Browsing by Author "Knepley, Matthew G."
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Item Efficient mesh management in Firedrake using PETSc-DMPlex(Society for Industrial and Applied Mathematics, 2016) Lange, Michael; Mitchell, Lawrence; Knepley, Matthew G.; Gorman, Gerard J.The use of composable abstractions allows the application of new and established algorithms to a wide range of problems, while automatically inheriting the benefits of well-known performance optimizations. This work highlights the composition of the PETSc DMPlex domain topology abstraction with the Firedrake automated finite element system to create a PDE solving environment that combines expressiveness, flexibility, and high performance. We describe how Firedrake utilizes DMPlex to provide the indirection maps required for finite element assembly, while supporting various mesh input formats and runtime domain decomposition. In particular, we describe how DMPlex and its accompanying data structures allow the generic creation of user-defined discretizations, while utilizing data layout optimizations that improve cache coherency and ensure overlapped communication during assembly computation.Item Hybridizable Discontinuous Galerkin Methods for Flow and Transport: Applications, Solvers, and High Performance Computing(2019-04-16) Fabien, Maurice S; Riviere, Beatrice M.; Knepley, Matthew G.This thesis proposal explores e cient computational methods for the approximation of solutions to partial di erential equations that model ow and transport phenomena in porous media. These problems can be challenging to solve as the governing equations are coupled, nonlinear, and material properties are often highly varying and discontinuous. The high-order implicit hybridizable discontinuous method (HDG) is utilized for the discretization, which signi cantly reduces the computational cost. To our knowledge, HDG methods have not been previously applied to this class of complex problems in porous media. The HDG method is high-order accurate, locally mass-conservative, allows us to use unstructured complicated meshes, and enables the use of static condensation. We demonstrate that the HDG method is able to e ciently generate high- delity simulations of ow and transport phenomena in porous media. Several challenging benchmarks are used to verify and validate the method in heterogeneous porous media. High-order methods give rise to less sparse discretization matrices, which is problematic for linear solvers. To address the issue of less sparse discretization matrices (compared to low-order methods), we develop and deployed a novel nested multigrid method. It is based on a combination of p-multgrid, h-multigrid and algebraic multigrid. The method is demonstrated to be algorithmically e cient, achieving convergences rates of at most 0:2. We also show how to implement the multigrid technique in many-core parallel architectures. Parallel computing is a critical step in the simulation process, as it allows us to consider larger problems, and potentially generate simulations faster. Traditional performance measures like FLOPs or run-time are not entirely appropriate for nite element problems, as they ignore solution accuracy. A new accuracy-inclusive performance measure has been investigated as a part of my research. This performance measure, called the Time-Accuracy-Size spectrum (TAS), allows us to have a more complete assessment of how e cient our algorithms are. Utilizing TAS also enables a systematic way of determining which discretization is best suited for a given application.Item Landau Collision Integral Solver with Adaptive Mesh Refinement on Emerging Architectures(Society for Industrial and Applied Mathematics, 2017) Adams, Mark F.; Hirvijoki, Eero; Knepley, Matthew G.; Brown, Jed; Isaac, Tobin; Mills, RichardThe Landau collision integral is an accurate model for the small-angle dominated Coulomb collisions in fusion plasmas. We investigate a high order accurate, fully conservative, finite element discretization of the nonlinear multispecies Landau integral with adaptive mesh refinement using the PETSc library (ŭlwww.mcs.anl.gov/petsc). We develop algorithms and techniques to efficiently utilize emerging architectures with an approach that minimizes memory usage and movement and is suitable for vector processing. The Landau collision integral is vectorized with Intel AVX-512 intrinsics and the solver sustains as much as 22% of the theoretical peak flop rate of the Second Generation Intel Xeon Phi (``Knights Landing'') processor.Item Numerical Analysis of Nonlinear Boundary Integral Equations Arising in Molecular Biology(2019-04-18) Klotz, Thomas S; Rivière, Béatrice M.; Knepley, Matthew G.The molecular electrostatics problem, which asks for the potential generated by a charged solute suspended in a dielectric solvent, is of great importance in computational biology. Poisson models, which treat the solvent as a dielectric continuum, have inherent inaccuracies which can ruin energy predictions. These inaccuracies are primarily due to the inability of continuum models to capture the structure of solvent molecules in close proximity to the solute. A common approach to overcome these inaccuracies is to adjust the dielectric boundary by changing atomic radii. This adjustment procedure can accurately reproduce the expected solvation free energy, but fails to predict thermodynamic behavior. The Solvation Layer Interface Method (SLIC) replaces the standard dielectric boundary condition in Poisson models with a nonlinear boundary condition which accounts for the small-scale arrangement of solvent molecules close to the dielectric interface. Remarkably, SLIC retains the accuracy of Poisson models and furthermore predicts solvation entropies and heat capacities, while removing the need to adjust atomic radii. In this thesis, we perform foundational numerical analysis for the SLIC model. The first major result is a proof that a solution exists for the nonlinear boundary integral equation arising in the SLIC model. We are able to do this by proving existence for an auxiliary equation whose solutions correspond to the SLIC model's equation. Next, we prove that solutions to the SLIC model are unique for spherical geometries, which are common in biological solutes. Finally, we have experimented with nonlinear solvers for the nonlinear BIE, such as Anderson Acceleration, as well as two discretization techniques, in order to provide scalable numerical methods which can be applied to a variety of problems in drug design and delivery.Item Segmental refinement: A multigrid technique for data locality(Society for Industrial and Applied Mathematics, 2016) Adams, Mark F.; Brown, Jed; Knepley, Matthew G.; Samtaney, RaviWe investigate a domain decomposed multigrid technique, termed segmental refinement, for solving general nonlinear elliptic boundary value problems. We extend the method first proposed in 1994 by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid is eliminated on fine grids, with modest amounts of extra work and storage, while maintaining the asymptotic exactness of full multigrid. We observe an accuracy dependence on the segmental refinement subdomain size, which was not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores on a Cray XC30.