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Due to the limitations of current voltage sensing techniques, optimal filtering of noisy, undersampled voltage signals on dendritic trees is a key problem in computational cellular neuroscience. These limitations lead to two sources of difficulty: 1) voltage data is incomplete (in the sense of only capturing a small portion of the full spatiotemporal signal) and 2) these data are available in only limited quantities for a single neuron. In this paper we use a Kalman filtering framework to develop optimal experimental design for voltage sampling. Our approach is to use a simple greedy algorithm with lazy evaluation to minimize the expected mean-square error of the estimated spatiotemporal voltage signal. We take advantage of some particular features of the dendritic filtering problem to efficiently calculate the estimator covariance by approximating it as a low-rank perturbation to the steady-state (zero-SNR) solution. We test our framework with simulations of real dendritic branching structures and compare the quality of both time-invariant and time-varying sampling schemes. The lazy evaluation proved critical to making the optimization tractable. In the time-invariant case improvements ranged from 30-100% over simpler methods, with larger gains for smaller numbers of observations. Allowing for time-dependent sampling produced up to an additional 30% improvement.