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Probabilistic ideas and tools have recently begun to permeate into severalfields where they had traditionally not played a major role, including fieldssuch as numerical linear algebra and optimization. One of the key ways in whichthese ideas influence these fields is via the development and analysis ofrandomized algorithms for solving standard and new problems of these fields.Such methods are typically easier to analyze, and often lead to faster and/ormore scalable and versatile methods in practice.  This thesis explores the design and analysis of new randomized iterativemethods for solving linear systems and inverting matrices. The methods arebased on a novel sketch-and-project framework. By sketching we mean, to startwith a difficult problem and then randomly generate a simple problem thatcontains all the solutions of the original problem. After sketching theproblem, we calculate the next iterate by projecting our current iterate ontothe solution space of the sketched problem.

Sketch and Project: Randomized Iterative Methods for Linear Systems and Inverting Matrices