s	3.5
PCA: $N=25$	7.63 s	×	0.62 s	0.62 s	2.1
PCA: $N=50$	7.63 s	×	0.98 s	0.98 s	2.0
CNN: Projection loss	~10 hr	0.18 s	13.46 s	13.52 s	64.5
CNN: Residual loss	~10 hr	0.40 s	13.07 s	13.47 s	63.3
Jacobi	×	0.57 ms	39.30 s	39.30 s	153.8
Gauss seidel	×	17.59 ms	136.82 s	136.82 s	107.3
Ref. 17	×	5.80 ms	24.87 s	24.87 s	118.7
SOR: $\omega =0.25$	×	16.40 ms	170.17 s	170.17 s	132.0
SOR: $\omega =0.50$	×	17.55 ms	152.21 s	152.21 s	120.2
SOR: $\omega =0.75$	×	17.10 ms	140.56 s	140.56 s	111.7
SOR: $\omega =1.25$	×	18.33 ms	138.67 s	138.67 s	107.5
SOR: $\omega =1.50$	×	20.55 ms	169.73 s	169.73 s	120.4
SOR: $\omega =1.75$	×	21.37 ms	349.27 s	349.27 s	235.9
ILU: Drop tol. = 10−1	×	6.62 s	13.90 s	20.52 s	66.2
ILU: Drop tol. = 10−2	×	7.01 s	4.59 s	11.60 s	28.6
ILU: Drop tol. = 10−3	×	9.29 s	0.74 s	10.03 s	3.0

The training time refers to the amount of time taken to compute the principal components or train the convolutional neural network using the training data. This is done only once, and the same principal components or trained neural network are used for augmenting GMRES over the evaluation data. The setup time is the time taken for computing the augmenting vectors or for computing the preconditioner for an evaluation data sample. The GMRES time is the time taken for running GMRES to reach a threshold residual of $r_{\mathrm{th}}=0.04$ over the evaluation data. The total solve time is computed by summing the setup time and the GMRES time. The number of iterations shown are the number of GMRES iterations required to reduce the residual to below $r_{\mathrm{th}}=0.04$. The setup time, GMRES time and number of iterations shown in the table are averaged values computed over 50 randomly chosen evaluation data samples. The evaluation times reported are for a machine with 16 CPU cores and 60 GB RAM.