Jacob Schroder is a computational mathematician at the Center for Applied Scientific Computing. The core direction of his research is numerical analysis and scientific computing. His specific focus is on high-performance computing, iterative solvers for large sparse (non)linear systems, their associated preconditioning, and numerical PDEs. He approaches his research both from a software perspective centered on providing these methods to the broader community and also from a theoretical perspective centered on the development of new methods. He is a member of the Scalable Linear Solvers (hypre) project and the Parallel Time Integration with Multigrid (XBraid) project.

Jacob earned his Ph.D. in computer science from the University of Illinois at Urbana-Champaign under the direction of Prof. Luke Olson. His dissertation resulted in new methods for smoothed aggregation-based algebraic multigrid (AMG), which proved effective for a variety of problems, e.g., anisotropic diffusion, Helmholtz, elasticity and Euler flow. Next, he joined University of Colorado at Boulder for one year as a postdoc under Profs. Thomas Manteuffel and Stephen McCormick. Jacob joined LLNL in September 2011.

His current work focuses both on classical spatial multigrid solvers and on parallel-in-time methods using a multigrid reduction strategy. Regarding classical spatial multigrid, Jacob focuses on improving the parallel efficiency of algebraic multigrid methods and also on basic multigrid research such as new adaptive multigrid methods. Jacob's parallel-in-time work applies a multigrid reduction scheme to the time dimension, thus allowing for the parallelization of serial time-stepping methods. The result is a method that circumvents the sequential time integration bottleneck and takes advantage of the coming massive increase in concurrency at exascale. This approach can even be applied to non-PDE evolution problems, such as powergrid simulations and neural network training.

**Areas: **Numerical analysis, computational science, high performance computing

**Keywords: **Iterative methods, preconditioning, multigrid, neural networks, numerical PDEs, parallel-in-time

- J. B. Schroder, Parallelizing Over Artificial Neural Network Training Runs with Multigrid. arXiv preprint arXiv:1708.02276 (2017). LLNL-JRNL-736173.
- J. B. Schroder, M. Lecouvez, R. D. Falgout, C. S. Woodward, P. Top, Parallel-in-Time Solution of Power Systems with Scheduled Events. 2018 Power and Energy Society General Meeting (PESGM), IEEE, (submitted), (2018). LLNL- CONF-740658.
- H. De Sterck, R. D. Falgout, A. J. M Howse, S. P. MacLachlan, and J. B. Schroder, Parallel-in-Time Multigrid with Adaptive Spatial Coarsening for the Linear Advection and Inviscid Burgers Equations. SIAM J. Sci. Comput., (submitted), (2017). Supplementary Materials. LLNL-JRNL-737050.
- A. Hessenthaler, D. Nordsletten, O. Roehrle, J. B. Schroder, and R. D. Falgout, Convergence of the multigrid-reduction-in-time algorithm for the linear elasticity equations. Numer. Linear Algebra Appl., (accepted), (2017). LLNL-JRNL-731168.
- S. Gunther, N. R. Gauger, and J. B. Schroder, A Non-Intrusive Parallel-in-Time Adjoint Solver with the XBraid Library. Computing and Visualization in Science, Springer, (submitted), (2017). LLNL-JRNL-730159.
- A. J. M. Howse, in collaboration with H. De Sterck, R. D. Falgout, S. P. Machlachlan, and J. B. Schroder,
*Multigrid Reduction in Time with Adaptive Spatial Coarsening for the Linear Advection Equation.**Student Paper Winner.*Eighteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March, 2017. LLNL-PROC-716758 - H. Gahvari, V. A. Dobrev, R. D. Falgout, Tz. V. Kolev, J. B. Schroder, M. Schulz and U. M. Yang,
*A Performance Model for Allocating the Parallelism in a Multigrid-in-Time Solver*. The 7th International Workshop on Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems (PMBS16), Supercomputing 16. LLNL-CONF-701995. - T. A. Manteuffel, L. N. Olson, J. B. Schroder and B. S. Southworth,
*A Root-Node Based Algebraic Multigrid Method.*SIAM J. Sci. Comput., pp. 723-756. 39 (2017). LLNL-JRNL-695797. - A. Bienz, R.D. Falgout, W. Gropp, L.N. Olson and J.B. Schroder,
*Reducing Parallel Communication in Algebraic Multigrid Through Sparsification.*SIAM J. Sci. Comput., pp. 332-357. 38 (2016). LLNL-JRNL-673388. Supplementary material. - R. D. Falgout, T. A. Manteuffel, B. O’Neill and J. B. Schroder,
*Multigrid Reduction in Time for Nonlinear Parabolic Problem: A Case Study.*SIAM J. Sci. Comput., pp. 298-322. 39 (2017). LLNL-JRNL-692258. - V. Dobrev, Tz. Kolev, N. A. Petersson and J. B. Schroder,
*Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT).*SIAM J. Sci. Comput., pp. 501-527. 39 (2017). LLNL-JRNL-692418. - R.D. Falgout, T.A. Manteuffel, J.B. Schroder and B. Southworth,
*Parallel-in-Time for Moving Meshes.*Technical Report. LLNL-TR-681918. - R.D. Falgout, S. Friedhoff, Tz.V. Kolev, S.P. MacLachlan, J.B. Schroder and S. Vandewalle, Multigrid Methods with Space-Time Concurrency. Computing and Visualization in Science, Springer, (2017). LLNL-JRNL-678572.
- C. Ketelsen, T. Manteuffel and J. B. Schroder,
*Least-Squares Finite-Element Discretization of the Neutron Transport Equation in Spherical Geometry.*SIAM J. Sci. Comput., pp. 71-89. 37 (2015). LLNL-JRNL-656198. - R. D. Falgout, A. Katz, Tz. V. Kolev, J. B. Schroder, A. Wissink and U. M. Yang,
*Parallel Time Integration with Multigrid Reduction for a Compressible Fluid Dynamics Application.*Technical Report. LLNL-JRNL-663416. - R. D. Falgout, S. Friedhoff, Tz. V. Kolev, S. P. MacLachlan and J. B. Schroder,
*Parallel Time Integration with Multigrid.*SIAM J. Sci. Comput., pp. 635-661. 36 (2014). LLNL-JRNL-645325. - R. D. Falgout and J. B. Schroder,
*Non-Galerkin Coarse Grids for Algebraic Multigrid.*SIAM Journal on Scientific Computing, pp. 309-334. 36 (2014). LLNL-JRNL-641635. - S. Friedhoff, R. Falgout, T. Kolev, S. MacLachlan and J. Schroder,
*A Multigrid-In-Time Algorithm for Solving Evolution Equations in Parallel. Student paper winner.*Sixteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March, 2013. - J. B. Schroder,
*Smoothed Aggregation Solvers for Anisotropic Diffusion.*Numer. Linear Algebra Appl. pp. 296–312. 19 (2012). - L. N. Olson, J. B. Schroder and R. S. Tuminaro,
*A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization*. SIAM J. Sci. Comput., 33:966-991, 2011. - L. N. Olson and J. B. Schroder,
*Smoothed Aggregation Multigrid Solvers for High-Order Discontinuous Galerkin Methods for Elliptic Problems*. J. Comput. Phys., 230:6959-6976, 2011. - L. N. Olson and J. B. Schroder,
*Components of a More Robust Multilevel Solver for Emerging Architectures and Complex Applications.*In SciDAC 2011 (2011). - J. B. Schroder,
*Generalizing Smoothed Aggregation-Based Algebraic Multigrid*. Ph.D. Thesis. University of Illinois at Urbana-Champaign, Department of Computer Science, 2010. - L. N. Olson and J. B. Schroder,
*Smoothed Aggregation for Helmholtz Problems*. Numer. Linear Algebra Appl., 17:361-386, 2010. - L. N. Olson, J. B. Schroder and R. S. Tuminaro,
*A New Perspective on Strength Measures in Algebraic Multigrid*. Numer. Linear Algebra Appl., 17:713-733, 2010. - J. B. Schroder, R. S. Tuminaro and L. N. Olson,
*Generalized Strength-of-Connection in Algebraic Multigrid*. CSRI Summer Proceedings 2007. pp. 12–26. (2007). - V. E. Howle, J. B. Schroder and R. S. Tuminaro,
*The Effect of Boundary Conditions within Pressure- Convection Diffusion Preconditioners*. Sandia National Laboratory Technical Report #2006-4466. July 2006.

PyAMG is a highly usable open source Python/C++ implementation of both classical algebraic multigrid and smoothed aggregation-based algebraic multigrid solvers. Thousands of downloads from over a hundred countries.

Hypre is a benchmark library of high performance preconditioners that features parallel multigrid methods for both structured and unstructured grid problems. Thousands of downloads from over a hundred countries.

XBraid is a C/MPI implementation of the multigrid reduction in time (MGRIT) approach. It is a non-intrusive, scalable and parallel library for multigrid in time. 200+ downloads in the last year.

*The Lake City Algebraic Multigrid Summit*. University of Colorado at Boulder. 10/2010, 9/2011, 10/2012, 9/2013, 10/2014, 9/2015, 10/2016, 9/2017.*XBraid Tutorial*, 18th Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March, 28th, 2017.*Parallel Time Integration Minisymposium, Co-Organizer.*18th Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March, 2017.*Space-Time Multigrid Methods Minisymposium, Co-Organizer.*SIAM Conference on Computational Science and Engineering, Atlanta, Georgia. March, 2017.*Space-Time Adaptive Meshing with the XBraid Library.*Twenty-Fourth International Conference on Domain Decomposition Methods, Svalbard, Norway. February, 2017.*Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT).*Fifth Workshop on Parallel-in-Time Integration, Banff International Research Station, Canada. November, 2016.*Multigrid Reduction in Time: Recent Theoretical Results.*Fourteenth Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. March 22, 2016.*Multigrid Reduction in Time (MGRIT): A Flexible and Scalable Approach to Parallel-in-Time.*Monash Workshop on Numerical PDEs, Melbourne, Australia. February 17, 2016.*Multigrid Reduction in Time (MGRIT): An Overview.*Fourth Workshop on Parallel-in-Time Integration, Dresden, Germany. May 27, 2015.*A General Purpose Parallel-in-Time Approach.*SIAM Conference on Computational Science and Engineering (CS&E), Salt Lake City, Utah. March 18, 2015.*Multigrid Reduction in Time: A Flexible and Non-Intrusive Method.*Third Workshop on Parallel-in-Time Integration, Juelich, Germany. May 27, 2014.*Theoretical Advances Regarding Non-Galerkin Coarse Grid Operators for AMG.*Thirteenth Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. April 9, 2014.*Non-Galerkin Coarse-Grid Operators for Parallel Algebraic Multigrid.*Sixteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March 19, 2013.*Energy-Minimization Interpolation for Adaptive Algebraic Multigrid.*SIAM Conference on Applied Linear Algebra, Valencia, Spain. June 18–22, 2012.*Non-Galerkin Coarse-Grid Operators for Parallel Algebraic Multigrid.*Twelfth Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. March 27, 2012.*PyAMG Tutorial.*Twelfth Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. March 28, 2012.*A General Energy-Minimization Strategy for Interpolation in Algebraic Multigrid.*Seventh International Congress on Industrial and Applied Mathematics, Vancouver, Canada. July 21, 2011.*PyAMG Tutorial*. Fifteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March 31, 2011.*Smoothed Aggregation Solvers for Anisotropic Diffusion.*Fifteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March 28, 2011.*Generalizing Smoothed Aggregation-Based Algebraic Multigrid.*Tech-X Corporation, Boulder, Colorado. July 29, 2010.*A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization.*Eleventh Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. April 5, 2010.*Smoothed Aggregation Multigrid for Helmholtz Problems*. Fourteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, Colorado. March 23, 2009.*A General Strength-of-Connection Concept in AMG.*Tenth Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado. April 7, 2008.*Stability and Load Balancing in a NASA Global Circulation Model.*Southeast ACM Conference, Gatlinburg, Tennessee. November 22, 2003.