17,084 research outputs found
A fast, simple, and stable Chebyshev-Legendre transform using an asymptotic formula
A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree polynomial in operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid
Landau Collision Integral Solver with Adaptive Mesh Refinement on Emerging Architectures
The Landau collision integral is an accurate model for the small-angle
dominated Coulomb collisions in fusion plasmas. We investigate a high order
accurate, fully conservative, finite element discretization of the nonlinear
multi-species Landau integral with adaptive mesh refinement using the PETSc
library (www.mcs.anl.gov/petsc). We develop algorithms and techniques to
efficiently utilize emerging architectures with an approach that minimizes
memory usage and movement and is suitable for vector processing. The Landau
collision integral is vectorized with Intel AVX-512 intrinsics and the solver
sustains as much as 22% of the theoretical peak flop rate of the Second
Generation Intel Xeon Phi, Knights Landing, processor
Simultaneous floating-point sine and cosine for VLIW integer processors
Accepted for publication in the proceedings of the 23rd IEEE International Conference on Application-specific Systems, Architectures and Processors (ASAP 2012).International audienceGraphics and signal processing applications often require that sines and cosines be evaluated at a same floating-point argument, and in such cases a very fast computation of the pair of values is desirable. This paper studies how 32-bit VLIW integer architectures can be exploited in order to perform this task accurately for IEEE single precision. We describe software implementations for sinf, cosf, and sincosf over [-pi/4,pi/4] that have a proven 1-ulp accuracy and whose latency on STMicroelectronics' ST231 VLIW integer processor is 19, 18, and 19 cycles, respectively. Such performances are obtained by introducing a novel algorithm for simultaneous sine and cosine that combines univariate and bivariate polynomial evaluation schemes
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Max-Sliced Wasserstein Distance and its use for GANs
Generative adversarial nets (GANs) and variational auto-encoders have
significantly improved our distribution modeling capabilities, showing promise
for dataset augmentation, image-to-image translation and feature learning.
However, to model high-dimensional distributions, sequential training and
stacked architectures are common, increasing the number of tunable
hyper-parameters as well as the training time. Nonetheless, the sample
complexity of the distance metrics remains one of the factors affecting GAN
training. We first show that the recently proposed sliced Wasserstein distance
has compelling sample complexity properties when compared to the Wasserstein
distance. To further improve the sliced Wasserstein distance we then analyze
its `projection complexity' and develop the max-sliced Wasserstein distance
which enjoys compelling sample complexity while reducing projection complexity,
albeit necessitating a max estimation. We finally illustrate that the proposed
distance trains GANs on high-dimensional images up to a resolution of 256x256
easily.Comment: Accepted to CVPR 201
- …