We propose a novel algorithm based on inexact GMRES methods for linear response calculations in density functional theory. Such calculations require iteratively solving a nested linear problem $\mathcal{E} \delta\rho = b$ to obtain the variation of the electron density $\delta \rho$. Notably each application of the dielectric operator $\mathcal{E}$ in turn requires the iterative solution of multiple linear systems, the Sternheimer equations. We develop computable bounds to estimate the accuracy of the density variation given the tolerances to which the Sternheimer equations have been solved. Based on this result we suggest reliable strategies for adaptively selecting the convergence tolerances of the Sternheimer equations, such that each application of $\mathcal{E}$ is no more accurate than needed. Experiments on challenging materials systems of practical relevance demonstrate our strategies to achieve superlinear convergence as well as a reduction of computational time by about 40% while preserving the accuracy of the returned response solution. Our algorithm seamlessly combines with standard preconditioning approaches known from the context of self-consistent field problems making it a promising framework for efficient response solvers based on Krylov subspace techniques.

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