17,015 research outputs found
Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
A conservative flux postprocessing algorithm is presented for both
steady-state and dynamic flow models. The postprocessed flux is shown to have
the same convergence order as the original flux. An arbitrary flux
approximation is projected into a conservative subspace by adding a piecewise
constant correction that is minimized in a weighted norm. The application
of a weighted norm appears to yield better results for heterogeneous media than
the standard norm which has been considered in earlier works. We also
study the effect of different flux calculations on the domain boundary. In
particular we consider the continuous Galerkin finite element method for
solving Darcy flow and couple it with a discontinuous Galerkin finite element
method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table
Compressive Wave Computation
This paper considers large-scale simulations of wave propagation phenomena.
We argue that it is possible to accurately compute a wavefield by decomposing
it onto a largely incomplete set of eigenfunctions of the Helmholtz operator,
chosen at random, and that this provides a natural way of parallelizing wave
simulations for memory-intensive applications.
This paper shows that L1-Helmholtz recovery makes sense for wave computation,
and identifies a regime in which it is provably effective: the one-dimensional
wave equation with coefficients of small bounded variation. Under suitable
assumptions we show that the number of eigenfunctions needed to evolve a sparse
wavefield defined on N points, accurately with very high probability, is
bounded by C log(N) log(log(N)), where C is related to the desired accuracy and
can be made to grow at a much slower rate than N when the solution is sparse.
The PDE estimates that underlie this result are new to the authors' knowledge
and may be of independent mathematical interest; they include an L1 estimate
for the wave equation, an estimate of extension of eigenfunctions, and a bound
for eigenvalue gaps in Sturm-Liouville problems.
Numerical examples are presented in one spatial dimension and show that as
few as 10 percents of all eigenfunctions can suffice for accurate results.
Finally, we argue that the compressive viewpoint suggests a competitive
parallel algorithm for an adjoint-state inversion method in reflection
seismology.Comment: 45 pages, 4 figure
A Compact Linear Programming Relaxation for Binary Sub-modular MRF
We propose a novel compact linear programming (LP) relaxation for binary
sub-modular MRF in the context of object segmentation. Our model is obtained by
linearizing an -norm derived from the quadratic programming (QP) form of
the MRF energy. The resultant LP model contains significantly fewer variables
and constraints compared to the conventional LP relaxation of the MRF energy.
In addition, unlike QP which can produce ambiguous labels, our model can be
viewed as a quasi-total-variation minimization problem, and it can therefore
preserve the discontinuities in the labels. We further establish a relaxation
bound between our LP model and the conventional LP model. In the experiments,
we demonstrate our method for the task of interactive object segmentation. Our
LP model outperforms QP when converting the continuous labels to binary labels
using different threshold values on the entire Oxford interactive segmentation
dataset. The computational complexity of our LP is of the same order as that of
the QP, and it is significantly lower than the conventional LP relaxation
Graph- and finite element-based total variation models for the inverse problem in diffuse optical tomography
Total variation (TV) is a powerful regularization method that has been widely
applied in different imaging applications, but is difficult to apply to diffuse
optical tomography (DOT) image reconstruction (inverse problem) due to complex
and unstructured geometries, non-linearity of the data fitting and
regularization terms, and non-differentiability of the regularization term. We
develop several approaches to overcome these difficulties by: i) defining
discrete differential operators for unstructured geometries using both finite
element and graph representations; ii) developing an optimization algorithm
based on the alternating direction method of multipliers (ADMM) for the
non-differentiable and non-linear minimization problem; iii) investigating
isotropic and anisotropic variants of TV regularization, and comparing their
finite element- and graph-based implementations. These approaches are evaluated
on experiments on simulated data and real data acquired from a tissue phantom.
Our results show that both FEM and graph-based TV regularization is able to
accurately reconstruct both sparse and non-sparse distributions without the
over-smoothing effect of Tikhonov regularization and the over-sparsifying
effect of L regularization. The graph representation was found to
out-perform the FEM method for low-resolution meshes, and the FEM method was
found to be more accurate for high-resolution meshes.Comment: 24 pages, 11 figures. Reviced version includes revised figures and
improved clarit
A Novel Euler's Elastica based Segmentation Approach for Noisy Images via using the Progressive Hedging Algorithm
Euler's Elastica based unsupervised segmentation models have strong
capability of completing the missing boundaries for existing objects in a clean
image, but they are not working well for noisy images. This paper aims to
establish a Euler's Elastica based approach that properly deals with random
noises to improve the segmentation performance for noisy images. We solve the
corresponding optimization problem via using the progressive hedging algorithm
(PHA) with a step length suggested by the alternating direction method of
multipliers (ADMM). Technically, all the simplified convex versions of the
subproblems derived from the major framework of PHA can be obtained by using
the curvature weighted approach and the convex relaxation method. Then an
alternating optimization strategy is applied with the merits of using some
powerful accelerating techniques including the fast Fourier transform (FFT) and
generalized soft threshold formulas. Extensive experiments have been conducted
on both synthetic and real images, which validated some significant gains of
the proposed segmentation models and demonstrated the advantages of the
developed algorithm
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