  A method is described for the extrapolation of perturbative expansions in
powers of asymptotically small coupling parameters or other variables onto the
region of finite variables and even to the variables tending to infinity. The
method involves the combination of ideas from renormalization group theory,
approximation theory, dynamical theory, and optimal control theory. The
extrapolation is realized by means of self-similar factor approximants, whose
control parameters can be uniquely defined. The method allows to find the
large-variable behavior of sought functions knowing only their small-variable
expansions. Convergence and accuracy of the method are illustrated by explicit
examples, including the so-called zero-dimensional field theory and anharmonic
oscillator. Strong-coupling behavior of Gell-Mann-Low functions in
multicomponent field theory, quantum electrodynamics, and quantum
chromodynamics is found, being based on their weak-coupling perturbative
expansions.
