Single-Iteration Sobolev Descent for Linear Initial Value Problems

Sobolev descent has long been established as an efficient method for numerically solving boundary value problems, ordinary differential equations and partial differential equations in a small number of iterations. We demonstrate that for any linear ordinary differential equation with initial value conditions sufficient to assure a unique solution, there exists a Hilbert space in which gradient descent will converge to the solution in one iteration. We provide two elementary examples, one initial value problem and one boundary value problem, demonstrating the effectiveness of the theory in numerical settings. As there are ample efficient numerical methods for solving such problems, the significance of the paper is in the approach and the question it raises. Namely, do such spaces exist for wider classes of differential equations?