  In many practical applications of numerical methods a substantial increase in
efficiency can be obtained by using local grid refinement, since the solution
is generally smooth in large parts of the domain and large gradients occur only
locally. Fast evaluation of integral transforms on such an adaptive grid
requires an algorithm that relies on the smoothness of the continuum kernel
only, independent of its discrete form. A multilevel algorithm with this
property was presented in [A. Brandt and C.H. Venner, SIAM J. Sci. Stat.
Comput. 19 (1998) pp.468-492] [Bra1998]. Ref. [Bra1998] shows that already on a
uniform grid the new algorithm is more efficient than earlier fast evaluation
algorithms, and elaborates the application to one-dimensional transforms. The
present work analyses the extension and implementation of the algorithm for
multidimensional transforms. The analysis conveys that the multidimensional
extension is nontrivial, on account of the occurence of nonlocal corrections.
However, by virtue of the asymptotic smoothness properties of the continuum
kernel, these corrections can again be evaluated fast. By recursion, it is then
possible to obtain the optimal work estimates indicated in [Bra1998].
Currently, only uniform grids are considered. Detailed numerical results will
be presented for a two dimensional model problem. The results demonstrate that
with the new algorithm the evaluation of multidimensional transforms is also
more efficient than with previous algorithms.
