Specifically when the parameters can not be deter- mined completely by the experimental data, we carry out the Laplace approximations in the directions or- thogonal to the null space of the corresponding Ja- cobian matrix, so that the information gain (K–L di- vergence) can be reduced to an integration against the marginal density of the transformed parameters which are not determined by the experiments. This in- tegration can be numerically estimated by projecting the prior samples or quadratures onto the underdeter- mined manifold via the gradient directions.
To deal with the issue of dimensionality in cer- tain complex problems, we can use a sparse quadra- ture for the integration over the prior pdf. We demon- strate the accuracy, efficiency and robustness of the proposed method via several nonlinear numerical ex- amples, including the designs of the scalar parameter in an one–dimensional cubic polynomial function, the design of the same scalar in a slightly modified func- tion with two indistinguishable parameters, the reso- lution width and measurement time for a blurred sin- gle peak spectrum, and the sensor optimization for a impedance tomography problem.