Topology optimization is nowadays a mature and widely accepted computational design methodology. Manufacturing processes have also progressed and can realize optimized designs with complex geometries. Consequently, topology optimization is pushed towards very high computational resolution so that novel design concepts can be found that bring significant weight reduction and excellent performance. Hence there is a growing interest in efficient computational schemes that can generate high-resolution optimized designs using reasonable computational cost. This leads to an increasing popularity of parallel computing where iterative methods are utilized for solving the state and adjoint equations. But top-class hardware and iterative solvers alone are not enough: algorithmic developments are necessary as well to achieve significant reduction in computational time. Iterative solution of the state and adjoint problems can be terminated prematurely, yielding an approximation of the state or adjoint vector at a reduced computational cost. Various criteria for early termination have been investigated in the context of topology optimization. In this talk, we will focus on techniques that monitor the predicted accuracy of the design sensitivities – implying that inexact design sensitivities are utilized for optimization. We will relate the notion of inexact design sensitivities to one-shot approaches, in which single steps towards solving the state, adjoint and optimization are performed sequentially. Results for minimum compliance topology optimization have already showed significant computational savings, and we aim to extend this success to the cases of stress-based problems and nonlinear structural responses.