Gradient-Based Dimension Reduction of Multivariate Vector-Valued Functions

We propose a gradient-based method for detecting and exploiting low-dimensional input parameter dependence of multivariate functions. The methodology consists in minimizing an upper bound, obtained by Poincar\'e-type inequalities, on the approximation error. The resulting method can be used to approximate vector-valued functions (e.g., functions taking values in $\mathbb{R}^n$ or functions taking values in function spaces) and generalizes the notion of active subspaces associated with scalar-valued functions. A comparison with the truncated Karhunen-Lo\`eve decomposition shows that using gradients of the function can yield more effective dimension reduction. Numerical examples reveal that the choice of norm on the codomain of the function can have a significant impact on the function's low-dimensional approximation.