Interplay of colour kinematics duality and analytic calculation of multi-loop scattering amplitudes: one and two loops

In this talk, we review recent developments towards the calculation of multi-loop scattering amplitudes. In particular, we discuss how the colour-kinematics duality can provide new integral relations at one-loop level via the Loop-Tree duality formalism. On the other hand, in order to compute scattering amplitudes at one- and two-loop level, numerically and analytically, we describe the preliminary automation of the adaptive integrand decomposition algorithm. We show preliminary results on the analytic reduction of the $mu e$-elastic scattering at one- and two-loop level.