In many computer experiments, surrogates are used to assist in searching for certain target points. If the surrogates are defined by response function values evaluated by costly iterative processes, the computational burdens may impede the efficiency of regular surrogate-assisted methods. Instead of computing the fully convergent response function values, we propose to control the function evaluation iterations dynamically to save time on function evaluations without degrading the overall performance. Our new algorithms adaptively determine whether each of the function evaluation iterations should be paused, kept running, or restarted; we then use the approximate function values with various levels of accuracy to construct the surrogates. The numerical results show that the proposed algorithms achieve significant savings when solving super-level set searching problems that involve identifying positive Lyapunov exponents of a dynamical system.
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