  We present a method for dimensionally adaptive sparse trigonometric
interpolation of multidimensional periodic functions belonging to a smoothness
class of finite order. This method targets applications where periodicity must
be preserved and the precise anisotropy is not known a priori. To the authors'
knowledge, this is the first instance of a dimensionally adaptive sparse
interpolation algorithm that uses a trigonometric interpolation basis. The
motivating application behind this work is the adaptive approximation of a
multi-input model for a molecular potential energy surface (PES) where each
input represents an angle of rotation. Our method is based on an anisotropic
quasi-optimal estimate for the decay rate of the Fourier coefficients of the
model; a least-squares fit to the coefficients of the interpolant is used to
estimate the anisotropy. Thus, our adaptive approximation strategy begins with
a coarse isotropic interpolant, which is gradually refined using the estimated
anisotropic rates. The procedure takes several iterations where ever-more
accurate interpolants are used to generate ever-improving anisotropy rates. We
present several numerical examples of our algorithm where the adaptive
procedure successfully recovers the theoretical "best" convergence rate,
including an application to a periodic PES approximation. An open-source
implementation of our algorithm resides in the Tasmanian UQ library developed
at Oak Ridge National Laboratory.
