4,823 research outputs found
Sobolev gradients and image interpolation
We present here a new image inpainting algorithm based on the Sobolev
gradient method in conjunction with the Navier-Stokes model. The original model
of Bertalmio et al is reformulated as a variational principle based on the
minimization of a well chosen functional by a steepest descent method. This
provides an alternative of the direct solving of a high-order partial
differential equation and, consequently, allows to avoid complicated numerical
schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze
our algorithm in an infinite dimensional setting using an evolution equation
and obtain global existence and uniqueness results as well as the existence of
an -limit. Using a finite difference implementation, we demonstrate
using various examples that the Sobolev gradient flow, due to its smoothing and
preconditioning properties, is an effective tool for use in the image
inpainting problem
Thermodynamic phase-field model for microstructure with multiple components and phases: the possibility of metastable phases
A diffuse-interface model for microstructure with an arbitrary number of
components and phases was developed from basic thermodynamic and kinetic
principles and formalized within a variational framework. The model includes a
composition gradient energy to capture solute trapping, and is therefore suited
for studying phenomena where the width of the interface plays an important
role. Derivation of the inhomogeneous free energy functional from a Taylor
expansion of homogeneous free energy reveals how the interfacial properties of
each component and phase may be specified under a mass constraint. A diffusion
potential for components was defined away from the dilute solution limit, and a
multi-obstacle barrier function was used to constrain phase fractions. The
model was used to simulate solidification via nucleation, premelting at phase
boundaries and triple junctions, the intrinsic instability of small particles,
and solutal melting resulting from differing diffusivities in solid and liquid.
The shape of metastable free energy surfaces is found to play an important role
in microstructure evolution and may explain why some systems premelt at phase
boundaries and phase triple junctions while others do not.Comment: 14 pages, 8 figure
An adaptive fixed-mesh ALE method for free surface flows
In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.Postprint (published version
Augmentation with Projection: Towards an Effective and Efficient Data Augmentation Paradigm for Distillation
Knowledge distillation is one of the primary methods of transferring
knowledge from large to small models. However, it requires massive
task-specific data, which may not be plausible in many real-world applications.
Data augmentation methods such as representation interpolation, token
replacement, or augmentation with models are applied to tackle this problem.
However, these data augmentation methods either potentially cause shifts in
decision boundaries (representation interpolation), are not expressive enough
(token replacement), or introduce too much computational overhead (augmentation
with models). To this end, we propose AugPro (Augmentation with Projection), an
effective and efficient data augmentation method for distillation. Our method
builds on top of representation interpolation augmentation methods to maintain
the diversity of expressions and converts the augmented data to tokens to avoid
shifting decision boundaries. It uses simple operations that come with little
computational overhead. The results on multiple GLUE tasks show that our
methods can improve distillation performance by a large margin at a low time
cost. Codes are available at
https://github.com/google-research/google-research/tree/master/augpro.Comment: 20 pages, 5 figures. Accepted by ICLR 202
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