We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully nonlinear form. Second, we examine the general case where some data and/or unknowns may be functions of a continuous variable and where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodifferential equations). As particular cases of our nonlinear algorithm we find linear solutions well known in geophysics, like Jackson's (1979) solution for discrete problems or Backus and Gilbert's (1970) solution for continuous problems.

Gaia stellar measurements are currently revolutionizing our knowledge of the evolutionary history of the Milky Way. 3D maps of the interstellar dust provide complementary information and are a tool for a wide range of uses. We aimed at building 3D maps of the dust in the Local arm and surrounding regions. To do so, Gaia DR2 photometric data were combined with 2MASS measurements to derive extinction towards stars that possess accurate photometry and relative uncertainties on DR2 parallaxes smaller than 20%. We applied to the individual extinctions a new hierarchical inversion algorithm adapted to large datasets and to a inhomogeneous target distribution. Each step associates regularized Bayesian inversions along all radial directions and a subsequent inversion in 3D of all their results. Each inverted distribution serves as a prior for the subsequent step and the spatial resolution is progressively increased. We present the resulting 3D distribution of the dust in a 6 x 6 x 0.8 kpc 3 volume around the Sun. Its main features are found to be elongated along different directions that vary from below to above the mid-plane: the outer part of Carina-Sagittarius, mainly located above the mid-plane, the Local arm/Cygnus Rift around and above the mid-plane and the fragmented Perseus arm are oriented close to the direction of circular motion. The more than 2 kpc long spur (nicknamed the split) that extends between the Local Arm and Carina-Sagittarius, the compact near side of Carina-Sagittarius and the Cygnus Rift below the Plane are oriented along l∼40 to 55 • . Dust density images in vertical planes reveal in some regions a wavy pattern and show that the solar neighborhood within ∼500 pc remains atypical by its extent above and below the Plane. We show several comparisons with the locations of molecular clouds, HII regions, O stars and masers. The link between the dust concentration and these tracers is markedly different from one region to the other.

Aims. Three-dimensional (3D) maps of the Galactic interstellar matter (ISM) are a potential tool of wide use, but accurate and detailed maps are still lacking. One of the ways to construct the maps is to invert individual distance-limited ISM measurements, a method we have applied here to measurements of stellar color excess in the optical. Methods. We assembled color excess data together with the associated parallax or photometric distances to constitute a catalog of 23 000 sightlines for stars within 2.5 kpc. The photometric data are taken from Strömgren catalogs, the Geneva photometric database, and the Geneva-Copenhagen survey. We also included extinctions derived towards open clusters. We applied an inversion method based on a regularized Bayesian approach to this color excess dataset, a method previously used for mapping at closer distances. Results. We show the dust spatial distribution resulting from the inversion by means of planar cuts through the differential opacity 3D distribution, and by means of 2D maps of the integrated opacity from the Sun up to various distances. The mapping assigns locations to the nearby dense clouds and represents their distribution at the spatial resolution that is allowed by the dataset properties, i.e. 10 pc close to the Sun and increasing to 100 pc beyond 1 kpc. Biases toward nearby and/or weakly extincted stars make this dataset particularly appropriate to mapping the local and neighboring cavities and to locating faint, extended nearby clouds, which are both goals that are difficult or impossible with other mapping methods. The new maps reveal a 1 kpc wide empty region in the third quadrant in the continuation of the so-called CMa tunnel of the Local Cavity, a cavity that we identify as the Superbubble GSH238+00+09 detected in radio emission maps and that is found to be bounded by the Orion and Vela clouds. The maps also show an extended narrower tunnel in the opposite direction (l 70• ) that also extends the Local Bubble further and together with it forms a conspicuous cavity bounded by the main Lup, Sco, Oph, Aql, Lac, Cep, and Tau clouds and OB associations. This chain of cavities and surrounding dense regions constitute the first computed representation of the well known Gould belt/Lindblad ring structures. Finally, almost all off-plane faint features that appear in 2D dust maps have a counterpart in the 3D maps, providing the dust distribution in nearby tenuous clouds.