The ill-posed problem of solving linear equations in the space of vector-valued finite
Radon measures with Hilbert space data is considered. Approximate solutions are obtained
by minimizing the Tikhonov functional with a total variation penalty. The well-posedness
of this regularization method and further regularization properties are mentioned.
Furthermore, a flexible numerical minimization algorithm is proposed which converges
subsequentially in the weak* sense and with rate 𝒪(n-1)
in terms of the functional values. Finally, numerical results for sparse
deconvolution demonstrate the applicability for a finite-dimensional discrete data space
and infinite-dimensional solution space.