Fictitious null spaces for improving the solution of injective inverse problems

For linear ill-posed problems with nontrivial null spaces, Tikhonov regularization and truncated singular value decomposition (TSVD) typically yield solutions that are close to the minimum norm solution. Such a bias is not always desirable, and we have therefore in a series of papers developed a weighting procedure which produces solutions with a different and controlled bias. This methodology can also conveniently be invoked when sparsity regularization is employed. The purpose of the present work is to study the potential use of this weighting applied to injective operators. The image under a compact operator of the singular vectors/functions associated with very small singular values will be almost zero. Consequently, one may regard these singular vectors/functions to constitute a basis for a fictitious null space that allows us to mimic the previous weighting procedure. It turns out that this regularization by weighting can improve the solution of injective inverse problems compared with more traditional approaches. We present some analysis of this methodology and exemplify it numerically, using sparsity regularization, for three PDE-driven inverse problems: the inverse heat conduction problem, the Cauchy problem for Laplace's equation, and the (linearized) Electrical Impedance Tomography problem with experimental data.