Adaptive experiment design for probabilistic integration

Probabilistic integration is a Bayesian inference technique for numerical integration, and has received much attention in the community of scientific and engineering computations. The most appealing advantages are the ability to improve the integration accuracy by making full use of the spatial correlation information among the design points, and the treatment of discretization error as a source of epistemic uncertainty being explicitly propagated to the integration results. This paper aims to develop an adaptive algorithm for further improving the efficiency and accuracy of the probabilistic integration when it is applied to the time-consuming computer simulators. A learning function is first extracted from the posterior variance of the integration and is shown to be especially useful for identifying the design point, by adding which to the training data set, the most reduction of the posterior variance of integration can be achieved. Based on this learning function, an adaptive experiment design algorithm is then developed for actively producing optimal design points. Results of the experiment tests and engineering application show that, with the same number of design points, the developed design strategy always produce more accurate and robust integration results, than the three kinds of commonly used random sampling design strategies (i.e., Monte Carlo design, Latin-hypercube design and Sobol sequence).