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1. Multidimensional WENO-AO Reconstructions Using a Simplified Smoothness Indicator and Applications to Conservation Laws

   https://doi.org/10.1007/s10915-023-02319-x  

   Huang, Chieh-Sen; Arbogast, Todd; Tian, Chenyu (August 2023, Journal of Scientific Computing)

   Abstract Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator$$\sigma ^{\textrm{S}}= \frac{1}{N_{\textrm{S}}-1}\sum _{j} ({\bar{u}}_{j} - {\bar{u}}_{m})^2$$ ${\sigma }^{\text{S}}=\frac{1}{N_{\text{S}}-1}{\sum }_j{({\overline{u}}_j-{\overline{u}}_m)}^2$, where$$N_{\textrm{S}}$$ $N_{\text{S}}$is the number of mesh elements in the stencil,$${\bar{u}}_j$$ ${\overline{u}}_j$is the local function average over mesh elementj, and indexmgives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved. 

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2. Reconstruction of implicit surfaces from fluid particles using convolutional neural networks

   https://doi.org/10.1111/cgf.15181  

   Zhao, C; Shinar, T; Schroeder, C (December 2024, Computer Graphics Forum)

   In this paper, we present a novel network‐based approach for reconstructing signed distance functions from fluid particles. The method uses a weighting kernel to transfer particles to a regular grid, which forms the input to a convolutional neural network. We propose a regression‐based regularization to reduce surface noise without penalizing high‐curvature features. The reconstruction exhibits improved spatial surface smoothness and temporal coherence compared with existing state of the art surface reconstruction methods. The method is insensitive to particle sampling density and robustly handles thin features, isolated particles, and sharp edges. 

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3. Application-oriented Analysis of Material Interface Reconstruction Algorithms in Time-varying Bijel Simulations

   https://doi.org/10.2312/evs.20221104  

   Bao, Xueyi; Karthikeyan, Nikhil; Schiller, Ulf D.; Iuricich, Federico (January 2022, The Eurographics Association)

   Multimaterial interface reconstruction has been investigated over the years both from visualization and analytical point of view using different metrics. When focusing on visualization, interface continuity and smoothness are used to quantify interface quality. When the end goal is interface analysis, metrics closer to the physical properties of the material are preferred (e.g., curvature, tortuosity). In this paper, we re-evaluate three Multimaterial Interface Reconstruction (MIR) algorithms, already integrated in established visualization frameworks, under the lens of application-oriented metrics. Specifically, we analyze interface curvature, particle-interface distance, and medial axis-interface distance in a time-varying bijel simulation. Our analysis shows that the interface presenting the best visual qualities is not always the most useful for domain scientists when evaluating the material properties. 

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4. Principal curve approaches for inferring 3D chromatin architecture

   https://doi.org/10.1093/biostatistics/kxaa046  

   Tuzhilina, Elena; Hastie, Trevor J; Segal, Mark R (November 2020, Biostatistics)

   null (Ed.)

   Summary Three-dimensional (3D) genome spatial organization is critical for numerous cellular processes, including transcription, while certain conformation-driven structural alterations are frequently oncogenic. Genome architecture had been notoriously difficult to elucidate, but the advent of the suite of chromatin conformation capture assays, notably Hi-C, has transformed understanding of chromatin structure and provided downstream biological insights. Although many findings have flowed from direct analysis of the pairwise proximity data produced by these assays, there is added value in generating corresponding 3D reconstructions deriving from superposing genomic features on the reconstruction. Accordingly, many methods for inferring 3D architecture from proximity data have been advanced. However, none of these approaches exploit the fact that single chromosome solutions constitute a one-dimensional (1D) curve in 3D. Rather, this aspect has either been addressed by imposition of constraints, which is both computationally burdensome and cell type specific, or ignored with contiguity imposed after the fact. Here, we target finding a 1D curve by extending principal curve methodology to the metric scaling problem. We illustrate how this approach yields a sequence of candidate solutions, indexed by an underlying smoothness or degrees-of-freedom parameter, and propose methods for selection from this sequence. We apply the methodology to Hi-C data obtained on IMR90 cells and so are positioned to evaluate reconstruction accuracy by referencing orthogonal imaging data. The results indicate the utility and reproducibility of our principal curve approach in the face of underlying structural variation. 

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5. Higher-Order Total Variation Classes on Grids: Minimax Theory and Trend Filtering Methods.

   Sadhanala, Veeranjaneyulu; Wang, Yu-Xiang; Sharpnack, James L; Tibshirani, Ryan J (December 2017, Advances in neural information processing systems)

   We consider the problem of estimating the values of a function over n nodes of a d-dimensional grid graph (having equal side lengths) from noisy observations. The function is assumed to be smooth, but is allowed to exhibit different amounts of smoothness at different regions in the grid. Such heterogeneity eludes classical measures of smoothness from nonparametric statistics, such as Holder smoothness. Meanwhile, total variation (TV) smoothness classes allow for heterogeneity, but are restrictive in another sense: only constant functions count as perfectly smooth (achieve zero TV). To move past this, we define two new higher-order TV classes, based on two ways of compiling the discrete derivatives of a parameter across the nodes. We relate these two new classes to Holder classes, and derive lower bounds on their minimax errors. We also analyze two naturally associated trend filtering methods; when d=2, each is seen to be rate optimal over the appropriate class. 

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