# Jacob Schroder - Applied Search Candidate Colloquium

### Event Description:

**Speaker: Dr. Jacob Schroder
Title: Parallel Multigrid Solvers in Space and Time for Future Architectures**

**Abstract:**

Multigrid solvers are popular and effective approaches for solving large sparse systems of equations, which often come from discretized partial differential equations. While multigrid is effective over a broad class of problems, the massive parallelism of exascale computing presents pressing challenges, three of which will be addressed. The first challenge is posed by the serial time integration bottleneck, which is caused by the fact that future growth in high-performance computing is coming from more cores, not faster clock speeds. Previously, faster clock speeds decreased the runtime per time step, thus allowing for either faster simulations, or for more time steps without increasing the overall runtime. However, clock speeds are now stagnate and the bottleneck has already begun to affect some fields, such as power grid simulations. The presented approach utilizes multigrid to simultaneously compute multiple time steps in parallel and has the ability to dramatically decrease overall time to solution. Several application areas for this multigrid in time approach will be considered, e.g., parabolic problems, fluid dynamics, optimization, and neural network training.

Two further challenges concern spatial multigrid, and in particular, algebraic multigrid (AMG). One regards the need to reduce communication for parallel AMG. The efficiency of AMG relies on its multilevel structure of successively coarser representations of the problem; however, these coarse level matrices suffer from fill-in and associated communication. Moreover, as the problem size increases, more levels are constructed, leading to increased fill-in and possible scalability issues at exascale. The proposed approach reduces this fill-in and the associated communication by eliminating unnecessary entries in coarse level matrices, while preserving spectral equivalence with the original coarse level matrix. The other challenge concerns the generality of AMG and the need to extend AMG beyond traditional restrictions to certain subclasses of symmetric positive definite (SPD) matrices. A generalized approach to algebraic coarsening and interpolation is presented that allows AMG to be applied to areas such as high-order discontinuous Galerkin discretizations, neutron transport, and the indefinite Helmholtz problem.

### Event Contact

**Contact Name: **Deborah Sulsky

**Contact Phone: **505-277-4613

**Contact Email: **sulsky@math.unm.edu