Tropical sampling is a mathematically streamlined version of the

geometric sector decomposition approach to evaluate algebraic integrals

numerically. If the structure of the Newton polytope of the integrand is

precomputed or known from first principles, then tropical sampling

offers the advantage of a significantly improved performance.  In the

case of Euclidean Feynman integrals these polytopes are generalized

permutahedra, a class of polytopes which is well-studied.  Employing

this knowledge leads to a highly efficient and practical numerical

Feynman integration algorithm which can compute scalar Euclidean Feynman

integrals with arbitrary kinematics up to loop order ~20 on present-day

hardware.