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We introduce a hybrid filter that incorporates a mathematically accurate moment-based filter with a data driven filter for discontinuous Galerkin approximations to PDE solutions that contain discontinuities. Numerical solutions of PDEs suffer from an $\mathcal{O}(1)$ error in the neighborhood about discontinuities, especially for shock waves that arise in inviscid compressible flow problems. While post-processing filters, such as the Smoothness-Increasing Accuracy-Conserving (SIAC) filter, can improve the order of error in smooth regions, the $\mathcal{O}(1)$ error in the vicinity of a discontinuity remains. To alleviate this, we combine the SIAC filter with a data-driven filter based on Convolutional Neural Networks. The data-driven filter is specifically focused on improving the errors in discontinuous regions and therefore {\it only includes top-hat functions in the training dataset.} For both filters, a consistency constraint is enforced, while the SIAC filter additionally satisfies $r-$moments. This hybrid filter approach allows for maintaining the accuracy guaranteed by the theory in smooth regions while the hybrid SIAC-data-driven approach reduces the $\ell_2$ and $\ell_\infty$ errors about discontinuities. Thus, overall the global errors are reduced. We examine the performance of the hybrid filter about discontinuities for the one-dimensional Euler equations for the Lax, Sod, and sine-entropy (Shu-Osher) shock-tube problems.

In this article we consider the extension of the (L)SIAC-MRA enhancement procedure to nonuniform meshes. We demonstrate that error reduction can be obtained on perturbed quadrilateral and Delaunay meshes, and investigate the effect of limited resolution and its impact on the procedure for various function types. We show that utilizing mesh-based localized kernel scalings, which were shown to reduce approximation errors for LSIAC filters, improve the performance of the LSIAC-MRA enhancement procedure. Lastly, we demonstrate the usefulness of enhanced approximations generated by (L)SIAC-MRA in mesh adaptivity applications, and show that SIAC reconstruction can be used in identification of regions of high error in steady-state DG approximations.

The simulation of plasma physics is computationally expensive because the underlying physical system is of high dimensions, requiring three spatial dimensions and three velocity dimensions. One popular numerical approach is Particle-In-Cell (PIC) methods owing to its ease of implementation and favorable scalability in high-dimensional problems. An unfortunate drawback of the method is the introduction of statistical noise resulting from the use of finitely many particles. In this paper we examine the application of the Smoothness-Increasing Accuracy-Conserving (SIAC) family of convolution kernel filters as denoisers for moment data arising from PIC simulations. We show that SIAC filtering is a promising tool to denoise PIC data in the physical space as well as capture the appropriate scales in the Fourier space. Furthermore, we demonstrate how the application of the SIAC technique reduces the amount of information necessary in the computation of quantities of interest in plasma physics such as the Bohm speed.

This paper considers the discontinuous Galerkin (DG) methods for solving the Vlasov-Maxwell (VM) system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. This work studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. In particular, we prove the superconvergence of order $(2k+\frac{1}{2})$ in the negative order norm for the probability distribution function and the electromagnetic fields when piecewise polynomial degree $k$ is used. Numerical tests including Landau damping, two-stream instability and streaming Weibel instabilities are considered showing the performance of the post-processor.

In this article we introduce Line Smoothness-Increasing Accuracy-Conserving Multi-Resolution Analysis\linebreak (LSIAC-MRA). This is a procedure for exploiting convolution kernel post-processors for obtaining more accurate multi-dimensional multi-resolution analysis (MRA) in terms of error reduction. This filtering-projection tool allows for the transition of data between different resolutions while simultaneously decreasing errors in the fine grid approximation. It specifically allows for defining detail multi-wavelet coefficients when translating coarse data onto finer meshes. These coefficients are usually not defined in such cases. We show how to analytically evaluate the resulting convolutions and express the filtered approximation in a new basis. This is done by combining the filtering procedure with projection operators that allow for computational implementation of this scale transition procedure. Further, this procedure can be applied to piecewise constant approximations to functions, as it provides error reduction. We demonstrate the effectiveness of this technique in two and three dimensions., Comment: Submitted to SISC

In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that `tweaks' a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Enhancing filter and Superconvergent Patch Recovery. Extensive numerical tests are conducted that confirm our analytic findings., Comment: 25 pages, 17 figures

Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. These advantages can be exploited during flow visualization, for example by applying the SIAC filter to the DG data before streamline computations [Steffan {\it et al.}, IEEE-TVCG 14(3): 680-692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [Walfisch {\it et al.}, JOMP 38(2);164-184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the SIAC line filter and explore how the orientation, structure and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community.

In Vuik and Ryan (2014) we studied the use of troubled-cell indicators for discontinuity detection in nonlinear hyperbolic partial differential equations and introduced a new multiwavelet technique to detect troubled cells. We found that these methods perform well as long as a suitable, problem-dependent parameter is chosen. This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically. The choice of the parameter has impact on the approximation: it determines the strictness of the troubled-cell indicator. An inappropriate choice of the parameter will result in detection (and limiting) of too few or too many elements. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In this paper we will see that for each troubled-cell indicator the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting the outliers of a vector (one dimension) or matrix (two dimensions). This is done using Tukey's boxplot approach to detect which coefficients in a vector are straying far beyond others (Tukey, 1977). We provide an algorithm that can be applied to various troubled-cell indication variables. Using this technique the problem-dependent parameter that the original indicator requires is no longer necessary as the parameter will be chosen automatically.

In this paper, we introduce a new global troubled-cell indicator for the discontinuous Galerkin (DG) method in one- and two-dimensions. This is done by taking advantage of the global expression of the DG method and re-expanding it in terms of a multiwavelet basis, which is a sum of the global average and finer details on different levels. Examining the higher level difference coefficients acts as a troubled-cell indicator, thus avoiding unnecessary increased computational cost of a new expansion. In two-dimensions the multiwavelet decomposition uses combinations of scaling functions and multiwavelets in the x- and y-directions for improved troubled-cell indication. By using such a troubled-cell indicator, we are able to reduce the computational cost by avoiding limiting in smooth regions. We present numerical examples in one- and two-dimensions and compare our troubled-cell indicator to the subcell resolution technique of Harten (1989) and the shock detector of Krivodonova, Xin, Remacle, Chevaugeon, and Flaherty (2004), which were previously investigated by Qiu and Shu (2005).