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We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is the first step in reliably merging hybrid skeletal formulations and localized orthogonal decomposition to unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.

We propose a topological mapping and localization system able to operate on real human colonoscopies, despite significant shape and illumination changes. The map is a graph where each node codes a colon location by a set of real images, while edges represent traversability between nodes. For close-in-time images, where scene changes are minor, place recognition can be successfully managed with the recent transformers-based local feature matching algorithms. However, under long-term changes -- such as different colonoscopies of the same patient -- feature-based matching fails. To address this, we train on real colonoscopies a deep global descriptor achieving high recall with significant changes in the scene. The addition of a Bayesian filter boosts the accuracy of long-term place recognition, enabling relocalization in a previously built map. Our experiments show that ColonMapper is able to autonomously build a map and localize against it in two important use cases: localization within the same colonoscopy or within different colonoscopies of the same patient. Code: //github.com/jmorlana/ColonMapper.

A new approach based on censoring and moment criterion is introduced for parameter estimation of count distributions when the probability generating function is available even though a closed form of the probability mass function and/or finite moments do not exist.

The purpose of the study was to find the true comfort of the wearer by conceptualizing, formulating, and proving the relation between physiological and emotional parameters with clothing fit and fabric. A mixed-method research design was used, and the findings showed that physiological indicators such as heart rate are closely linked with user comfort. However, a significant change in emotional response indicated a definite relationship between different fabric and fit types. The research was conducted to discover the relation between true comfort parameters and clothing, which is unique to the field. The findings help us understand how fabric types and clothing fit types can affect physiological and emotional responses, providing the consumer with satisfactory clothing with the suitable properties needed.

This paper delves into the equivalence problem of Smith forms for multivariate polynomial matrices. Generally speaking, multivariate ($n \geq 2$) polynomial matrices and their Smith forms may not be equivalent. However, under certain specific condition, we derive the necessary and sufficient condition for their equivalence. Let $F\in K[x_1,\ldots,x_n]^{l\times m}$ be of rank $r$, $d_r(F)\in K[x_1]$ be the greatest common divisor of all the $r\times r$ minors of $F$, where $K$ is a field, $x_1,\ldots,x_n$ are variables and $1 \leq r \leq \min\{l,m\}$. Our key findings reveal the result: $F$ is equivalent to its Smith form if and only if all the $i\times i$ reduced minors of $F$ generate $K[x_1,\ldots,x_n]$ for $i=1,\ldots,r$.

We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $\mathcal{O}(\varepsilon)$ around the interface; this allows the equations to be posed uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretisation of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We in turn prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.

We provide a unified continuum formulation of linearized mechanics, Stokes' flow and poromechanics in terms of a conservation structure. Starting from this formulation, we construct corresponding simple and robust finite volume discretizations for these physical systems, based only on co-located, cell-centered variables. These discretizations have a minimal discretization stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We show well-posedness of a weak statement of the continuous formulation in appropriate Hilbert spaces, and identify the appropriate weighted norms for the problem. For the discrete approximations, we prove stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments in 3D support the theoretical results, and provide additional insight into the practical performance of the discretization.

We systematically validate the static local mesh refinement capabilities of a recently proposed IMEX-DG scheme implemented in the framework of the deal.II library. Non-conforming meshes are employed in atmospheric flow simulations to increase the resolution around complex orography. A number of numerical experiments based on classical benchmarks with idealized as well as real orography profiles demonstrate that simulations with the refined mesh are stable for long lead times and no spurious effects arise at the interfaces of mesh regions with different resolutions. Moreover, correct values of the momentum flux are retrieved and the correct large-scale orographic response is established. Hence, large-scale orography-driven flow features can be simulated without loss of accuracy using a much lower total amount of degrees of freedom. In a context of spatial resolutions approaching the hectometric scale in numerical weather prediction models, these results support the use of locally refined, non-conforming meshes as a reliable and effective tool to greatly reduce the dependence of atmospheric models on orographic wave drag parametrizations.

The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay $\tau$ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Gr\"onwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Gr\"onwall inequality, we obtain the pointwise-in-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval $((i-1)\tau,i\tau]$, although the convergence in time is low for $i=1$, it will be improved as the increasing $i$, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing $i$. Finally, we present some numerical tests to verify the developed theory.

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.