23,233 research outputs found
Square-rich fixed point polynomial evaluation on FPGAs
Polynomial evaluation is important across a wide range of application domains, so significant work has been done on accelerating its computation. The conventional algorithm, referred to as Horner's rule, involves the least number of steps but can lead to increased latency due to serial computation. Parallel evaluation algorithms such as Estrin's method have shorter latency than Horner's rule, but achieve this at the expense of large hardware overhead. This paper presents an efficient polynomial evaluation algorithm, which reforms the evaluation process to include an increased number of squaring steps. By using a squarer design that is more efficient than general multiplication, this can result in polynomial evaluation with a 57.9% latency reduction over Horner's rule and 14.6% over Estrin's method, while consuming less area than Horner's rule, when implemented on a Xilinx Virtex 6 FPGA. When applied in fixed point function evaluation, where precision requirements limit the rounding of operands, it still achieves a 52.4% performance gain compared to Horner's rule with only a 4% area overhead in evaluating 5th degree polynomials
An Algebraic Framework for the Real-Time Solution of Inverse Problems on Embedded Systems
This article presents a new approach to the real-time solution of inverse
problems on embedded systems. The class of problems addressed corresponds to
ordinary differential equations (ODEs) with generalized linear constraints,
whereby the data from an array of sensors forms the forcing function. The
solution of the equation is formulated as a least squares (LS) problem with
linear constraints. The LS approach makes the method suitable for the explicit
solution of inverse problems where the forcing function is perturbed by noise.
The algebraic computation is partitioned into a initial preparatory step, which
precomputes the matrices required for the run-time computation; and the cyclic
run-time computation, which is repeated with each acquisition of sensor data.
The cyclic computation consists of a single matrix-vector multiplication, in
this manner computation complexity is known a-priori, fulfilling the definition
of a real-time computation. Numerical testing of the new method is presented on
perturbed as well as unperturbed problems; the results are compared with known
analytic solutions and solutions acquired from state-of-the-art implicit
solvers. The solution is implemented with model based design and uses only
fundamental linear algebra; consequently, this approach supports automatic code
generation for deployment on embedded systems. The targeting concept was tested
via software- and processor-in-the-loop verification on two systems with
different processor architectures. Finally, the method was tested on a
laboratory prototype with real measurement data for the monitoring of flexible
structures. The problem solved is: the real-time overconstrained reconstruction
of a curve from measured gradients. Such systems are commonly encountered in
the monitoring of structures and/or ground subsidence.Comment: 24 pages, journal articl
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
Accelerating Parametric Probabilistic Verification
We present a novel method for computing reachability probabilities of
parametric discrete-time Markov chains whose transition probabilities are
fractions of polynomials over a set of parameters. Our algorithm is based on
two key ingredients: a graph decomposition into strongly connected subgraphs
combined with a novel factorization strategy for polynomials. Experimental
evaluations show that these approaches can lead to a speed-up of up to several
orders of magnitude in comparison to existing approache
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
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