1,176 research outputs found
A Case Study in Mechanically Deriving Dense Linear Algebra Code
Abstract Design by Transformation (DxT) is a top-down approach to mechanically derive high-performance algorithms for dense linear algebra. We use DxT to derive the implementation of a representative matrix operation, two-sided Trmm. We start with a knowledge base of transformations that were encoded for a simpler set of operations, the level-3 BLAS, and add only a few transformations to accommodate the more complex two-sided Trmm. These additions explode the search space of our prototype system, DxTer, requiring the novel techniques defined in this paper to eliminate large segments of the search space that contain suboptimal algorithms. Performance results for the mechanically optimized implementations on 8,192 cores of a BlueGene/P architecture are given
General phase spaces: from discrete variables to rotor and continuum limits
We provide a basic introduction to discrete-variable, rotor, and
continuous-variable quantum phase spaces, explaining how the latter two can be
understood as limiting cases of the first. We extend the limit-taking
procedures used to travel between phase spaces to a general class of
Hamiltonians (including many local stabilizer codes) and provide six examples:
the Harper equation, the Baxter parafermionic spin chain, the Rabi model, the
Kitaev toric code, the Haah cubic code (which we generalize to qudits), and the
Kitaev honeycomb model. We obtain continuous-variable generalizations of all
models, some of which are novel. The Baxter model is mapped to a chain of
coupled oscillators and the Rabi model to the optomechanical radiation pressure
Hamiltonian. The procedures also yield rotor versions of all models, five of
which are novel many-body extensions of the almost Mathieu equation. The toric
and cubic codes are mapped to lattice models of rotors, with the toric code
case related to U(1) lattice gauge theory.Comment: 22 pages, 3 figures; part of special issue on Rabi model; v2 minor
change
An Introduction to Mechanized Reasoning
Mechanized reasoning uses computers to verify proofs and to help discover new
theorems. Computer scientists have applied mechanized reasoning to economic
problems but -- to date -- this work has not yet been properly presented in
economics journals. We introduce mechanized reasoning to economists in three
ways. First, we introduce mechanized reasoning in general, describing both the
techniques and their successful applications. Second, we explain how mechanized
reasoning has been applied to economic problems, concentrating on the two
domains that have attracted the most attention: social choice theory and
auction theory. Finally, we present a detailed example of mechanized reasoning
in practice by means of a proof of Vickrey's familiar theorem on second-price
auctions
The Linear Algebra Mapping Problem
We observe a disconnect between the developers and the end users of linear
algebra libraries. On the one hand, the numerical linear algebra and the
high-performance communities invest significant effort in the development and
optimization of highly sophisticated numerical kernels and libraries, aiming at
the maximum exploitation of both the properties of the input matrices, and the
architectural features of the target computing platform. On the other hand, end
users are progressively less likely to go through the error-prone and time
consuming process of directly using said libraries by writing their code in C
or Fortran; instead, languages and libraries such as Matlab, Julia, Eigen and
Armadillo, which offer a higher level of abstraction, are becoming more and
more popular. Users are given the opportunity to code matrix computations with
a syntax that closely resembles the mathematical description; it is then a
compiler or an interpreter that internally maps the input program to lower
level kernels, as provided by libraries such as BLAS and LAPACK. Unfortunately,
our experience suggests that in terms of performance, this translation is
typically vastly suboptimal.
In this paper, we first introduce the Linear Algebra Mapping Problem, and
then investigate how effectively a benchmark of test problems is solved by
popular high-level programming languages. Specifically, we consider Matlab,
Octave, Julia, R, Armadillo (C++), Eigen (C++), and NumPy (Python); the
benchmark is meant to test both standard compiler optimizations such as common
subexpression elimination and loop-invariant code motion, as well as linear
algebra specific optimizations such as optimal parenthesization of a matrix
product and kernel selection for matrices with properties. The aim of this
study is to give concrete guidelines for the development of languages and
libraries that support linear algebra computations
Type-Directed Program Synthesis and Constraint Generation for Library Portability
Fast numerical libraries have been a cornerstone of scientific computing for
decades, but this comes at a price. Programs may be tied to vendor specific
software ecosystems resulting in polluted, non-portable code. As we enter an
era of heterogeneous computing, there is an explosion in the number of
accelerator libraries required to harness specialized hardware. We need a
system that allows developers to exploit ever-changing accelerator libraries,
without over-specializing their code.
As we cannot know the behavior of future libraries ahead of time, this paper
develops a scheme that assists developers in matching their code to new
libraries, without requiring the source code for these libraries.
Furthermore, it can recover equivalent code from programs that use existing
libraries and automatically port them to new interfaces. It first uses program
synthesis to determine the meaning of a library, then maps the synthesized
description into generalized constraints which are used to search the program
for replacement opportunities to present to the developer.
We applied this approach to existing large applications from the scientific
computing and deep learning domains. Using our approach, we show speedups
ranging from 1.1 to over 10 on end to end performance when
using accelerator libraries.Comment: Accepted to PACT 201
Fluid statics of a self-gravitating perfect-gas isothermal sphere
We open the paper with introductory considerations describing the motivations
of our long-term research plan targeting gravitomagnetism, illustrating the
fluid-dynamics numerical test case selected for that purpose, that is, a
perfect-gas sphere contained in a solid shell located in empty space
sufficiently away from other masses, and defining the main objective of this
study: the determination of the gravitofluid-static field required as initial
field () in forthcoming fluid-dynamics calculations. The determination of
the gravitofluid-static field requires the solution of the isothermal-sphere
Lane-Emden equation. We do not follow the habitual approach of the literature
based on the prescription of the central density as boundary condition; we
impose the gravitational field at the solid-shell internal wall. As the
discourse develops, we point out differences and similarities between the
literature's and our approach. We show that the nondimensional formulation of
the problem hinges on a unique physical characteristic number that we call
gravitational number because it gauges the self-gravity effects on the gas'
fluid statics. We illustrate and discuss numerical results; some peculiarities,
such as gravitational-number upper bound and multiple solutions, lead us to
investigate the thermodynamics of the physical system, particularly entropy and
energy, and preliminarily explore whether or not thermodynamic-stability
reasons could provide justification for either selection or exclusion of
multiple solutions. We close the paper with a summary of the present study in
which we draw conclusions and describe future work.Comment: 32 pages, 26 figure
06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality
From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Exact theory of dense amorphous hard spheres in high dimension. III. The full RSB solution
In the first part of this paper, we derive the general replica equations that
describe infinite-dimensional hard spheres at any level of replica symmetry
breaking (RSB) and in particular in the fullRSB scheme. We show that these
equations are formally very similar to the ones that have been derived for spin
glass models, thus showing that the analogy between spin glasses and structural
glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a
strong sense in the mean field limit. We also suggest how the computation could
be generalized in an approximate way to finite dimensional hard spheres. In the
second part of the paper, we discuss the solution of these equations and we
derive from it a number of physical predictions. We show that, below the
Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase
exists and we locate the boundary of the fullRSB phase. Most importantly, we
show that the fullRSB solution predicts correctly that jammed packings are
isostatic, and allows one to compute analytically the critical exponents
associated with the jamming transition, which are missed by the 1RSB solution.
We show that these predictions compare very well with numerical results.Comment: 58 pages, 13 figures -- Final version to appear on JSTA
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