Asynchronous execution is an appealing approach in computational linear algebra, particularly for making better use of modern heterogeneous hardware. At the same time, its practical implementation on systems combining CPUs and GPUs remains a nontrivial task. In this talk, we consider the implementation of an asynchronous solver on an AMD-based system as a representative case study. The presentation will provide an overview of the key computational concepts underlying asynchronous methods, along with a discussion of the programming models and software tools required to effectively deploy them on AMD platforms. Particular emphasis will be placed on the interaction between algorithmic structure and hardware capabilities, as well as on the considerations that arise when mapping abstract computational ideas to concrete implementations. Through this example, we aim to illustrate the broader challenges and opportunities associated with asynchronous approaches in large-scale scientific computing, offering insights into both the methodological framework and the practical aspects of developing such methods on modern GPU-accelerated systems.