Non-blind　image　deblurring　is　typically　formulated　as　a　linear　least-squares　problem　regularized　by　natural　priors　on　the　corresponding　sharp　picture’s　gradients,　which　can　be　solved,　for　example,　using　a　half-quadratic　splitting　method　with　Richardson　fixed-point　iterations　for　its　least-squares　updates　and　a　proximal　operator　for　the　auxiliary　variable　updates.　We　propose　to　precondition　the　Richardson　solver　using　approximate　inverse　filters　of　the　(known)　blur　and　natural　image　prior　kernels.　Using　convolutions　instead　of　a　generic　linear　preconditioner　allows　extremely　efficient　parameter　sharing　across　the　image,　and　leads　to　significant　gains　in　accuracy　and/or　speed　compared　to　classical　FFT　and　conjugate-gradient　methods.　More　importantly,　the　proposed　architecture　is　easily　adapted　to　learning　both　the　preconditioner　and　the　proximal　operator　using　CNN　embeddings.　This　yields　a　simple　and　efficient　algorithm　for　non-blind　image　deblurring　which　is　fully　interpretable,　can　be　learned　end　to　end,　and　whose　accuracy　matches　or　exceeds　the　state　of　the　art,　quite　significantly,　in　the　non-uniform　case.　©　2020,　Springer　Nature　Switzerland　AG.