Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View

Mirzargar, Mahsa; Ryan, Jennifer K.; Kirby, Robert M.

doi:10.1007/s10915-015-0081-9

Cited by 16 publications

(20 citation statements)

References 31 publications

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“…Traditionally, the filter employed 2k + 1 B-splines of order k + 1 (i.e., degree k) since this configuration can raise the convergence order of the field up to 2k + 1 [3]. However, alternative configurations can still lead to error reduction and smoothness recovery, even when building kernels using fewer B-splines and of lower order [20]. Here, we present the kernel in a general form and provide a brief introduction to this post-processing technique in order to understand its extension to the LSIAC family.…”

Section: The One-dimensional Siac Family Of Filtersmentioning

confidence: 99%

“…For a more detailed explanation on setting up knots for B-Splines and finding the coeffients of the kernel, we refer the reader to [19,20]. First, we compute the width of the symmetric LSIAC filter by considering both the characteristic length of the filter H and the footprint of the linear combination of B-splines that make up the kernel; we then shift the center of LSIAC kernel to the evaluation position P, and then rotate the LSIAC filter by the given direction r. If the line segment does not cross any domain domain boundaries, we proceed to the next step.…”

Section: Implementation Of Lsiacmentioning

confidence: 99%

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On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions

Jallepalli

Docampo-Sánchez‬

Ryan

et al. 2018

IEEE Trans. Visual. Comput. Graphics

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Fig. 1. 3D NACA wing example of vorticity before (left of the dashed line) and after (right) the application of our proposed approach.Abstract-As the finite element method (FEM) and the finite volume method (FVM), both traditional and high-order variants, continue their proliferation into various applied engineering disciplines, it is important that the visualization techniques and corresponding data analysis tools that act on the results produced by these methods faithfully represent the underlying data. To state this in another way: the interpretation of data generated by simulation needs to be consistent with the numerical schemes that underpin the specific solver technology. As the verifiable visualization literature has demonstrated: visual artifacts produced by the introduction of either explicit or implicit data transformations, such as data resampling, can sometimes distort or even obfuscate key scientific features in the data. In this paper, we focus on the handling of elemental continuity, which is often only C 0 continuous or piecewise discontinuous, when visualizing primary or derived fields from FEM or FVM simulations. We demonstrate that traditional data handling and visualization of these fields introduce visual errors. In addition, we show how the use of the recently proposed line-SIAC filter provides a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous.

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Section: The One-dimensional Siac Family Of Filtersmentioning

confidence: 99%

Section: Implementation Of Lsiacmentioning

confidence: 99%

On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions

Jallepalli

Docampo-Sánchez‬

Ryan

et al. 2018

IEEE Trans. Visual. Comput. Graphics

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“…([SRV11] additionally required quadruple precision for computing the DG output.) Indeed, the coefficients of the boundary filters [RS03, SRV11, MRK12, RLKV15, MRK15] are computed by inverting a matrix whose entries are determined by Gaussian quadrature; and, as pointed out in [RLKV15], SRV filter matrices are close to singular.…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…Most recently these boundary filters have been simplified and improved by replacing numerical approximation with symbolic formulas, both in the uniform symmetric case [MRK15] and in the general case [Pet15]. For general knot sequences, [NP16] introduced a factored symbolic characterization of spline filters that identified the existing boundary filters as position-dependent SIAC spline filters (PSIAC filters).…”

Section: Introductionmentioning

confidence: 99%

Explicit Least-Degree Boundary Filters for Discontinuous Galerkin

Nguyen

Peters

2017

SIAM J. Sci. Comput.

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Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation. The paper introduces a new class of PSIAC filters NP0 that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP0 filters outperform the more complex known boundary filters. NP0 filters typically reduce the L∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP0 filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute.

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“…However, there is promising relations to image processing [13,23] as well as potential in LES filtering [7,8].…”

Section: Applicationsmentioning

confidence: 99%

Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

Ryan

2015

Lecture Notes in Computational Science and Engineering

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There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of SmoothnessIncreasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering.

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Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View

Cited by 16 publications

References 31 publications

On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions

On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions

Explicit Least-Degree Boundary Filters for Discontinuous Galerkin

Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

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