Certified dimension reduction of the input parameter space of multivariate functions

Approximation of multivariate functions is a difficult task when the number of input parameters is large. Identifying the directions where the function does not significantly vary is a key preprocessing step to reduce the complexity of the problem we have at hand. We propose a gradient based method that permits to detect such a low-dimensional structure of a function. Our methodology consists in minimizing an upper-bound of the approximation error obtained using Poincaré-type inequalities, and it generalizes the Active Subspace method. Numerical examples reveals the importance of the choice of the metric to measure errors and compare it with the commonly used truncated Karhunen-Loeve decomposition. 
We also show that the same methodology can be applied for the reduction of the dimension of Bayesian inverse problems. By seeking an approximation of the likelihood function by a ridge function, the resulting method exploits the fact that the data are not informative over the whole parameter space but only on a low-dimensional subspace.