Optimal Solutions of Well-Posed Linear Systems via Low-Precision Right-Preconditioned GMRES with Forward and Backward Stabilization
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Linear systems in applications are typically well-posed, and yet the coefficient matrices may be nearly singular in that the condition number κ(A) may be close to 1/ε_w, where ε_w denotes the unit roundoff of the working precision. It is well known that iterative refinement (IR) can make the forward error independent of κ(A) if κ(A) is sufficiently smaller than 1/ε_w and the residual is computed in higher precision. We propose a new iterative method, called Forward-and-Backward Stabilized Minimal Residual or FBSMR, by conceptually hybridizing right-preconditioned GMRES (RP-GMRES) with quasi-minimization. We develop FBSMR based on a new theoretical framework of essential-forward-and-backward stability (EFBS), which extends the backward error analysis to consider the intrinsic condition number of a well-posed problem. We stabilize the forward and backward errors in RP-GMRES to achieve EFBS by evaluating a small portion of the algorithm in higher precision while evaluating the preconditioner in lower precision. FBSMR can achieve optimal accuracy in terms of both forward and backward errors for well-posed problems with unpolluted matrices, independently of κ(A). With low-precision preconditioning, FBSMR can reduce the computational, memory, and energy requirements over direct methods with or without IR. FBSMR can also leverage parallelization-friendly classical Gram-Schmidt in Arnoldi iterations without compromising EFBS. We demonstrate the effectiveness of FBSMR using both random and realistic linear systems.
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