This paper introduces a novel method for the efficient second-order accurate computation of normal fields from volume fractions on unstructured polyhedral meshes. Locally, i.e. in each mesh cell, an averaged normal is reconstructed by fitting a plane in a least squares sense to the volume fraction data of neighboring cells while implicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by employing a Newton-type method. Moreover, applying the Reynolds transport theorem allows to assess the regularity of the derivatives. Since the divergence theorem implies that the volume fraction can be cast as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes. We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented, where we obtain second-order convergence for the normal alignment and symmetric volume difference. Moreover, the findings are substantiated by completely new deep insights into the minimization procedure.
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We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.
Ethereum Improvement Proposal (EIP) 1559 was recently implemented to transform Ethereum's transaction fee market. EIP-1559 utilizes an algorithmic update rule with a constant learning rate to estimate a base fee. The base fee reflects prevailing network conditions and hence provides a more reliable oracle for current gas prices. Using on-chain data from the period after its launch, we evaluate the impact of EIP-1559 on the user experience and market performance. Our empirical findings suggest that although EIP-1559 achieves its goals on average, short-term behavior is marked by intense, chaotic oscillations in block sizes (as predicted by our recent theoretical dynamical system analysis [1]) and slow adjustments during periods of demand bursts (e.g., NFT drops). Both phenomena lead to unwanted inter-block variability in mining rewards. To address this issue, we propose an alternative base fee adjustment rule in which the learning rate varies according to an additive increase, multiplicative decrease (AIMD) update scheme. Our simulations show that the latter robustly outperforms the EIP-1559 protocol under various demand scenarios. These results provide evidence that variable learning rate mechanisms may constitute a promising alternative to the default EIP-1559-based format and contribute to the ongoing discussion on the design of more efficient transaction fee markets.
Recent years have witnessed the improving performance of Chinese Named Entity Recognition (NER) from proposing new frameworks or incorporating word lexicons. However, the inner composition of entity mentions in character-level Chinese NER has been rarely studied. Actually, most mentions of regular types have strong name regularity. For example, entities end with indicator words such as "company" or "bank" usually belong to organization. In this paper, we propose a simple but effective method for investigating the regularity of entity spans in Chinese NER, dubbed as Regularity-Inspired reCOgnition Network (RICON). Specifically, the proposed model consists of two branches: a regularity-aware module and a regularityagnostic module. The regularity-aware module captures the internal regularity of each span for better entity type prediction, while the regularity-agnostic module is employed to locate the boundary of entities and relieve the excessive attention to span regularity. An orthogonality space is further constructed to encourage two modules to extract different aspects of regularity features. To verify the effectiveness of our method, we conduct extensive experiments on three benchmark datasets and a practical medical dataset. The experimental results show that our RICON significantly outperforms previous state-of-the-art methods, including various lexicon-based methods.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
Gaussian process regression is increasingly applied for learning unknown dynamical systems. In particular, the implicit quantification of the uncertainty of the learned model makes it a promising approach for safety-critical applications. When using Gaussian process regression to learn unknown systems, a commonly considered approach consists of learning the residual dynamics after applying some generic discretization technique, which might however disregard properties of the underlying physical system. Variational integrators are a less common yet promising approach to discretization, as they retain physical properties of the underlying system, such as energy conservation and satisfaction of explicit kinematic constraints. In this work, we present a novel structure-preserving learning-based modelling approach that combines a variational integrator for the nominal dynamics of a mechanical system and learning residual dynamics with Gaussian process regression. We extend our approach to systems with known kinematic constraints and provide formal bounds on the prediction uncertainty. The simulative evaluation of the proposed method shows desirable energy conservation properties in accordance with general theoretical results and demonstrates exact constraint satisfaction for constrained dynamical systems.
We introduce Universal Solution Manifold Network (USM-Net), a novel surrogate model, based on Artificial Neural Networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. Our method employs a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Indeed, we encode geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. Our approach is non-intrusive and modular, as we select a data-driven loss function. The latter can also be modified by considering additional constraints, thus leveraging available physical knowledge. Our approach can also accommodate a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.
This paper studies node classification in the inductive setting, i.e., aiming to learn a model on labeled training graphs and generalize it to infer node labels on unlabeled test graphs. This problem has been extensively studied with graph neural networks (GNNs) by learning effective node representations, as well as traditional structured prediction methods for modeling the structured output of node labels, e.g., conditional random fields (CRFs). In this paper, we present a new approach called the Structured Proxy Network (SPN), which combines the advantages of both worlds. SPN defines flexible potential functions of CRFs with GNNs. However, learning such a model is nontrivial as it involves optimizing a maximin game with high-cost inference. Inspired by the underlying connection between joint and marginal distributions defined by Markov networks, we propose to solve an approximate version of the optimization problem as a proxy, which yields a near-optimal solution, making learning more efficient. Extensive experiments on two settings show that our approach outperforms many competitive baselines.
In this paper, a third order gas kinetic scheme is developed on the three dimensional hybrid unstructured meshes for the numerical simulation of compressible inviscid and viscous flows. A third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A simple strategy is adopted for the selection of big stencil and sub-stencils. Incorporate with the two-stage fourth-order temporal discretization and lower-upper symmetric Gauss-Seidel methods, both explicit and implicit high-order gas-kinetic schemes are developed. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.
In the pooled data problem we are given a set of $n$ agents, each of which holds a hidden state bit, either $0$ or $1$. A querying procedure returns for a query set the sum of the states of the queried agents. The goal is to reconstruct the states using as few queries as possible. In this paper we consider two noise models for the pooled data problem. In the noisy channel model, the result for each agent flips with a certain probability. In the noisy query model, each query result is subject to random Gaussian noise. Our results are twofold. First, we present and analyze for both error models a simple and efficient distributed algorithm that reconstructs the initial states in a greedy fashion. Our novel analysis pins down the range of error probabilities and distributions for which our algorithm reconstructs the exact initial states with high probability. Secondly, we present simulation results of our algorithm and compare its performance with approximate message passing (AMP) algorithms that are conjectured to be optimal in a number of related problems.
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