  We consider increasing stability in the inverse Schr\"{o}dinger potential
problem with power type nonlinearities at a large wavenumber. Two linearization
approaches, with respect to small boundary data and small potential function,
are proposed and their performance on the inverse Schr\"{o}dinger potential
problem is investigated. It can be observed that higher order linearization for
small boundary data can provide an increasing stability for an arbitrary power
type nonlinearity term if the wavenumber is chosen large. Meanwhile,
linearization with respect to the potential function leads to increasing
stability for a quadratic nonlinearity term, which highlights the advantage of
nonlinearity in solving the inverse Schr\"{o}dinger potential problem. Noticing
that both linearization approaches can be numerically approximated, we provide
several reconstruction algorithms for the quadratic and general power type
nonlinearity terms, where one of these algorithms is designed based on boundary
measurements of multiple wavenumbers. Several numerical examples shed light on
the efficiency of our proposed algorithms.
