A multi-resolution approach to dynamic programming is presented, which reduces the computational effort of solving multistage optimization problems with long horizons and short decision intervals. The approach divides an optimization horizon into a series of subhorizons, discretized at different state space and temporal resolutions, enabling a reduced computational complexity compared to a single-resolution approach. The method is applied to optimizing the operation of a residential energy storage system, using real 1-min demand and rooftop PV generation data. The multi-resolution approach reduces the required computation time, allowing optimization to be rerun more frequently, increasing the robustness of the receding-horizon-control approach to forecast errors. In an empirical study, this increases the cost-saving offered by a 2 kWh behind-the-meter battery energy storage system by 120% on average, compared to an approach using a single fine-grained resolution.