  We introduce a new method to evaluate algebraic integrals over the simplex
numerically. This new approach employs techniques from tropical geometry and
exceeds the capabilities of existing numerical methods by an order of
magnitude. The method can be improved further by exploiting the geometric
structure of the underlying integrand. As an illustration of this, we give a
specialized integration algorithm for a class of integrands that exhibit the
form of a generalized permutahedron. This class includes integrands for
scattering amplitudes and parametric Feynman integrals with tame kinematics. A
proof-of-concept implementation is provided with which Feynman integrals up to
loop order 17 can be evaluated.
