Solving sparse linear systems is crucial in scientific computing. Sparse Conjugate Gradient (CG) is one of the most well-known iterative solvers with high efficiency and low storage requirements. However, the performance of sparse CG solvers implemented on storage-compute separated architectures is greatly limited by the irregular memory access and the large amount of data transmission.

Algebraic multigrid (AMG) methods are particularly efficient to solve a wide range of sparse linear systems, due to their good flexibility and adaptability. Even though modern parallel devices, such as GPUs, brought massive parallelism to AMG, the latest major hardware features, i.e., tensor core units and their low precision compute power, have not been exploited to accelerate AMG. This paper proposes AmgT, a new AMG solver that utilizes the tensor core and mixed precision ability of the latest GPUs during multiple phases of the AMG algorithm. Considering that the sparse general matrix-matrix multiplication (SpGEMM) and sparse matrix-vector multiplication (SpMV) are extensively used in the setup and solve phases, respectively, we propose a novel method based on a new unified sparse storage format that leverages tensor cores and their variable precision. Our method improves both the performance of GPU kernels, and also reduces the cost of format conversion in the whole data flow of AMG. To better utilize the algorithm components in existing libraries, the data format and compute kernels of the AmgT solver are incorporated into the HYPRE library. The experimental results on NVIDIA A100, H100 and AMD MI210 GPUs show that our AmgT outperforms the original GPU version of HYPRE by a factor of on geomean $1.46 \times, 1.32 \times$ and $2.24 \times$ (up to $2.10 \times, 2.06 \times$ and $3.67 \times$), respectively.

Conjugate gradient (CG) and biconjugate gradient stabilized (BiCGSTAB) are effective methods used for solving sparse linear systems. We in this paper propose Mille-feuille, a new solver for accelerating CG and BiCGSTAB on GPUs. We first analyze the two methods and list three findings related to the use of mixed precision, the reduction of kernel synchronization costs, and the awareness of partial convergence during the iteration steps. Then, (1) to enable tile-grained mixed precision, we develop a tiled sparse format; (2) to reduce synchronization costs, we leverage atomic operations that make the whole solving procedure work within a single GPU kernel; (3) to support a partial convergence-aware mixed precision strategy, we enable tile-wise on-chip dynamic precision conversion within the single kernel at runtime. The experimental results on an NVIDIA A100 and an AMD MI210 show that the Mille-feuille solver outperforms baseline implementations using the vendor-support cuSPARSE/hipSPARSE as well as two state-of-the-art libraries PETSc and Ginkgo by a factor of on average 3.03x/2.68x, 5.37 x, 4.36x (up to $8.77 \mathrm{x} / 7.14 x$, 16.54x, 15.69x) in CG, on average 2.65x/2.32x, 3.57x, 3.78x (up to 7.51x/6.63x, 16.64x, 11.73x) in BiCGSTAB, on average 3.82x/3.47x (up to 40.38x/47.75x) in preconditioned CG (PCG), on average 1.79x/1.63x (up to 45.63x/44.34x) in preconditioned BiCGSTAB (PBiCGSTAB), respectively.