Hybrid Digital-Analog Approximate Inverse Preconditioning for Krylov Methods

Abstract

Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid digital-analog setting. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while preconditioner applications are performed through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We compare various block Jacobi preconditioning schemes including exact block inverses, sparse approximate inverses, Monte Carlo approximate inverses (MCAI), damping, and nested block Jacobi schemes. Numerical experiments with realistic analog matrix-vector simulations show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.