MLAPM - a C code for cosmological simulations

We present a computer code written in C that is designed to simulate structure formation from collisionless matter. The code is purely grid-based and uses a recursively refined Cartesian grid to solve Poisson's equation for the potential, rather than obtaining the potential from a Green's function. Refinements can have arbitrary shapes and in practice closely follow the complex morphology of the density field that evolves. The timestep shortens by a factor two with each successive refinement. It is argued that an appropriate choice of softening length is of great importance and that the softening should be at all points an appropriate multiple of the local inter-particle separation. Unlike tree and P3M codes, multigrid codes automatically satisfy this requirement. We show that at early times and low densities in cosmological simulations, the softening needs to be significantly smaller relative to the inter-particle separation than in virialized regions. Tests of the ability of the code's Poisson solver to recover the gravitational fields of both virialized halos and Zel'dovich waves are presented, as are tests of the code's ability to reproduce analytic solutions for plane-wave evolution. The times required to conduct a LCDM cosmological simulation for various configurations are compared with the times required to complete the same simulation with the ART, AP3M and GADGET codes. The power spectra, halo mass functions and halo-halo correlation functions of simulations conducted with different codes are compared.Comment: 20 pages, 20 figures, MNRAS in press, the code can be downloaded at http://www-thphys.physics.ox.ac.uk/users/MLAPM

Afivo: a framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods

Afivo is a framework for simulations with adaptive mesh refinement (AMR) on quadtree (2D) and octree (3D) grids. The framework comes with a geometric multigrid solver, shared-memory (OpenMP) parallelism and it supports output in Silo and VTK file formats. Afivo can be used to efficiently simulate AMR problems with up to about $10^{8}$ unknowns on desktops, workstations or single compute nodes. For larger problems, existing distributed-memory frameworks are better suited. The framework has no built-in functionality for specific physics applications, so users have to implement their own numerical methods. The included multigrid solver can be used to efficiently solve elliptic partial differential equations such as Poisson's equation. Afivo's design was kept simple, which in combination with the shared-memory parallelism facilitates modification and experimentation with AMR algorithms. The framework was already used to perform 3D simulations of streamer discharges, which required tens of millions of cells

An odyssey into local refinement and multilevel preconditioning III: Implementation and numerical experiments

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems, which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on Bramble, Pasciak, and Xu (BPX)-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis (HB) method, and their use in the local mesh refinement setting. In parts I and II of this trilogy, it was shown that both BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes of locally refined two- and three-dimensional meshes based on fairly standard (and easily implementable) red and red-green mesh refinement algorithms. In this third part of the trilogy, we describe in detail the implementation of these types of algorithms, including detailed discussions of the data structures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard data types, available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. We have successfully used these data structure ideas for both MATLAB and C implementations using the FEtk, an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement

Reversible Recursive Instance-level Object Segmentation

In this work, we propose a novel Reversible Recursive Instance-level Object Segmentation (R2-IOS) framework to address the challenging instance-level object segmentation task. R2-IOS consists of a reversible proposal refinement sub-network that predicts bounding box offsets for refining the object proposal locations, and an instance-level segmentation sub-network that generates the foreground mask of the dominant object instance in each proposal. By being recursive, R2-IOS iteratively optimizes the two sub-networks during joint training, in which the refined object proposals and improved segmentation predictions are alternately fed into each other to progressively increase the network capabilities. By being reversible, the proposal refinement sub-network adaptively determines an optimal number of refinement iterations required for each proposal during both training and testing. Furthermore, to handle multiple overlapped instances within a proposal, an instance-aware denoising autoencoder is introduced into the segmentation sub-network to distinguish the dominant object from other distracting instances. Extensive experiments on the challenging PASCAL VOC 2012 benchmark well demonstrate the superiority of R2-IOS over other state-of-the-art methods. In particular, the $\text{AP}^r$ over $20$ classes at $0.5$ IoU achieves $66.7\%$, which significantly outperforms the results of $58.7\%$ by PFN~\cite{PFN} and $46.3\%$ by~\cite{liu2015multi}.Comment: 9 page

Refined invariants of finite-dimensional Jacobi algebras

We define and study refined Gopakumar-Vafa invariants of contractible curves in complex algebraic 3-folds, alongside the cohomological Donaldson--Thomas theory of finite-dimensional Jacobi algebras. These Gopakumar-Vafa invariants can be constructed one of two ways: as cohomological BPS invariants of contraction algebras controlling the deformation theory of these curves, as defined by Donovan and Wemyss, or by feeding the moduli spaces that Katz used to define genus zero Gopakumar-Vafa invariants into the machinery developed by Joyce et al. The conjecture that the two definitions give isomorphic results is a special case of a kind of categorified version of the strong rationality conjecture due to Pandharipande and Thomas, that we discuss and propose a means of proving. We prove the positivity of the cohomological/refined BPS invariants of all finite-dimensional Jacobi algebras. This result supports this strengthening of the strong rationality conjecture, as well as the conjecture of Brown and Wemyss stating that all finite-dimensional Jacobi algebras for appropriate symmetric quivers are isomorphic to contraction algebras.Comment: 29 page

Effect of different grain structures on centerline macrosegregation during direct-chill casting

This is the post-print version of the final paper published in Acta Materialia. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2008 Elsevier B.V.Duplex grain structure consisting of coarse-cell and fine-cell dendritic grains is frequently found in the central portion of direct-chill cast billets and ingots. Coarse-cell grains are usually considered as free-floating crystals settled to the bottom of the billet sump. These grains are assumed to be solute-lean and contribute to the negative centerline segregation. In this paper the contribution of coarse-cell and fine-cell grains to macrosegregation is for the first time studied experimentally by direct measurements of their composition. It is shown that the coarse-cell, floating grains are depleted of solute and the areas of their accumulation contribute to the negative macrosegregation. The areas of fine-cell grains can be either enriched in solute or be close to the nominal composition. It is argued that their composition results from the interplay between thermo-solutal and shrinkage-induced flows. The roles of casting speed and grain refining are also under scrutiny in this paper.Netherlands Institute for Metals Researc