A general algorithm for decimating unstructured discretized data sets is presented. The discretized space may be a planar triangulation, a general 3D surface triangulation, or a 3D tetrahedrization. The decimation algorithm enforces Dirichlet boundary conditions, uses only existing vertices, and assumes manifold geometry. Local dynamic vertex removal is performed without history information while preserving the initial topology and boundary geometry. The tessellation at each step of the algorithm is preserved and, in the pathological case, every interior vertex is a candidate for removal. The research focuses on how to remove a vertex from an existing unstructured n-dimensional tessellation, not on the formulation of decimation criteria. Criteria for removing a candidate vertex may be based on geometric properties or any scalar governing function specific to the application. Use of scalar functions to adaptively control or optimize tessellation resolution is particularly applicable to the computer graphics, computational fluids, and structural analysis disciplines. Potential applications in the geologic exploration and medical or industrial imaging fields are promising.