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Design optimization of dielectric resonator antennas (DRAs) is a challenging task that normally involves electromagnetic (EM) simulation. Automated simulation-based DRA design realized as an optimization problem with a discrete EM solver directly embedded into the optimization algorithm is often very time consuming due to computational demands of discrete high-fidelity simulations. Here, we demonstrate a computationally efficient optimization approach to DRA design that exploits space mapping as the optimization engine, and kriging interpolation, used to create a fast surrogate model of the DRA under design. The kriging model is set up using simulation data of the coarse-discretization EM model which is evaluated by the same solver as the one that simulates the high-fidelity model of the DRA. Two DRA cases are demonstrated; in each case the optimal design is found at the computational cost corresponding only to a few high-fidelity simulations of the antenna structure.