In this paper, we provide a theoretical analysis for a preconditioned steepest descent (PSD) iterative solver that improves the computational time of a finite difference numerical scheme for the Cahn-Hilliard equation with Flory-Huggins energy potential. In the numerical design, a convex splitting approach is applied to the chemical potential such that the logarithmic and the surface diffusion terms are treated implicitly while the expansive concave term is treated with an explicit update. The nonlinear and singular nature of the logarithmic energy potential makes the numerical implementation very challenging. However, the positivity-preserving property for the logarithmic arguments, unconditional energy stability, and optimal rate error estimates have been established in a recent work and it has been shown that successful solvers ensure a similar positivity-preserving property at each iteration stage. Therefore, in this work, we will show that the PSD solver ensures a positivity-preserving property at each iteration stage. The PSD solver consists of first computing a search direction (involved with solving a Poisson-like equation) and then takes a one-parameter optimization step over the search direction in which the Newton iteration becomes very powerful. A theoretical analysis is applied to the PSD iteration solver and a geometric convergence rate is proved for the iteration. In particular, the strict separation property of the numerical solution, which indicates a uniform distance between the numerical solution and the singular limit values of $\pm 1$ for the phase variable, plays an essential role in the iteration convergence analysis. A few numerical results are presented to demonstrate the robustness and efficiency of the PSD solver.
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In this paper, we investigate the conditions under which link analysis algorithms prevent minority groups from reaching high ranking slots. We find that the most common link-based algorithms using centrality metrics, such as PageRank and HITS, can reproduce and even amplify bias against minority groups in networks. Yet, their behavior differs: one one hand, we empirically show that PageRank mirrors the degree distribution for most of the ranking positions and it can equalize representation of minorities among the top ranked nodes; on the other hand, we find that HITS amplifies pre-existing bias in homophilic networks through a novel theoretical analysis, supported by empirical results. We find the root cause of bias amplification in HITS to be the level of homophily present in the network, modeled through an evolving network model with two communities. We illustrate our theoretical analysis on both synthetic and real datasets and we present directions for future work.
In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et. al. (2019, 2022) to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach. We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository. In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.
In this paper, we propose a novel method for 3D scene and object reconstruction from sparse multi-view images. Different from previous methods that leverage extra information such as depth or generalizable features across scenes, our approach leverages the scene properties embedded in the multi-view inputs to create precise pseudo-labels for optimization without any prior training. Specifically, we introduce a geometry-guided approach that improves surface reconstruction accuracy from sparse views by leveraging spherical harmonics to predict the novel radiance while holistically considering all color observations for a point in the scene. Also, our pipeline exploits proxy geometry and correctly handles the occlusion in generating the pseudo-labels of radiance, which previous image-warping methods fail to avoid. Our method, dubbed Ray Augmentation (RayAug), achieves superior results on DTU and Blender datasets without requiring prior training, demonstrating its effectiveness in addressing the problem of sparse view reconstruction. Our pipeline is flexible and can be integrated into other implicit neural reconstruction methods for sparse views.
This paper explores the potential of Physics-Informed Neural Networks (PINNs) to serve as Reduced Order Models (ROMs) for simulating the flow field within stirred tank reactors (STRs). We solve the two-dimensional stationary Navier-Stokes equations within a geometrically intricate domain and explore methodologies that allow us to integrate additional physical insights into the model. These approaches include imposing the Dirichlet boundary conditions (BCs) strongly and employing domain decomposition (DD), with both overlapping and non-overlapping subdomains. We adapt the Extended Physics-Informed Neural Network (XPINN) approach to solve different sets of equations in distinct subdomains based on the diverse flow characteristics present in each region. Our exploration results in a hierarchy of models spanning various levels of complexity, where the best models exhibit l1 prediction errors of less than 1% for both pressure and velocity. To illustrate the reproducibility of our approach, we track the errors over repeated independent training runs of the best identified model and show its reliability. Subsequently, by incorporating the stirring rate as a parametric input, we develop a fast-to-evaluate model of the flow capable of interpolating across a wide range of Reynolds numbers. Although we exclusively restrict ourselves to STRs in this work, we conclude that the steps taken to obtain the presented model hierarchy can be transferred to other applications.
In this paper, we study a new problem, Film Removal (FR), which attempts to remove the interference of wrinkled transparent films and reconstruct the original information under films for industrial recognition systems. We first physically model the imaging of industrial materials covered by the film. Considering the specular highlight from the film can be effectively recorded by the polarized camera, we build a practical dataset with polarization information containing paired data with and without transparent film. We aim to remove interference from the film (specular highlights and other degradations) with an end-to-end framework. To locate the specular highlight, we use an angle estimation network to optimize the polarization angle with the minimized specular highlight. The image with minimized specular highlight is set as a prior for supporting the reconstruction network. Based on the prior and the polarized images, the reconstruction network can decouple all degradations from the film. Extensive experiments show that our framework achieves SOTA performance in both image reconstruction and industrial downstream tasks. Our code will be released at \url{//github.com/jqtangust/FilmRemoval}.
In this paper, we study a facility location problem within a competitive market context, where customer demand is predicted by a random utility choice model. Unlike prior research, which primarily focuses on simple constraints such as a cardinality constraint on the number of selected locations, we introduce routing constraints that necessitate the selection of locations in a manner that guarantees the existence of a tour visiting all chosen locations while adhering to a specified tour length upper bound. Such routing constraints find crucial applications in various real-world scenarios. The problem at hand features a non-linear objective function, resulting from the utilization of random utilities, together with complex routing constraints, making it computationally challenging. To tackle this problem, we explore three types of valid cuts, namely, outer-approximation and submodular cuts to handle the nonlinear objective function, as well as sub-tour elimination cuts to address the complex routing constraints. These lead to the development of two exact solution methods: a nested cutting plane and nested branch-and-cut algorithms, where these valid cuts are iteratively added to a master problem through two nested loops. We also prove that our nested cutting plane method always converges to optimality after a finite number of iterations. Furthermore, we develop a local search-based metaheuristic tailored for solving large-scale instances and show its pros and cons compared to exact methods. Extensive experiments are conducted on problem instances of varying sizes, demonstrating that our approach excels in terms of solution quality and computation time when compared to other baseline approaches.
In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the distance selection problem is to find the $k$-th smallest interpoint distance among all pairs of points of $P$. The previously best deterministic algorithm solves the problem in $O(n^{4/3} \log^2 n)$ time [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm to $O(n^{4/3} \log n)$ time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fr\'{e}chet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014] by a factor of roughly $\log^2(m+n)$ (resp., $(m+n)^{\epsilon}$), where $m$ and $n$ are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future.
In this letter, we proved a matrix identity of Hankel matrices that seems unrevealed before, generated from the moments of Gaussian distributions. In particular, we derived the Cholesky decompositions of the Hankel matrices in closed-forms, and showed some interesting connections between them. The results have potential applications in such as optimizing a nonlinear (NL) distortion function that maximizes the receiving gain in wireless communication systems.
Current studies on human locomotion focus mainly on solid ground walking conditions. In this paper, we present a biomechanic comparison of human walking locomotion on solid ground and sand. A novel dataset containing 3-dimensional motion and biomechanical data from 20 able-bodied adults for locomotion on solid ground and sand is collected. We present the data collection methods and report the sensor data along with the kinematic and kinetic profiles of joint biomechanics. A comprehensive analysis of human gait and joint stiffness profiles is presented. The kinematic and kinetic analysis reveals that human walking locomotion on sand shows different ground reaction forces and joint torque profiles, compared with those patterns from walking on solid ground. These gait differences reflect that humans adopt motion control strategies for yielding terrain conditions such as sand. The dataset also provides a source of locomotion data for researchers to study human activity recognition and assistive devices for walking on different terrains.
In this paper, we propose a novel multi-task learning architecture, which incorporates recent advances in attention mechanisms. Our approach, the Multi-Task Attention Network (MTAN), consists of a single shared network containing a global feature pool, together with task-specific soft-attention modules, which are trainable in an end-to-end manner. These attention modules allow for learning of task-specific features from the global pool, whilst simultaneously allowing for features to be shared across different tasks. The architecture can be built upon any feed-forward neural network, is simple to implement, and is parameter efficient. Experiments on the CityScapes dataset show that our method outperforms several baselines in both single-task and multi-task learning, and is also more robust to the various weighting schemes in the multi-task loss function. We further explore the effectiveness of our method through experiments over a range of task complexities, and show how our method scales well with task complexity compared to baselines.
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