  This work is concerned with approximating a trivariate function defined on a
tensor-product domain via function evaluations. Combining tensorized Chebyshev
interpolation with a Tucker decomposition of low multilinear rank yields
function approximations that can be computed and stored very efficiently. The
existing Chebfun3 algorithm [Hashemi and Trefethen, SIAM J. Sci. Comput., 39
(2017)]uses a similar format but the construction of the approximation proceeds
indirectly, via a so called slice-Tucker decomposition. As a consequence,
Chebfun3 sometimes uses unnecessarily many function evaluations and does not
fully benefit from the potential of the Tucker decomposition to reduce,
sometimes dramatically, the computational cost. We propose a novel algorithm
Chebfun3F that utilizes univariate fibers instead of bivariate slices to
construct the Tucker decomposition. Chebfun3F reduces the cost for the
approximation in terms of the number of function evaluations for nearly all
functions considered, typically by 75%, and sometimes by over 98%.
