Neural Acceleration of Incomplete Cholesky Preconditioners

Abstract

The solution of a sparse system of linear equations is ubiquitous in scientific applications. Iterative methods, such as the Preconditioned Conjugate Gradient method (PCG), are normally chosen over direct methods due to memory and computational complexity constraints. However, the efficiency of these methods depends on the preconditioner utilized. The development of the preconditioner normally requires some insight into the sparse linear system and the desired trade-off of generating the preconditioner and the reduction in the number of iterations. Incomplete factorization methods tend to be black box methods to generate these preconditioners but may fail for a number of reasons. These reasons include numerical issues that require searching for adequate scaling, shifting, and fill-in while utilizing a difficult to parallelize algorithm. With a move towards heterogeneous computing, many sparse applications find GPUs that are optimized for dense tensor applications like training neural networks being underutilized. In this work, we demonstrate that a simple artificial neural network trained either at compile time or in parallel to the running application on a GPU can provide an incomplete sparse Cholesky factorization that can be used as a preconditioner. This generated preconditioner is as good or better in terms of reduction of iterations than the one found using multiple preconditioning techniques such as scaling and shifting. Moreover, the generated method also works and never fails to produce a preconditioner that does not reduce the iteration count