1,341 research outputs found
Maximum-principle preserving space-time isogeometric analysis
In this work we propose a nonlinear stabilization technique for
convection-diffusion-reaction and pure transport problems discretized with
space-time isogeometric analysis. The stabilization is based on a
graph-theoretic artificial diffusion operator and a novel shock detector for
isogeometric analysis. Stabilization in time and space directions are performed
similarly, which allow us to use high-order discretizations in time without any
CFL-like condition. The method is proven to yield solutions that satisfy the
discrete maximum principle (DMP) unconditionally for arbitrary order. In
addition, the stabilization is linearity preserving in a space-time sense.
Moreover, the scheme is proven to be Lipschitz continuous ensuring that the
nonlinear problem is well-posed. Solving large problems using a space-time
discretization can become highly costly. Therefore, we also propose a
partitioned space-time scheme that allows us to select the length of every time
slab, and solve sequentially for every subdomain. As a result, the
computational cost is reduced while the stability and convergence properties of
the scheme remain unaltered. In addition, we propose a twice differentiable
version of the stabilization scheme, which enjoys the same stability properties
while the nonlinear convergence is significantly improved. Finally, the
proposed schemes are assessed with numerical experiments. In particular, we
considered steady and transient pure convection and convection-diffusion
problems in one and two dimensions
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
Isogeometric Analysis in advection-diffusion problems: tension splines approximation
We present a novel approach, within the new paradigm of isogeometric analysis
introduced by Hughes et al., to deal with advection dominated
advection-diffusion problems. The key ingredient is the use of Galerkin approximating
spaces of functions with high smoothness, as in IgA based on
classical B-splines, but particularly well suited to describe sharp layers involving
very strong gradients
Weighted Quasi Interpolant Spline Approximations: Properties and Applications
Continuous representations are fundamental for modeling sampled data and
performing computations and numerical simulations directly on the model or its
elements. To effectively and efficiently address the approximation of point
clouds we propose the Weighted Quasi Interpolant Spline Approximation method
(wQISA). We provide global and local bounds of the method and discuss how it
still preserves the shape properties of the classical quasi-interpolation
scheme. This approach is particularly useful when the data noise can be
represented as a probabilistic distribution: from the point of view of
nonparametric regression, the wQISA estimator is robust to random
perturbations, such as noise and outliers. Finally, we show the effectiveness
of the method with several numerical simulations on real data, including curve
fitting on images, surface approximation and simulation of rainfall
precipitations
Data-driven quasi-interpolant spline surfaces for point cloud approximation
In this paper we investigate a local surface approximation, the Weighted
Quasi Interpolant Spline Approximation (wQISA), specifically designed for large
and noisy point clouds. We briefly describe the properties of the wQISA
representation and introduce a novel data-driven implementation, which combines
prediction capability and complexity efficiency. We provide an extended
comparative analysis with other continuous approximations on real data,
including different types of surfaces and levels of noise, such as 3D models,
terrain data and digital environmental data
Invariant higher-order variational problems II
Motivated by applications in computational anatomy, we consider a
second-order problem in the calculus of variations on object manifolds that are
acted upon by Lie groups of smooth invertible transformations. This problem
leads to solution curves known as Riemannian cubics on object manifolds that
are endowed with normal metrics. The prime examples of such object manifolds
are the symmetric spaces. We characterize the class of cubics on object
manifolds that can be lifted horizontally to cubics on the group of
transformations. Conversely, we show that certain types of non-horizontal
geodesics on the group of transformations project to cubics. Finally, we apply
second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics
on the group of transformations. This leads to a reduced form of the equations
that reveals the obstruction for the projection of a cubic on a transformation
group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome
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