Part 1 of this paper provides a comprehensive guide to generating unconstrained, simplicial, four-dimensional (4D), hypervolume meshes. While a general procedure for constructing unconstrained n-dimensional Delaunay meshes is well-known, many of the explicit implementation details are missing from the relevant literature for cases in which n >= 4. This issue is especially critical for the case in which n = 4, as the resulting meshes have important space-time applications. As a result, the purpose of this paper is to provide explicit descriptions of the key components in a 4D mesh-generation algorithm: namely, the point-insertion process, geometric predicates, element quality metrics, and bistellar flips. This paper represents a natural continuation of the work which was pioneered by Anderson et al. in "Surface and hypersurface meshing techniques for space-time finite element methods", Computer-Aided Design, 2023. In this previous paper, hypersurface meshes were generated using a novel, trajectory-tracking procedure. In the current paper, we are interested in generating coarse, 4D hypervolume meshes (boundary meshes) which are formed by sequentially inserting points from an existing hypersurface mesh. In the latter portion of this paper, we present numerical experiments which demonstrate the viability of this approach for a simple, convex domain. Although, our main focus is on the generation of hypervolume boundary meshes, the techniques described in this paper are broadly applicable to a much wider range of 4D meshing methods. We note that the more complex topics of constrained hypervolume meshing, and boundary recovery for non-convex domains will be covered in Part 2 of the paper.
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In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.
We introduce Optimistix: a nonlinear optimisation library built in JAX and Equinox. Optimistix introduces a novel, modular approach for its minimisers and least-squares solvers. This modularity relies on new practical abstractions for optimisation which we call search and descent, and which generalise classical notions of line search, trust-region, and learning-rate algorithms. It provides high-level APIs and solvers for minimisation, nonlinear least-squares, root-finding, and fixed-point iteration. Optimistix is available at //github.com/patrick-kidger/optimistix.
The context of this paper is the creation of large uniform archaeological datasets from heterogeneous published resources, such as find catalogues - with the help of AI and Big Data. The paper is concerned with the challenge of consistent assemblages of archaeological data. We cannot simply combine existing records, as they differ in terms of quality and recording standards. Thus, records have to be recreated from published archaeological illustrations. This is only a viable path with the help of automation. The contribution of this paper is a new workflow for collecting data from archaeological find catalogues available as legacy resources, such as archaeological drawings and photographs in large unsorted PDF files; the workflow relies on custom software (AutArch) supporting image processing, object detection, and interactive means of validating and adjusting automatically retrieved data. We integrate artificial intelligence (AI) in terms of neural networks for object detection and classification into the workflow, thereby speeding up, automating, and standardising data collection. Objects commonly found in archaeological catalogues - such as graves, skeletons, ceramics, ornaments, stone tools and maps - are detected. Those objects are spatially related and analysed to extract real-life attributes, such as the size and orientation of graves based on the north arrow and the scale. We also automate recording of geometric whole-outlines through contour detection, as an alternative to landmark-based geometric morphometrics. Detected objects, contours, and other automatically retrieved data can be manually validated and adjusted. We use third millennium BC Europe (encompassing cultures such as 'Corded Ware' and 'Bell Beaker', and their burial practices) as a 'testing ground' and for evaluation purposes; this includes a user study for the workflow and the AutArch software.
We analyze a Discontinuous Galerkin method for a problem with linear advection-reaction and $p$-type diffusion, with Sobolev indices $p\in (1, \infty)$. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on local P\'eclet numbers. A set of numerical tests supports the theoretical derivations.
Motivated by the important statistical role of sparsity, the paper uncovers four reparametrizations for covariance matrices in which sparsity is associated with conditional independence graphs in a notional Gaussian model. The intimate relationship between the Iwasawa decomposition of the general linear group and the open cone of positive definite matrices allows a unifying perspective. Specifically, the positive definite cone can be reconstructed without loss or redundancy from the exponential map applied to four Lie subalgebras determined by the Iwasawa decomposition of the general linear group. This accords geometric interpretations to the reparametrizations and the corresponding notion of sparsity. Conditions that ensure legitimacy of the reparametrizations for statistical models are identified. While the focus of this work is on understanding population-level structure, there are strong methodological implications. In particular, since the population-level sparsity manifests in a vector space, imposition of sparsity on relevant sample quantities produces a covariance estimate that respects the positive definite cone constraint.
Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems
We present an information-theoretic lower bound for the problem of parameter estimation with time-uniform coverage guarantees. Via a new a reduction to sequential testing, we obtain stronger lower bounds that capture the hardness of the time-uniform setting. In the case of location model estimation, logistic regression, and exponential family models, our $\Omega(\sqrt{n^{-1}\log \log n})$ lower bound is sharp to within constant factors in typical settings.
The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. For a fixed $H$, the input of the optimization problem LHomVD($H$) is a graph $G$ with lists $L(v)$, and the task is to find a set $X$ of vertices having minimum size such that $(G-X,L)$ has a list homomorphism to $H$. We define analogously the edge-deletion variant LHomED($H$). This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs $H$ that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed $H$. Second, as our main result, we determine for every graph $H$ for which the problem is NP-hard, the smallest possible constant $c_H$ such that the problem can be solved in time $c^t_H\cdot n^{O(1)}$ if a tree decomposition of $G$ having width $t$ is given in the input.Let $i(H)$ be the maximum size of a set of vertices in $H$ that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD($H$), we show that the smallest possible constant is $i(H)+1$ for every $H$. The situation is more complex for the edge-deletion version. For every $H$, one can solve LHomED($H$) in time $i(H)^t\cdot n^{O(1)}$ if a tree decomposition of width $t$ is given. However, the existence of a specific type of decomposition of $H$ shows that there are graphs $H$ where LHomED($H$) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than $i(H)$. Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed $H$.
This paper introduces the generalized Hausman test as a novel method for detecting non-normality of the latent variable distribution of unidimensional Item Response Theory (IRT) models for binary data. The test utilizes the pairwise maximum likelihood estimator obtained for the parameters of the classical two-parameter IRT model, which assumes normality of the latent variable, and the quasi-maximum likelihood estimator obtained under a semi-nonparametric framework, allowing for a more flexible distribution of the latent variable. The performance of the generalized Hausman test is evaluated through a simulation study and it is compared with the likelihood-ratio and the M2 test statistics. Additionally, various information criteria are computed. The simulation results show that the generalized Hausman test outperforms the other tests under most conditions. However, the results obtained from the information criteria are somewhat contradictory under certain conditions, suggesting a need for further investigation and interpretation.
This paper focuses on the inverse elastic impedance and the geometry problem by a Cauchy data pair on the access part of the boundary in a two-dimensional case. Through the decomposition of the displacement, the problem is transform the solution of into a coupled boundary value problem that involves two scalar Helmholtz equations. Firstly, a uniqueness result is given, and a non-iterative algorithm is proposed to solve the data completion problem using a Cauchy data pair on a known part of the solution domain's boundary. Next, we introduce a Newton-type iterative method for reconstructing the boundary and the impedance function using the completion data on the unknown boundary, which is governed by a specific type of boundary conditions. Finally, we provide several examples to demonstrate the effectiveness and accuracy of the proposed method.
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