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We describe a method to numerically compute multi-loop integrals, dependingon one dimensionless parameter $x$ and the dimension $d$, in the wholekinematic range of $x$. The method is based on differential equations, which,however, do not require any special form, and series expansions around singularand regular points. This method provides results well suited for fast numericalevaluation and sufficiently precise for phenomenological applications. We applythe approach to four-loop on-shell integrals and compute the coefficientfunction of eight colour structures in the relation between the mass of a heavyquark defined in the $\overline{\rm MS}$ and the on-shell scheme allowing for asecond non-zero quark mass. We also obtain analytic results for these eightcoefficient functions in terms of harmonic polylogarithms and iteratedintegrals. This allows for a validation of the numerical accuracy.

A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $\overline{\rm MS}$-pole quark mass relation