6,031 research outputs found
Differentiable Programming Tensor Networks
Differentiable programming is a fresh programming paradigm which composes
parameterized algorithmic components and trains them using automatic
differentiation (AD). The concept emerges from deep learning but is not only
limited to training neural networks. We present theory and practice of
programming tensor network algorithms in a fully differentiable way. By
formulating the tensor network algorithm as a computation graph, one can
compute higher order derivatives of the program accurately and efficiently
using AD. We present essential techniques to differentiate through the tensor
networks contractions, including stable AD for tensor decomposition and
efficient backpropagation through fixed point iterations. As a demonstration,
we compute the specific heat of the Ising model directly by taking the second
order derivative of the free energy obtained in the tensor renormalization
group calculation. Next, we perform gradient based variational optimization of
infinite projected entangled pair states for quantum antiferromagnetic
Heisenberg model and obtain start-of-the-art variational energy and
magnetization with moderate efforts. Differentiable programming removes
laborious human efforts in deriving and implementing analytical gradients for
tensor network programs, which opens the door to more innovations in tensor
network algorithms and applications.Comment: Typos corrected, discussion and refs added; revised version accepted
for publication in PRX. Source code available at
https://github.com/wangleiphy/tensorgra
Design and Evaluation of Approximate Logarithmic Multipliers for Low Power Error-Tolerant Applications
In this work, the designs of both non-iterative and iterative approximate logarithmic multipliers (LMs) are studied to further reduce power consumption and improve performance. Non-iterative approximate LMs (ALMs) that use three inexact mantissa adders, are presented. The proposed iterative approximate logarithmic multipliers (IALMs) use a set-one adder in both mantissa adders during an iteration; they also use lower-part-or adders and approximate mirror adders for the final addition. Error analysis and simulation results are also provided; it is found that the proposed approximate LMs with an appropriate number of inexact bits achieve a higher accuracy and lower power consumption than conventional LMs using exact units. Compared with conventional LMs with exact units, the normalized mean error distance (NMED) of 16-bit approximate LMs is decreased by up to 18% and the power-delay product (PDP) has a reduction of up to 37%. The proposed approximate LMs are also compared with previous approximate multipliers; it is found that the proposed approximate LMs are best suitable for applications allowing larger errors, but requiring lower energy consumption and low power. Approximate Booth multipliers fit applications with less stringent power requirements, but also requiring smaller errors. Case studies for error-tolerant computing applications are provided
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
High speed simulation of flexible multibody dynamics
A multiflexible body dynamics code intended for fast turnaround control design trades is described. Nonlinear rigid body dynamics and linearized flexible dynamics combine to provide efficient solution of the equations of motion. Comparison with results from the DISCOS code provide verification of accuracy
Random spherical hyperbolic diffusion
The paper starts by giving a motivation for this research and justifying the
considered stochastic diffusion models for cosmic microwave background
radiation studies. Then it derives the exact solution in terms of a series
expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy
problem with random initial conditions is studied. All assumptions are stated
in terms of the angular power spectrum of the initial conditions. An
approximation to the solution is given and analysed by finitely truncating the
series expansion. The upper bounds for the convergence rates of the
approximation errors are derived. Smoothness properties of the solution and its
approximation are investigated. It is demonstrated that the sample H\"older
continuity of these spherical fields is related to the decay of the angular
power spectrum. Numerical studies of approximations to the solution and
applications to cosmic microwave background data are presented to illustrate
the theoretical results.Comment: 30 pages, 15 figures. Updated file. Some misprints are correcte
Shapes and Shears, Stars and Smears: Optimal Measurements for Weak Lensing
We present the theoretical and analytical bases of optimal techniques to
measure weak gravitational shear from images of galaxies. We first characterize
the geometric space of shears and ellipticity, then use this geometric
interpretation to analyse images. The steps of this analysis include:
measurement of object shapes on images, combining measurements of a given
galaxy on different images, estimating the underlying shear from an ensemble of
galaxy shapes, and compensating for the systematic effects of image distortion,
bias from PSF asymmetries, and `"dilution" of the signal by the seeing. These
methods minimize the ellipticity measurement noise, provide calculable shear
uncertainty estimates, and allow removal of systematic contamination by PSF
effects to arbitrary precision. Galaxy images and PSFs are decomposed into a
family of orthogonal 2d Gaussian-based functions, making the PSF correction and
shape measurement relatively straightforward and computationally efficient. We
also discuss sources of noise-induced bias in weak lensing measurements and
provide a solution for these and previously identified biases.Comment: Version accepted to AJ. Minor fixes, plus a simpler method of shape
weighting. Version with full vector figures available via
http://www.astro.lsa.umich.edu/users/garyb/PUBLICATIONS
A Study on Efficient Designs of Approximate Arithmetic Circuits
Approximate computing is a popular field where accuracy is traded with energy. It can benefit applications such as multimedia, mobile computing and machine learning which are inherently error resilient. Error introduced in these applications to a certain degree is beyond human perception. This flexibility can be exploited to design area, delay and power efficient architectures. However, care must be taken on how approximation compromises the correctness of results. This research work aims to provide approximate hardware architectures with error metrics and design metrics analyzed and their effects in image processing applications.
Firstly, we study and propose unsigned array multipliers based on probability statistics and with approximate 4-2 compressors, full adders and half adders. This work deals with a new design approach for approximation of multipliers. The partial products of the multiplier are altered to introduce varying probability terms. Logic complexity of approximation is varied for the accumulation of altered partial products based on their probability. The proposed approximation is utilized in two variants of 16-bit multipliers. Synthesis results reveal that two proposed multipliers achieve power savings of 72% and 38% respectively compared to an exact multiplier. They have better precision when compared to existing approximate multipliers. Mean relative error distance (MRED) figures are as low as 7.6% and 0.02% for the proposed approximate multipliers, which are better than the previous state-of-the-art works. Performance of the proposed multipliers is evaluated with geometric mean filtering application, where one of the proposed models achieves the highest peak signal to noise ratio (PSNR).
Second, approximation is proposed for signed Booth multiplication. Approximation is introduced in partial product generation and partial product accumulation circuits. In this work, three multipliers (ABM-M1, ABM-M2, and ABM-M3) are proposed in which the modified Booth algorithm is approximated. In all three designs, approximate Booth partial product generators are designed with different variations of approximation. The approximations are performed by reducing the logic complexity of the Booth partial product generator, and the accumulation of partial products is slightly modified to improve circuit performance. Compared to the exact Booth multiplier, ABM-M1 achieves up to 15% reduction in power consumption with an MRED value of 7.9 Ă— 10-4. ABM-M2 has power savings of up to 60% with an MRED of 1.1 Ă— 10-1. ABM-M3 has power savings of up to 50% with an MRED of 3.4 Ă— 10-3. Compared to existing approximate Booth multipliers, the proposed multipliers ABM-M1 and ABM-M3 achieve up to a 41% reduction in power consumption while exhibiting very similar error metrics. Image multiplication and matrix multiplication are used as case studies to illustrate the high performance of the proposed approximate multipliers.
Third, distributed arithmetic based sum of products units approximation is analyzed. Sum of products units are key elements in many digital signal processing applications. Three approximate sum of products models which are based on distributed arithmetic are proposed. They are designed for different levels of accuracy. First model of approximate sum of products achieves an improvement up to 64% on area and 70% on power, when compared to conventional unit. Other two models provide an improvement of 32% and 48% on area and 54% and 58% on power, respectively, with a reduced error rate compared to the first model. Third model achieves MRED and normalized mean error distance (NMED) as low as 0.05% and 0.009%. Performance of approximate units is evaluated with a noisy image smoothing application, where the proposed models are capable of achieving higher PSNR than existing state of the art techniques.
Fourth, approximation is applied in division architecture. Two approximation models are proposed for restoring divider. In the first design, approximation is performed at circuit level, where approximate divider cells are utilized in place of exact ones by simplifying the logic equations. In the second model, restoring divider is analyzed strategically and number of restoring divider cells are reduced by finding the portions of divisor and dividend with significant information. An approximation factor is used in both designs. In model 1, the design with p=8 has a 58% reduction in both area and power consumption compared to exact design, with a Q-MRED of 1.909 Ă— 10-2 and Q-NMED of 0.449 Ă— 10-2. The second model with an approximation factor p=4 has 54% area savings and 62% power savings compared to exact design. The proposed models are found to have better error metrics compared to existing designs, with better performance at similar error values. A change detection image processing application is used for real time assessment of proposed and existing approximate dividers and one of the models achieves a PSNR of 54.27 dB
Multipliers for Floating-Point Double Precision and Beyond on FPGAs
International audienceThe implementation of high-precision floating-point applications on reconfigurable hardware requires a variety of large multipliers: Standard multipliers are the core of floating-point multipliers; Truncated multipliers, trading resources for a well-controlled accuracy degradation, are useful building blocks in situations where a full multiplier is not needed. This work studies the automated generation of such multipliers using the embedded multipliers and adders present in DSP blocks of current FPGAs. The optimization of such multipliers is expressed as a tiling problem where a tile represents a hardware multiplier and super-tiles are the wiring of several hardware multipliers making efficient use of the DSP internal resources. This tiling technique is shown to adapt to full or truncated multipliers. It addresses arbitrary precisions including single, double but also in the quadruple precision introduced by the IEEE-754-2008 standard and currently unsupported by processor hardware. An open-source implementation is provided in the FloPoCo project
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