Monte Carlo Methods and Partial Differential Equations: Algorithms and Implications for High-Performance Computing

We give a brief overview of the history of the Monte Carlo method for the numerical solution of partial differential equations (PDEs) focusing on the Feynman-Kac formula for the proababilistic representation of the solution of the PDEs. We then take the example of solving the linearized Poisson-Boltzmann equation to compare and contrast standard deterministic numerical approaches with the Monte Carlo method. Monte Carlo methods have always been popular due to the ease of finding computational work that can be done in parallel. We look at how to extract parallelism from Monte Carlo methods, and some newer ideas based on Monte Carlo domain decomposition that extract even more parallelism. In light of this, we look at the implications of using Monte Carlo to on high-performance architectures and algorithmic resilience.

Speaker Bio: Michael Mascagni was born in Bologna, Italy of one American and one Italian parent. However, by the age of four he was in the US confusing his Lake Forest, Illinois kindergarten teacher (and American grandmother) by correctly answering her Italian. This confusion continued though High School in Clinton, Iowa and College at the University of Iowa. At Iowa, he obtained a B.S.E. in Biomedical Engineering, and a B.S. in Mathematics and was a member of Phi Beta Kappa and Tau Beta Pi, the top liberal arts and engineering academic honor societies. After graduation, he seriously considered going to Medical School, and declined two such offers to instead study Neurobiology at the Rockefeller University, which is located on the East River the middle of Manhattan. Since "the Rock" is such a small and specialized University, he also took graduate classes uptown, at Columbia University, and downtown, at the Courant Institute of Mathematical Sciences of New York University. Eventually, downtown trumped midtown and uptown, and Mathematics trumped Neurobiology, and he obtained his M.S. and Ph.D. in Mathematics from Courant. Perhaps more importantly, his social contacts at "the Rock" introduced him to his future wife, Becky Fandrei, at a party during an historic snow storm in New York. They were married between the Masters and Doctoral degrees. Upon graduation, he obtained a post-doctoral research position in the Mathematical Research Branch of an institute of the National Institutes of Health, in Bethesda, MD and they moved to Washington, DC. It was during this period his research moved away from modeling the nervous system to studying the high-performance computing implications of the algorithms he developed and used. He was one of the first to use random number-based algorithms on the massively parallel Connection Machine at the Naval Research Lab in DC. In fact, after two years at NIH he moved to the Institute for Defense Analyses' Supercomputing Research Center in Bowie, MD. This organization works for the National Security Agency, and it was here that his interests in parallel computing, random number generation, number theory, and discrete mathematics were nurtured. During his time at SRC his oldest sons Alexander and Marcus were born. After many happy years at SRC, he decided to rejoin academia, and went to the University of Southern Mississippi to run the Graduate Program in Scientific Computing. After a few years there, a desire to join a Computer Science department arose, and he moved to Florida State University as an Associate Professor of Computer Science, he has since been promoted to Full Professor. While at FSU his youngest son, Evan, was born, Dr. Mascagni is on the editorial board of three journals in his field, and is a member of the ACM (Association of Computing Machinery), SIAM (Society of Applied Mathematics), and IMACS (International Association of Mathematics and Computers in Simulation). He is also a member of the Board of Directors of IMACS. He has approximately 100 refereed technical papers that have appeared in a wide variety of publications in areas including Applied Mathematics, Computer Science, Simulation Science, Monte Carlo Methods, Computational Science, High-Performance Computing, Scientific Computing, Computational Physics, and Computational Neuroscience. He has been a visiting professor at the University of Padova in Italy, the University of Salzburg in Austria, and the Swiss Federal Technical Institute-Zürich in Switzerland, and is a consultant to industry and government. He has made technical presentations in 18 countries and in most of the 50 U.S. states. At present Dr. Mascagni's research group consists of post-doctoral associates, graduate students, and undergraduate workers. The areas they work on are parallel and distributed computing, Grid computing, random number generation, Monte Carlo methods, computational number theory and discrete algorithms, and applications to materials science, biochemistry, electrostatics, and finance. His hobbies include swimming and biking, which arise from his misspent youth at NYU where he commuted by bicycle and swam each day, and playing the violin. In fact, he has been the concertmaster of the Big Bend Community Orchestra in Tallahassee, has played with several orchestras in places like Washington, DC and Salzburg, Austria. By the way, he is always looking for others to play chamber music.

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