  In spite of the large number of papers appeared in the past which are devoted
to the lattice Boltzmann (LB) methods, basic aspects of the theory still remain
unchallenged. An unsolved theoretical issue is related to the construction of a
discrete kinetic theory which yields \textit{exactly} the fluid equations,
i.e., is non-asymptotic (here denoted as \textit{LB inverse kinetic theory}).
The purpose of this paper is theoretical and aims at developing an inverse
kinetic approach of this type. In principle infinite solutions exist to this
problem but the freedom can be exploited in order to meet important
requirements. In particular, the discrete kinetic theory can be defined so that
it yields exactly the fluid equation also for arbitrary non-equilibrium (but
suitably smooth) kinetic distribution functions and arbitrarily close to the
boundary of the fluid domain. Unlike previous entropic LB methods the theorem
can be obtained without functional constraints on the class of the initial
distribution functions. Possible realizations of the theory and asymptotic
approximations are provided which permit to determine the fluid equations
\textit{with prescribed accuracy.} As a result, asymptotic accuracy estimates
of customary LB approaches and comparisons with the Chorin artificial
compressibility method are discussed.
