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In this paper, the surface of revolution discrete element method (SR-DEM) is introduced to simulate systems of particles with closed surfaces of revolution. Due to the cylindrical symmetry of a surface of revolution, the geometry of any cross-section about the axis of rotation remains the same. Taking advantage of this geometric feature, a node-to-cross-section contact algorithm is proposed for efficient contact detection between particles with a surface of revolution. In our SR-DEM framework, the contact algorithm is realized in a master-slave fashion: the master particle is approximated by its surface nodes, while the slave particle is represented by a signed distance field (SDF) of the cross-section about the axis of rotation. This hybrid formulation in both 2D and 3D space allows a very efficient contact calculation yet relatively simple code implementation. We then apply SR-DEM to simulate particle-particle, particle-wall impact, granular packing in a cylindrical container, and tablets in a rotating drum, to demonstrate SR-DEM's ability to predict the post-impact velocities, packing porosity, and dynamic angle of repose, respectively. Finally, we suggest a simple approach to find an optimal surface resolution, by increasing the number of surface nodes until some of the bulk properties that could characterize the system converge.

We study least-squares trace regression when the parameter is the sum of a $r$-low-rank matrix and a $s$-sparse matrix and a fraction $\epsilon$ of the labels is corrupted. For subgaussian distributions and feature-dependent noise, we highlight three needed design properties, each one derived from a different process inequality: a "product process inequality", "Chevet's inequality" and a "multiplier process inequality". These properties handle, simultaneously, additive decomposition, label contamination and design-noise interaction. They imply the near-optimality of a tractable estimator with respect to the effective dimensions $d_{eff,r}$ and $d_{eff,s}$ of the low-rank and sparse components, $\epsilon$ and the failure probability $\delta$. The near-optimal rate is $\mathsf{r}(n,d_{eff,r}) + \mathsf{r}(n,d_{eff,s}) + \sqrt{(1+\log(1/\delta))/n} + \epsilon\log(1/\epsilon)$, where $\mathsf{r}(n,d_{eff,r})+\mathsf{r}(n,d_{eff,s})$ is the optimal rate in average with no contamination. Our estimator is adaptive to $(s,r,\epsilon,\delta)$ and, for fixed absolute constant $c>0$, it attains the mentioned rate with probability $1-\delta$ uniformly over all $\delta\ge\exp(-cn)$. Without matrix decomposition, our analysis also entails optimal bounds for a robust estimator adapted to the noise variance. Our estimators are based on "sorted" versions of Huber's loss. We present simulations matching the theory. In particular, it reveals the superiority of "sorted" Huber's losses over the classical Huber's loss.

We consider a non-isothermal compositional gas liquid model for the simulation of well operations in geothermal processes. The model accounts for phase transitions assumed to be at thermodynamical equilibrium and is based on an hydrodynamical Drift Flux Model (DFM) combined with a No Pressure Wave approximation of the momentum equation. The focus of this work is on the design of a robust discretization accounting for slanted and multibranch wells with the ability to simulate both transient behavior such as well opening as well as coupled simulations at the time scale of the reservoir. It is based on a staggered finite volume scheme in space combined with a fully implicit Euler time integration. The construction of consistent and stable numerical fluxes is a key feature for a robust numerical method. It is achieved by combining a monotone flux approximation for the phase superficial velocities with an upwind approximation of the phase molar fractions, density and enthalpy. In order to facilitate the coupling of the well and reservoir models, the Newton linearization accounts for the elimination of the hydrodynamical unknowns leading to Jacobian systems using the same primary unknowns than those of the reservoir model. The efficiency of our approach is investigated on both stand alone well test cases without and with cross flow, and on a fully coupled well-reservoir simulation.

The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Tur\'an problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.

We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due to discontinuities in source term. The problem is posed on the unit square. The second derivative is multiplied by a singular perturbation parameter, $\epsilon$, while the nature of the first derivative term is such that flow is aligned with a boundary. These two facts mean that solutions tend to exhibit layers of both exponential and characteristic type. We solve the problem using a finite difference method, specially adapted to the discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that that the computed solution converges to the true one at a rate that is independent of the perturbation parameter, and is nearly first-order. We present numerical results that verify that these results are sharp.

We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu$ is large, the error is $O(h^{k+1}+\Delta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.

We introduce a novel sampler called the energy based diffusion generator for generating samples from arbitrary target distributions. The sampling model employs a structure similar to a variational autoencoder, utilizing a decoder to transform latent variables from a simple distribution into random variables approximating the target distribution, and we design an encoder based on the diffusion model. Leveraging the powerful modeling capacity of the diffusion model for complex distributions, we can obtain an accurate variational estimate of the Kullback-Leibler divergence between the distributions of the generated samples and the target. Moreover, we propose a decoder based on generalized Hamiltonian dynamics to further enhance sampling performance. Through empirical evaluation, we demonstrate the effectiveness of our method across various complex distribution functions, showcasing its superiority compared to existing methods.

Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1 \leq i \leq n$. The Randi\'c matrix of $G$, denoted by $R(G) = [r_{i,j}]$, is the $n \times n$ matrix whose $(i,j)$-entry $r_{i,j}$ is $r_{i,j} = 1/\sqrt{d_id_j}$ if $v_i$ and $v_j$ are adjacent in $G$ and 0 otherwise. A tree is a connected acyclic graph. A level-wise regular tree is a tree rooted at one vertex $r$ or two (adjacent) vertices $r$ and $r'$ in which all vertices with the minimum distance $i$ from $r$ or $r'$ have the same degree $m_i$ for $0 \leq i \leq h$, where $h$ is the height of $T$. In this paper, we give a complete characterization of the eigenvalues with their multiplicity of the Randi\'c matrix of level-wise regular trees. We prove that the eigenvalues of the Randi\'c matrix of a level-wise regular tree are the eigenvalues of the particular tridiagonal matrices, which are formed using the degree sequence $(m_0,m_1,\ldots,m_{h-1})$ of level-wise regular trees.

There is a folkloric belief that a depth-$\Theta(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.

We study the problem of estimating a function $T$ given independent samples from a distribution $P$ and from the pushforward distribution $T_\sharp P$. This setting is motivated by applications in the sciences, where $T$ represents the evolution of a physical system over time, and in machine learning, where, for example, $T$ may represent a transformation learned by a deep neural network trained for a generative modeling task. To ensure identifiability, we assume that $T = \nabla \varphi_0$ is the gradient of a convex function, in which case $T$ is known as an \emph{optimal transport map}. Prior work has studied the estimation of $T$ under the assumption that it lies in a H\"older class, but general theory is lacking. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure $P$ satisfy a Poincar\'e inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for H\"older transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when $P$ is the normal distribution and the transport map is given by an infinite-width shallow neural network.