Low-rank Quasi-newton Updates for Robust Jacobian Lagging in Newton Methods

Newton-Krylov methods are standard tools for solving nonlinear problems. A common approach is to “lag” the Jacobian when assembly or preconditioner setup is computationally expensive, in exchange for some degradation in the convergence rate and robustness. We show that this degradation may be partially mitigated by using the lagged Jacobian as an initial operator in a quasi-Newton method, which applies unassembled low-rank updates to the Jacobian until the next full reassembly. We demonstrate the effectiveness of this technique on problems in glaciology and elasticity.