This Intel Parallel Computing Center (IPCC) focuses on the optimization of core operations for scalable solvers. Barry Smith has implemented a PETSc communication API that automatically segregates on-node and off-node traffic in order to avoid possible MPI overhead. Andy Terrel, Karl Rupp, and Matt Knepley have developed algoritms for vectorization of low-order FEM on low memory/core architectures such as Intel KNL and Nvidia GPUs. Jed Brown, Barry Smith, and Dave May have demonstrated exceptional performance for \(Q_2\) and higher order elements. Mark Adams, Toby Isaac, and Matt Knepley are curerntly optimizing Full Approximation Scheme (FAS) multigrid, applied to PIC methods as part of the SITAR effort.

The RELACS Team is focused on developing nonlinear solvers for complex, multiphysics simulations, using both Partial Differential Equations (PDE) and Boundary Integral Equations (BIE) in collaboration with Jay Bardhan. We are also developing solvers for Linear Complementarity Problems (LCP) and engineering design problems. We focus on using true multilevel formulations in the Full Approximation Scheme (FAS) multigrid formulation. Theoretically, we are interested in characterizing the convergence of Newton's Method over multiple meshes, as well as composed nonlinear iterations arising from Nonlinear Preconditioning (NPC).

The Solver Interfaces for Tree Adaptive Refinement (SITAR) project aims to develop scalable and robust nonlinear solvers for multiphysics problems discretized on large, parallel structured AMR meshes. Toby Isaac, a core developer of p4est, has integrated AMR functionality into PETSc. Toby and Matt Knepley have implemented Full Approximation Scheme (FAS) multigrid for nonlinear problems for the AMR setting. Mark Adams and Dave May are adding a Particle-in-Cell (PIC) infrastructure to SITAR, targeting MHD plasma and Earth mantle dynamics simulations.

With the goal of reducing the dimension of difficult PDE problems, we are developing a new representation of multivariate continuous functions in terms of compositions of univariate functions, based upon the classical Kolmogorov representation. We are currently pursuing stronger regularity guarantees on the univariate functions, and also efficient algorithms for manipulating the functions inside iterative methods.

With Prof. Jaydeep Bardhan, we have developed a nonlinear continuum molecular electrostatics model capable of quantiative accuracy for a number of phenomena that are flatly wrong in current models, including charge-hydration asymmetry and calculation of entropies. To illustrate the importance of treating both steric asymmetry and the static molecular potential separately, we have recently computed the charging free energies (and profiles) of individual atoms in small molecules. The static potential derived from MD simulations in TIP3P water is positive, meaning that for small positive charges, the charging free energy is actually positive (unfavorable). Traditional dielectric models are completely unable to reproduce such positive charging free energies, regardless of atomic charge or radius. However, our model reproduces these energies with quantitative accuracy, and we note that these represent unfavorable electrostatic contributions to solvation by hydrophobic groups. Traditional implicit-solvent models predicate that hydrophobic groups contribute nothing; for instance, nonpolar solvation models are often parameterized under the assumption that alkanes have electrostatic solvation free energies equal to zero. We have found this to be not the case, and because our model has few parameters that do not involve the atom radii, we have been able to parameterize a complete and consistent implicit-solvent model, that is, parameterizing both the electrostatic and nonpolar terms simultaneously. We found that this model is remarkably accurate, comparable to explicit solvent simulations for solvation free energies and water-octanol transfer free energies.

In collaboration with Richard Katz, we are working to scale solvers for Magma Dynamics simulations to large, parallel machines. The current formulation is an index 1 Differential-Algebraic Equation (DAE) in which the algebraic constraints are an elliptic PDE for the mass and momentum conservation, coupled to the porosity through permability and viscosities. We discretize this using continuous Galerkin FEM and our Full Approximation Scheme (FAS) solver has proven effective for the elliptic constraints. The advection of porosity is a purely hyperbolic equation coupled to the fluid velocity, which we discretize using FVM. The fully coupled nonlinear problem is solved at each explicit timestep.