Continuous analogues of Krylov methods for differential operators
This work provides a theoretical framework for applying Krylov methods to unbounded differential operators, benefiting researchers in numerical analysis and scientific computing.
The paper derives continuous analogues of Krylov methods (CG, MINRES, GMRES) for solving boundary value problems with second-order differential operators, using projection operators for boundary conditions and an operator preconditioner for finite-step convergence. The methods are efficient when fast operator-function products are available.
Analogues of the conjugate gradient method, MINRES, and GMRES are derived for solving boundary value problems (BVPs) involving second-order differential operators. Two challenges arise: imposing the boundary conditions on the solution while building up a Krylov subspace, and guaranteeing convergence of the Krylov-based method on unbounded operators. Our approach employs projection operators to guarantee that the boundary conditions are satisfied, and we develop an operator preconditioner that ensures that an approximate solution is computed after a finite number of iterations. The developed Krylov methods are practical iterative BVP solvers that are particularly efficient when a fast operator-function product is available.