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Fig. 5. Transient waveforms of LU and GPU GMRES at port node n0_5480720_1102640 in ibmpg6t. The black curve with dots is from LU direct method. All other colored curves are results of GMRES with preconditioners set to different ILU threshold, i.e., from 0.1 to 3.0.

Figure 5 Transient waveforms of LU and GPU GMRES at port node n0_5480720_1102640 in ibmpg6t. The black curve with dots is from LU direct method. All other colored curves are results of GMRES with preconditioners set to different ILU threshold, i.e., from 0.1 to 3.0.

Related Figures (7)

Fig. 1. An RLC model of power grid network.
Fig. 1. An RLC model of power grid network.
Fig. 2. Sparsity pattern of a circuit matrix from a power grid problem.
Fig. 2. Sparsity pattern of a circuit matrix from a power grid problem.
Fig. 3. Comparison of sparsity patterns in complete and incomplete LU factors. The factors are derived from conductance matrix G in the example of “ibmpg|t.” In both two figures, the left parts are complete LU factors, and the right parts are incomplete LU factors. Number of nonzero elements, “nz is listed below each plot.
Fig. 3. Comparison of sparsity patterns in complete and incomplete LU factors. The factors are derived from conductance matrix G in the example of “ibmpg|t.” In both two figures, the left parts are complete LU factors, and the right parts are incomplete LU factors. Number of nonzero elements, “nz is listed below each plot.
Fig. 4. | The proposed GPU-accelerated parallel preconditioned GMRES solver. We also show the partitioning of the major computing tasks between CPU and GPU here. limited. For instance, SpMV has O(n) communication and O(n) computing, so it has 1 to 1 computing and commu- nication ratio (n is number of non-zero elements in the sparse matrices). Hence, it is important to reduce the data communication traffic for the proposed GPU-GMRES solver. As a result, how to wisely partition the data between CPU memory (host side) and GPU memory (device side) to minimize data traffic is crucial for GPU computing. In the sequel, we make some detailed analysis first for GMRES in Algorithm 1. Although GMRES tends to converge quickly for most circuit examples, i.e., the iteration number m <n, the space needed to store the subspace V,,,, with a size of n-by-m, i.e., m column vectors with n-length, is still big. Therefore, transferring the memory of the subspace vectors between CPU memory and GPU memory is not an efficient choice. In addition, every newly generated matrix-vector product needs to be orthogonalized with respect to all its previous basis vectors in the Arnoldi processes. To utilize the data intensive capability of GPU, we keep all the basis vectors of the subspace V,,, in GPU global memory. In this case, GPU is allowed to handle those operations, such as inner-product of basis vectors (dot) and vector subtraction (axpy), in parallel.
Fig. 4. | The proposed GPU-accelerated parallel preconditioned GMRES solver. We also show the partitioning of the major computing tasks between CPU and GPU here. limited. For instance, SpMV has O(n) communication and O(n) computing, so it has 1 to 1 computing and commu- nication ratio (n is number of non-zero elements in the sparse matrices). Hence, it is important to reduce the data communication traffic for the proposed GPU-GMRES solver. As a result, how to wisely partition the data between CPU memory (host side) and GPU memory (device side) to minimize data traffic is crucial for GPU computing. In the sequel, we make some detailed analysis first for GMRES in Algorithm 1. Although GMRES tends to converge quickly for most circuit examples, i.e., the iteration number m <n, the space needed to store the subspace V,,,, with a size of n-by-m, i.e., m column vectors with n-length, is still big. Therefore, transferring the memory of the subspace vectors between CPU memory and GPU memory is not an efficient choice. In addition, every newly generated matrix-vector product needs to be orthogonalized with respect to all its previous basis vectors in the Arnoldi processes. To utilize the data intensive capability of GPU, we keep all the basis vectors of the subspace V,,, in GPU global memory. In this case, GPU is allowed to handle those operations, such as inner-product of basis vectors (dot) and vector subtraction (axpy), in parallel.
Fig. 6. The error of GPU GMRES result compared to LU golden result. This curve is calculated at node n0_5480720_1102640 of ibmpg6t, whose waveform is shown in Fig. 5.
Fig. 6. The error of GPU GMRES result compared to LU golden result. This curve is calculated at node n0_5480720_1102640 of ibmpg6t, whose waveform is shown in Fig. 5.
in ratio and GPU GMRES time on each time step are show: by two curves.
in ratio and GPU GMRES time on each time step are show: by two curves.
Fig. 7. The impact of ILU threshold on fill-in ratio and GPU GMRES solving time. The blue curve in 3D space is GMRES solving time with respect to threshold and fill-in ratio, and the red curve on the bottom plane reflects the changes of fill-in ratio caused by different threshold values. All the measurements are from ibmpg4t.
Fig. 7. The impact of ILU threshold on fill-in ratio and GPU GMRES solving time. The blue curve in 3D space is GMRES solving time with respect to threshold and fill-in ratio, and the red curve on the bottom plane reflects the changes of fill-in ratio caused by different threshold values. All the measurements are from ibmpg4t.
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