Continuous analogues of Krylov subspace methods for differential operators

Analogues of the conjugate gradient method, minimum residual method, and generalized minimum residual method are derived for solving boundary value problems (BVPs) involving ordinary diferential equations. 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 satisfed, and we develop an operator preconditioner that ensures that an approximate solution is computed after a fnite number of iterations. The developed Krylov methods are practical iterative BVP solvers that are particularly efcient when a fast operator-function product is available. An extension to partial diferential operators is also presented.