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

If
r(t)
is the linear scattering response of an object to an excitation waveform
e(t)
, then
r(t) = (e \ast h) (t)
. One would like to deconvolve and solve for
h(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 form
E \cdot H = R
by the following problem. Minimize
\|h_{1}\|+ \ldots + \|h_{n}\|
subject to
R - \lambda \leq E \cdot H \leq R + \lambda
where
\lambda
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
\parallel E \cdot H - R \parallel^{2} +\lambda \parallel H \parallel^{2}
where again
\lambda
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.