  We propose a new approach for computing tunneling rates in quantum or thermal
field theory with multiple scalar fields. It is based on exact analytical
solutions of piecewise linear potentials with many segments that describes any
given potential to arbitrary precision. The method is first developed for the
single field case in 3 and 4 space-time dimensions and demonstrated on examples
of classical potentials as well as the calculation of quantum fluctuations. A
systematic expansion of the potential beyond the linear order is considered,
taking into account higher order corrections, which paves the way for multiple
scalar fields. We thereby provide a fast semi-analytical tool for evaluating
the bounce action for theories with an extended scalar sector.
