42,033 research outputs found
Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs
The chemical kinetics ODEs arising from operator-split reactive-flow
simulations were solved on GPUs using explicit integration algorithms. Nonstiff
chemical kinetics of a hydrogen oxidation mechanism (9 species and 38
irreversible reactions) were computed using the explicit fifth-order
Runge-Kutta-Cash-Karp method, and the GPU-accelerated version performed faster
than single- and six-core CPU versions by factors of 126 and 25, respectively,
for 524,288 ODEs. Moderately stiff kinetics, represented with mechanisms for
hydrogen/carbon-monoxide (13 species and 54 irreversible reactions) and methane
(53 species and 634 irreversible reactions) oxidation, were computed using the
stabilized explicit second-order Runge-Kutta-Chebyshev (RKC) algorithm. The
GPU-based RKC implementation demonstrated an increase in performance of nearly
59 and 10 times, for problem sizes consisting of 262,144 ODEs and larger, than
the single- and six-core CPU-based RKC algorithms using the
hydrogen/carbon-monoxide mechanism. With the methane mechanism, RKC-GPU
performed more than 65 and 11 times faster, for problem sizes consisting of
131,072 ODEs and larger, than the single- and six-core RKC-CPU versions, and up
to 57 times faster than the six-core CPU-based implicit VODE algorithm on
65,536 ODEs. In the presence of more severe stiffness, such as ethylene
oxidation (111 species and 1566 irreversible reactions), RKC-GPU performed more
than 17 times faster than RKC-CPU on six cores for 32,768 ODEs and larger, and
at best 4.5 times faster than VODE on six CPU cores for 65,536 ODEs. With a
larger time step size, RKC-GPU performed at best 2.5 times slower than six-core
VODE for 8192 ODEs and larger. Therefore, the need for developing new
strategies for integrating stiff chemistry on GPUs was discussed.Comment: 27 pages, LaTeX; corrected typos in Appendix equations A.10 and A.1
Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis
A numerical method for obtaining a crystalline flow starting from a general polygon is presented. A crystalline flow is a polygonal flow and can be regarded as a discrete version of a classical curvature flow. In some cases, new facets may be created instantaneously and their facet lengths are governed by a system of singular ordinary differential equations(ODEs). The proposed method solves the system of the ODEs numerically by using expanding selfsimilar solutions for newly created facets. The computation method is applied to a multi-scale analysis of a contour figure
Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis
AbstractA numerical method for obtaining a crystalline flow starting from a general polygon is presented. A crystalline flow is a polygonal flow and can be regarded as a discrete version of a classical curvature flow. In some cases, new facets may be created instantaneously and their facet lengths are governed by a system of singular ordinary differential equations (ODEs). The proposed method solves the system of the ODEs numerically by using expanding selfsimilar solutions for newly created facets. The computation method is applied to a multi-scale analysis of a contour figure
Improved Techniques for Maximum Likelihood Estimation for Diffusion ODEs
Diffusion models have exhibited excellent performance in various domains. The
probability flow ordinary differential equation (ODE) of diffusion models
(i.e., diffusion ODEs) is a particular case of continuous normalizing flows
(CNFs), which enables deterministic inference and exact likelihood evaluation.
However, the likelihood estimation results by diffusion ODEs are still far from
those of the state-of-the-art likelihood-based generative models. In this work,
we propose several improved techniques for maximum likelihood estimation for
diffusion ODEs, including both training and evaluation perspectives. For
training, we propose velocity parameterization and explore variance reduction
techniques for faster convergence. We also derive an error-bounded high-order
flow matching objective for finetuning, which improves the ODE likelihood and
smooths its trajectory. For evaluation, we propose a novel training-free
truncated-normal dequantization to fill the training-evaluation gap commonly
existing in diffusion ODEs. Building upon these techniques, we achieve
state-of-the-art likelihood estimation results on image datasets (2.56 on
CIFAR-10, 3.43/3.69 on ImageNet-32) without variational dequantization or data
augmentation.Comment: Accepted in ICML202
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