ColDICE: a parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation

Resolving numerically Vlasov-Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-dimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincar\'e invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli (1993) generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a "warm" dark matter simulation in six-dimensional phase-space that we use to check the parallel scaling of the code.Comment: Code and illustration movies available at: http://www.vlasix.org/index.php?n=Main.ColDICE - Article submitted to Journal of Computational Physic

Is Your Model Susceptible to Floating-Point Errors?

This paper provides a framework that highlights the features of computer models that make them especially vulnerable to floating-point errors, and suggests ways in which the impact of such errors can be mitigated. We focus on small floating-point errors because these are most likely to occur, whilst still potentially having a major influence on the outcome of the model. The significance of small floating-point errors in computer models can often be reduced by applying a range of different techniques to different parts of the code. Which technique is most appropriate depends on the specifics of the particular numerical situation under investigation. We illustrate the framework by applying it to six example agent-based models in the literature.Floating Point Arithmetic, Floating Point Errors, Agent Based Modelling, Computer Modelling, Replication

Large-Scale Simulations of Complex Turbulent Flows: Modulation of Turbulent Boundary Layer Separation and Optimization of Discontinuous Galerkin Methods for Next-Generation HPC Platforms

The separation of spatially evolving turbulent boundary layer flow near regions of adverse pressure gradients has been the subject of numerous studies in the context of flow control. Although many studies have demonstrated the efficacy of passive flow control devices, such as vortex generators (VGs), in reducing the size of the separated region, the interactions between the salient flow structures produced by the VG and those of the separated flow are not fully understood. Here, wall-resolved large-eddy simulation of a model problem of flow over a backward-facing ramp is studied with a submerged, wall-mounted cube being used as a canonical VG. In particular, the turbulent transport that results in the modulation of the separated flow over the ramp is investigated by varying the size, location of the VG, and the spanwise spacing between multiple VGs, which in turn are expected to modify the interactions between the VG-induced flow structures and those of the separated region. The horseshoe vortices produced by the cube entrain the freestream turbulent flow towards the plane of symmetry. These localized regions of high vorticity correspond to turbulent kinetic energy production regions, which effectively transfer energy from the freestream to the near-wall regions. Numerical simulations indicate that: (i) the gradients and the fluctuations, scale with the size of the cube and thus lead to more effective modulation for large cubes, (ii) for a given cube height the different upstream cube positions affect the behavior of the horseshoe vortex---when placed too close to the leading edge, the horseshoe vortex is not sufficiently strong to affect the large-scale structures of the separated region, and when placed too far, the dispersed core of the streamwise vortex is unable to modulate the flow over the ramp, (iii) if the spanwise spacing between neighboring VGs is too small, the counter-rotating vortices are not sufficiently strong to affect the large-scale structures of the separated region, and if the spacing is too large, the flow modulation is similar to that of an isolated VG. Turbulent boundary layer flows are inherently multiscale, and numerical simulations of such systems often require high spatial and temporal resolution to capture the unsteady flow dynamics accurately. While the innovations in computer hardware and distributed computing have enabled advances in the modeling of such large-scale systems, computations of many practical problems of interest are infeasible, even on the largest supercomputers. The need for high accuracy and the evolving heterogeneous architecture of the next-generation high-performance computing centers has impelled interest in the development of high-order methods. While the new class of recovery-assisted discontinuous Galerkin (RADG) methods can provide arbitrary high-orders of accuracy, the large number of degrees of freedom increases costs associated with the arithmetic operations performed and the amount of data transferred on-node. The purpose of the second part of this thesis is to explore optimization strategies to improve the parallel efficiency of RADG. A cache data-tiling strategy is investigated for polynomial orders 1 through 6, which enhances the arithmetic intensity of RADG to make better utilization of on-node floating-point capability. In addition, a power-aware compute framework is suggested by analyzing the power-performance trade-offs when changing from double to single-precision floating-point types---energy savings of 5 W per node are observed---which suggests that a transprecision framework will likely offer better power-performance balance on modern HPC platforms.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163206/1/suyashtn_1.pd

The AutoProof Verifier: Usability by Non-Experts and on Standard Code

Formal verification tools are often developed by experts for experts; as a result, their usability by programmers with little formal methods experience may be severely limited. In this paper, we discuss this general phenomenon with reference to AutoProof: a tool that can verify the full functional correctness of object-oriented software. In particular, we present our experiences of using AutoProof in two contrasting contexts representative of non-expert usage. First, we discuss its usability by students in a graduate course on software verification, who were tasked with verifying implementations of various sorting algorithms. Second, we evaluate its usability in verifying code developed for programming assignments of an undergraduate course. The first scenario represents usability by serious non-experts; the second represents usability on "standard code", developed without full functional verification in mind. We report our experiences and lessons learnt, from which we derive some general suggestions for furthering the development of verification tools with respect to improving their usability.Comment: In Proceedings F-IDE 2015, arXiv:1508.0338

GAMER: a GPU-Accelerated Adaptive Mesh Refinement Code for Astrophysics

We present the newly developed code, GAMER (GPU-accelerated Adaptive MEsh Refinement code), which has adopted a novel approach to improve the performance of adaptive mesh refinement (AMR) astrophysical simulations by a large factor with the use of the graphic processing unit (GPU). The AMR implementation is based on a hierarchy of grid patches with an oct-tree data structure. We adopt a three-dimensional relaxing TVD scheme for the hydrodynamic solver, and a multi-level relaxation scheme for the Poisson solver. Both solvers have been implemented in GPU, by which hundreds of patches can be advanced in parallel. The computational overhead associated with the data transfer between CPU and GPU is carefully reduced by utilizing the capability of asynchronous memory copies in GPU, and the computing time of the ghost-zone values for each patch is made to diminish by overlapping it with the GPU computations. We demonstrate the accuracy of the code by performing several standard test problems in astrophysics. GAMER is a parallel code that can be run in a multi-GPU cluster system. We measure the performance of the code by performing purely-baryonic cosmological simulations in different hardware implementations, in which detailed timing analyses provide comparison between the computations with and without GPU(s) acceleration. Maximum speed-up factors of 12.19 and 10.47 are demonstrated using 1 GPU with 4096^3 effective resolution and 16 GPUs with 8192^3 effective resolution, respectively.Comment: 60 pages, 22 figures, 3 tables. More accuracy tests are included. Accepted for publication in ApJ