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A mathematical methodology is proposed for the computational inverse design of nozzle shapes producing controlled geometric cavitation. The proposed methodology uses an unstructured RANS flow solver, with the ability to compute sensitivity derivatives via an Adjoint algorithm and independently of the shape parameterisation. The method is used to develop and evaluate conceptual shapes for nozzle hole cavitation reduction. The localised region at the hole inlet where geometric-cavitation is produced, is parameterised using its radius of curvature. The parameterisation method is an empirical curvature fit method suitable for the design and manufacturing of such nozzles. In order to validate the efficiency of the proposed method, the optimisation problem is handled as an inverse design one. The objective function is formed using a target pressure distribution where the negative pressure area is narrowed or even eliminated.