Real-time arbitrary waveform generation (AWG) is essential in various engineering and research applications. This paper introduces a novel AWG architecture using an NVIDIA graphics processing unit (GPU) and a commercially available high-speed digital-to-analog converter (DAC) card, both running on a desktop personal computer (PC). The GPU accelerates the "embarrassingly" data-parallel additive synthesis framework for AWG, and the DAC reconstructs the generated waveform in the analog domain at high speed. The AWG software is developed using the developer-friendly compute unified device architecture (CUDA) runtime application programming interface (API) from NVIDIA. With this architecture, we achieve a 586-fold increase in the speed of computing periodic radio-frequency (rf) arbitrary waveforms compared to a central processing unit (CPU). We also demonstrate two different pathways for dynamically controlling multi-tone rf waveforms, which we characterize by chirping individual single-frequency tones in the multi-tone waveforms. One pathway offers arbitrary simultaneous chirping of 1000 individual Nyquist-limited single-frequency tones at a sampling rate of 280 megasamples per second (MS/s) for a limited time duration of 35 ms. The other pathway offers simultaneous chirping of 340 individual Nyquist-limited single-frequency tones at 50 MS/s, or 55 individual tones at 280 MS/s for an arbitrary duration. Using the latter pathway, we demonstrate control over 5000-tone and 10,000-tone waveforms by chirping all of their constituent tones in groups of up to 100 tones. This AWG architecture is designed for creating large defect-free optical tweezer arrays of single neutral atoms or molecules for quantum simulation and quantum computation.