  Most existing inference methods for the uncertainty quantification of
nonlinear inverse problems need repetitive runs of the forward model which is
computationally expensive for high-dimensional problems, where the forward
model is expensive and the inference need more iterations. These methods are
generally based on the Bayes' rule and implicitly assume that the probability
distribution is unique, which is not the case for scenarios with Knightian
uncertainty. In the current study, we assume that the probability distribution
is uncertain, and establish a new inference method based on the nonlinear
expectation theory for 'direct' uncertainty quantification of nonlinear inverse
problems. The uncertainty of random parameters is quantified using the
sublinear expectation defined as the limits of an ensemble of linear
expectations estimated on samples. Given noisy observed data, the posterior
sublinear expectation is computed using posterior linear expectations with
highest likelihoods. In contrary to iterative inference methods, the new
nonlinear expectation inference method only needs forward model runs on the
prior samples, while subsequent evaluations of linear and sublinear
expectations requires no forward model runs, thus quantifying uncertainty
directly which is more efficient than iterative inference methods. The new
method is analysed and validated using 2D and 3D test cases of transient Darcy
flows.
