## Distinguished Seminar Series in Computational Science and Engineering

**March 3, 2022**

**Nonlinear Preconditioning for Implicit Solution of Discretized PDEs
**David Keyes

Professor, Applied Mathematics and Computational Science

Director, Extreme Computing Research Center

King Abdullah University of Science and Technology

**Recorded Seminar YouTube Link:
**https://youtu.be/CxIhzeSx1HQ

**Abstract:**

Nonlinear preconditioning refers to transforming a nonlinear algebraic system to a form for which Newton-type algorithms have improved success through quicker advance to the domain of quadratic convergence. We place these methods, which go back at least as far as the Additive Schwarz Preconditioned Inexact Newton (ASPIN, 2002), in the context of a proliferation distinguished by being left- or right-sided, multiplicative or additive, and partitioned by field, subdomain, or other criteria. We present the Nonlinear Elimination Preconditioned Inexact Newton (NEPIN, 2021), which is based on a heuristic “bad/good” heuristic splitting of equations and corresponding degrees of freedom. We augment basic forms of nonlinear preconditioning with three features of practical interest: a cascadic identification of the “bad” discrete equation set, an adaptive switchover to ordinary Newton as the domain of convergence is approached, and error bounds on output functionals of the solution. Various nonlinearly stiff algebraic and model PDE problems are considered for insight and we illustrate performance advantage and scaling potential on challenging two-phase flows in porous media. Joint work with Lulu Liu, Li Luo, Xiao-Chuan Cai, and others.

**Bio:**

David Keyes directs the Extreme Computing Research Center at the King Abdullah University of Science and Technology (KAUST), where he was a founding Dean in 2009 and currently serves in the Office of the President as Senior Associate. He is a professor in the programs of Applied Mathematics, Computer Science, and Mechanical Engineering. He is also an Adjunct Professor of Applied Mathematics and Applied Physics at Columbia University, where he formerly held the Fu Foundation Chair. He works at the interface between parallel computing and PDEs and statistics, with a focus on scalable algorithms that exploit data sparsity. Before joining KAUST, Keyes led multi-institutional scalable solver software projects in the SciDAC and ASCI programs of the US Department of Energy (DoE), ran university collaboration programs at US DoE and NASA institutes, and taught at Columbia, Old Dominion, and Yale Universities. He is a Fellow of SIAM, the AMS, and the AAAS. He has been awarded the Gordon Bell Prize from the ACM, the Sidney Fernbach Award from the IEEE Computer Society, and the SIAM Prize for Distinguished Service to the Profession. He earned a B.S.E. in Aerospace and Mechanical Sciences from Princeton in 1978 and a Ph.D. in Applied Mathematics from Harvard in 1984.

Nonlinear Preconditioning for Implicit Solution of Discretized PDEs

David Keyes (King Abdullah University of Science and Technology)