Two methods to deconvolve: L1-method using simplex algorithm and L2-method using least squares and parameter

If is the linear scattering response of an object to an excitation waveform , then . One would like to deconvolve and solve for , the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method replaces the discretized matrix form by the following problem. Minimize subject to where is a column vector chosen sufficiently small to yield acceptable residuals, yet large enough to make the problem well-conditioned. This problem is converted to a linear programming problem so that the simplex algorithm can be used. The second method is to minimize where again is chosen small enough to yield acceptable residuals and large enough to make the problem well-conditioned. The method will be demonstrated with a Hilbert matrix inversion problem, and also by the deconvolution of the impulse response of a simple target from measured data.