  Using a deterministic framework allows us to estimate a function with the
purpose of interpolating data in spatial statistics. Radial basis functions are
commonly used for scattered data interpolation in a d-dimensional space,
however, interpolation problems have to deal with dense matrices. For the case
of smoothing thin plate splines, we propose an efficient way to address this
problem by compressing the dense matrix by an hierarchical matrix
($\mathcal{H}$-matrix) and using the conjugate gradient method to solve the
linear system of equations. A simulation study was conducted to assess the
effectiveness of the spatial interpolation method. The results indicated that
employing an $\mathcal{H}$-matrix along with the conjugate gradient method
allows for efficient computations while maintaining a minimal error. We also
provide a sensitivity analysis that covers a range of smoothing and compression
parameter values, along with a Monte Carlo simulation aimed at quantifying
uncertainty in the approximated function. Lastly, we present a comparative
study between the proposed approach and thin plate regression using the "mgcv"
package of the statistical software R. The comparison results demonstrate
similar interpolation performance between the two methods.
