Accuracy of Response Surfaces over Active Subspaces Computed with Random Sampling∗

Given a function f that depends on m parameters, the problem is to identify an “active subspace” of dimension k ≪ m, where f is most sensitive to change, and then to approximate f by a response surface over this lower-dimensional active subspace. We present a randomized algorithm for determining k and computing an orthonormal basis for the active subspace. We also derive a tighter probabilistic bound on the number of samples required for approximating the active subspace to a user-specified accuracy. The bound does not explicitly depend on the total number m of parameters, and allows tuning of the failure probability. We discuss different error measures for response surfaces; and separate errors due to approximation over a subspace from errors due to construction of the response surface. The accuracy of the construction method for the response surface is of utmost importance. We design a test problem that makes it easy to construct active subspaces of any dimension k. Numerical experiments with k = 10 and a response surface constructed with sparse grid interpolation confirm the effectiveness of our error measures.