Linear Algebra with Disordered Sparse Matrices that have Spatial Structure: Theory and Computation

Dedication I dedicate this thesis to my father, Vincent Edward Sacksteder III, who shared with me his love for physics and for so many other things, and for me. One of my favorite science-related memories of him is of long hours during car trips when he would answer my many questions about science, industry, and lots of other subjects. Another is of him reading science fiction aloud to us. I would also like to remember my grandpa Richard Anthony Miller, who died on the day that this thesis was completed. He remains a strength to us all, and I'm very thankful for him and for the way he lived and died. And I should remember Dr. Phil Peters, who died a year after teaching me graduate level electromagnetism. 5 6 Acknowledgements I want most to thank God: the Father, his son Jesus our Saviour, and the Holy Spirit of love. I am thankful to them for the last five years which have probably been the happiest years yet in my life, for the past three and a half years living in the beautiful and holy city of Rome and traveling to many other extraordinary places, for good friends and community, for the privilege of both begining and ending this Ph.D, and for letting this Ph.D. be of a quality that I can be very proud of. But those are just the externals: the deeper reality is love. I am thankful for God's constant intimate friendship, love, and mercy, which are faithful even when I am not faithful, that have been enabling me to change slowly for the better, and that challenge me to share what I can with others. I believe that each person was created through God's deliberate choice to create that individual person and not another person, and that his choice was motivated by love, by seeing something immensely precious in the person he was choosing, and by wanting to share a friendship and a love with that person. I am very thankful for the choice that God made about me; for the gift of being alive. I also want to thank my advisor, Giorgio Parisi. Without his faithful support I probably would not have been permitted to pursue the research projects which developed into this thesis. He worked to understand what I was doing even when it didn't relate directly to his research, encouraged me, and gave …