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A fast algorithm for the evaluation of acoustic fields produced by given source distributions is developed with the aim of accelerating iterative boundary element method (BEM) solvers. The algorithm is based on field smoothing by phase and amplitude compensation, which allows for sampling of the fields radiated by finite-size source distributions over coarse nonuniform (spherical) grids (NGs). Subsequently, the fields at the desired target points can be obtained by an interpolation and phase and amplitude restoration. Combining this approach with the divide-and-conquer strategy, the total field is computed via a hierarchical decomposition of the source domain. In this computational scheme, the phase and amplitude compensated fields produced by neighboring subdomains are gradually aggregated through a multilevel process involving interpolation between increasingly dense NGs and the scatterer surface. This multilevel NG algorithm is used to reduce the computational cost of applying the field evaluation operator and its adjoint, as required in each iteration of the conjugate gradient solver based on the BEM-discretized integral representation of scattering problems. Accuracy and computational efficiency of the NG algorithm are demonstrated on representative examples of elongated, quasi-planar, and full 3-D scatterers.