Solving the inverse scattering problem for the Helmholtz wave equation without employing the Born or Rytov approximations is a challenging problem, but some slow iterative methods have been proposed. One such method suggested by us is based on solving systems of nonlinear algebraic equations that are derived by applying the method of moments to a sinc basis function expansion of the fields and scattering potential. In the past, we have solved these equations for a 2-D object of n by n pixels in a time proportional to n 5. In the present paper, we demonstrate a new method based on FFT convolution and the concept of backprojection which solves these equations in time proportional to n 3. log(n). Several numerical examples are given for images up to 7 by 7 pixels in size. Analogous algorithms to solve the Riccati wave equation in n 3. log(n) time are also suggested, but not verified. A method is suggested for interpolating measurements from one detector geometry to a new perturbed detector geometry whose measurement points fall on a FFT accessible, rectangular grid and thereby render many detector geometrics compatible for use by our fast methods.