Matrix-free Sparse Direct Solvers

Existing direct solvers for large linear systems, especially sparse ones, require the matrices to be explicitly available. In practical computations, often only matrix-vector products instead of the matrix are available, which makes iterative methods the only choice. Here, we derive matrix-free sparse direct solvers based on matrixvector products. Two stages are involved. The first stage is to reconstruct the sparse matrix with a multi-nested dissection ordering. This idea recursively reorders the matrix and the separators with nested dissection, so that a compact probing strategy can reconstruct the matrix entries via the simultaneous recovery of multiple blocks with a small number of vectors. For discretized matrices in 2D or 3D, each dimension thus corresponds to one layer of nested dissection. The number of matrix-vector products required is O(logd N), where d is the dimension and N is the mesh dimension (e.g., 2D N ⇥ N or N ⇥ N ⇥ N mesh), and the reconstruction is thus said to be superfast. A simplified fast scheme can also be used, which uses O(N logd 1 N) In the second stage, the matrix is factorized in a randomized multifrontal method based on rank structures and randomized sampling. The overall solver costs about O(n) and O(n4/3) flops for some 2D and 3D problems, respectively, where n is the matrix size (e.g., n = N in 2D and N in 3D). The solver has a potential to work for varying parameters. For example, when the diagonal or few entries of the matrix change, we can reuse at least part of the previous factorizations, which is nearly impossible in classical direct factorizations. The multi-nested dissection idea also has other benefits such as in the structured solutions.