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Abstract

Iterative linear solvers are a key paradigm, developed within numerical linear algebra, for solving linear systems within a multitude of applications. Alongside this, mixed-precision is a growing novel field of methodology which utilizes non-standard precision configurations to modify algorithms and tune their characteristics. In this thesis, we propose a simple inter-iteration approach to applying mixed-precision concepts towards general iterative linear solvers, called precision-cascading. This approach involves executing an algorithm in a sequence of increasingly accurate hardware-supported precision formats, tightening precision based on tracked solver runtime metrics, to iteratively build towards an accurate calculated solution while extracting cost improvements from early low-accuracy high-speed precision formats. We contribute a formal articulation of this idea and a large robust GPU-accelerated code base to facilitate its experimental study. Using the code base, 36,480 linear solve experiments are executed and analyzed to understand the approach's comparative effectiveness, relative to a fixed-precision double control solver, within its application to the GMRES(m) algorithm. Experimentation supports the hypothesis that the precision-cascading approach can match solution accuracy and can improve computational cost relative to the performance of the control fixed-precision double approach. This specifically consists of high fractions of precision-cascading experiments complying with a strict fractional error threshold to solution accuracy from the corresponding control solver, and within such compliant experiments, precision-cascading experiments achieving up to 22% median computational cost improvements. Corollary hypotheses regarding the approach's effectiveness in utilizing low-accuracy high-speed precision formats, and regarding effectiveness of mixing precision along the iteration dimension, are also supported. Additional key insights into the convergence behaviour effects of ILU-preconditioning, differing phase sequences, and differing phase transition logic approaches, are also explored through the thesis experimentation.