Sparse linear solvers play a crucial role in transistor-level circuit simulation, especially for large-scale post-layout circuit simulation when considering complex parasitic effects. As semiconductor technology advances rapidly, the increasing sizes of circuits result in sparse linear solvers that require extended execution times and additional memory resources. Consequently, high-performance sparse linear solvers emerge as pivotal tools to facilitate rapid circuit simulation and verification. However, circuit matrices frequently exhibit high sparsity and non-uniform distributions of nonzero elements, compounding the challenge of achieving efficient acceleration. Recently, the flourishing developments in machine learning technology and the continuous enhancement of hardware capabilities have presented new opportunities for accelerating sparse linear solvers. This paper provides a perspective review of these technological advancements, while also highlighting the challenges and future opportunities in this evolving landscape.

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.

Sparse LU factorization is considered to be one of the most time-consuming components in circuit simulation, particularly when dealing with circuits of considerable size in the advanced process era. Sparse LU factorization can be expedited by utilizing the supernode structure, which partitions the matrix into dense sub-matrices, thereby improving computational performance by utilizing level-3 Basic Linear Algebra Subprograms (BLAS) General Matrix Multiplication (GEMM) operations. The sparse and irregular structure of circuit matrices often impedes the formation of supernodes or results in the formation of supernodes with many zero elements, which in turn poses challenges for exploiting GEMM operations. In this paper, by fully utilizing the density in sub-matrices and combining GEMM with the Dense-Sparse Matrix Multiplication (SpMM), we propose a density-aware adaptive matrix multiplication equipped with machine learning techniques to optimize performance of the most-time consuming matrix multiplication operator so as to accelerate the sparse LU factorization. Numerical experiment results show that among the 6 circuit matrices tested, the average performance of matrix multiplication in our algorithm can be improved by 5.35x (up to 9.35x) compared to the performance of using GEMM directly in Schur-complement updates. Compared with state-of-the-art solver SuperLU_DIST, our method shows a substantial performance improvement.

Sparse direct solvers play a vital role in large-scale high performance computing in science and engineering. Existing distributed sparse direct methods employ multifrontal/supernodal patterns to aggregate columns of nearly identical forms and to exploit dense basic linear algebra subprograms (BLAS) for computation. However, such a data layout may bring more unevenness when the structure of the input matrix is not ideal, and using dense BLAS may waste many floating-point operations on zero fill-ins.