Andy R. Terrel

Finite Element Integration on GPUs

We present a novel finite element integration method for low-order elements on GPUs. We achieve more than 100GF for element integration on first order discretizations of both the Laplacian and Elasticity operators on an NVIDIA GTX285, which has a nominal single precision peak flop rate of 1 TF/s and bandwidth of 159 GB/s, corresponding to a bandwidth limited peak of 40 GF/s.

Topological Optimization of the Evaluation of Finite Element Matrices

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization.

ForestClaw: Hybrid forest-of-octrees AMR for hyperbolic conservation laws

We present a new hybrid paradigm for parallel adaptive mesh refinement (AMR) that combines the scalability and lightweight architecture of tree-based AMR with the computational efficiency of patch-based solvers for hyperbolic conservation laws. The key idea is to interpret each leaf of the AMR hierarchy as one uniform compute patch in

From Equations to Code: Automated Scientific Computing

\sR^d

Common and unusual finite elements

with

Symbolic Statistics with SymPy

m^d

Finite elements for incompressible fluids

degrees of freedom, where

Scientific Software Communities

m

Discrete optimization of finite element matrix evaluation

is customarily between 8 and 32. Thus, computation on each patch can be optimized for speed, while we inherit the flexibility of adaptive meshes. In our work we choose to integrate with the p4est AMR library since it allows us to compose the mesh from multiple mapped octrees and enables the cubed sphere and other nontrivial multiblock geometries. We describe aspects of the parallel implementation and close with scalings for both MPI-only and OpenMP/MPI hybrid runs, where the largest MPI run executes on 16,384 CPU cores.

SymPy: symbolic computing in Python

Using domain-specific languages, scientific codes can let users work directly with equations and benefit from optimizations not available with general compilers.