A sparse grid surrogate model using hierarchical B-spline basis functions is used to approximate the objective function in an optimization-based inversion algorithm. The B-spline basis provides a smooth interpolant of the objective function and the gradient of the interpolant is readily available in closed-form. The latter is used in a gradient-based minimum search algorithm that results in the approximate solution of the inverse problem. The method is computationally more efficient than using gradient-free direct search methods, as illustrated by an example drawn from eddy-current nondestructive testing.