On Waring's problem for larger powers

Let $G(k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$-th powers. Then for all $k\in \mathbb N$, one has $$G(k)\le \lceil k(\log k+4.20032)\rceil .$$ Our new methods improve on all bounds available hitherto when $k\ge 14$.