A universal preconditioner for linear systems

We present a universal preconditioner $\Gamma$ that is applicable to all invertible linear problems $A x = y$ for which an approximate inverse is available. After preconditioning, the condition number depends on the norm of the discrepancy of this approximation instead of that of the original, potentially unbounded, system. We prove that our construct is the only universal approach that ensures that $\lVert 1-\Gamma^{-1} A \rVert<1$ in all cases, thus enabling the use of the highly memory efficient Richardson iteration. Its unique form permits the elimination of the forward problem from the preconditioned system, often halving the time required per iteration. We demonstrate and evaluate our approach for wave problems, diffusion problems, and the pantograph delay differential equation.