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

Ifr(t)is the linear scattering response of an object to an excitation waveforme(t), thenr(t) = (e ast h) (t). One would like to deconvolve and solve forh(t), 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 formE cdot H = Rby the following problem. Minimize|h_{1}|+ ldots + |h_{n}|subject toR - lambda leq E cdot H leq R + lambdawherelambdais 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 minimizeparallel E cdot H - R parallel^{2} +lambda parallel H parallel^{2}where againlambdais 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.