This paper develops a fast and scalable simultaneous variable selection and parameter estimation method for the Fine-Gray (1998) proportional subdistribution hazards (PSH) model for competing risks data. First we show that the broken adaptive ridge (BAR) estimator for the PSH model, defined as the limit of an $L_0$-based iteratively reweighted $L_2$-penalized regression algorithm, is consistent for both variable selection and parameter estimation. Then we develop novel algorithms to accelerate the BAR method in two steps. First we derive a cyclic coordinate-wise iteratively reweighted ridge regression algorithm which, for each coordinate, avoids actually performing iteratively reweighted ridge regression. Second, we introduce a two-way scanning algorithm to reduce the computation cost of the from the log-likelihood and its derived quantities such as gradients and Hessian matrix from order of $O(n^2)$ to $O(n)$. The resulting accelerated BAR method has yielded an impressive 100-1000 fold speedup over the original BAR method and some other competitors in empirical studies. Numerical illustrations will be given using both simulated and real data.