This work describes improvements by algorithmic and architectural means to black-box Bayesian inference over high-dimensional parameter spaces. The well-known adaptive Metropolis (AM) algorithm [H. Haario, E. Saksman, and J. Tamminen, Bernoulli, (2001), pp. 223--242] is extended herein to scale asymptotically uniformly with respect to the underlying parameter dimension for Gaussian targets, by respecting the variance of the target. The resulting algorithm, referred to as the dimension-independent adaptive Metropolis (DIAM) algorithm, also shows improved performance with respect to adaptive Metropolis on non-Gaussian targets. This algorithm is further improved, and the possibility of probing high-dimensional (with dimension $d \geq 1000$) targets is enabled, via GPU-accelerated numerical libraries and periodically synchronized concurrent chains (justified a posteriori). Asymptotically in dimension, this GPU implementation exhibits a factor of four improvement versus a competitive CPU-based Intel MKL (math kernel library) parallel version alone. Strong scaling to concurrent chains is exhibited, through a combination of longer time per sample batch (weak scaling) with fewer necessary samples to convergence. The algorithm performance is illustrated on several Gaussian and non-Gaussian target examples, in which the dimension may be in excess of one thousand.