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Targeting simulations on parallel hardware architectures, this paper presents computational kernels for efficient computations in mortar finite element methods. Mortar methods enable a variationally consistent imposition of coupling conditions at high accuracy, but come with considerable numerical effort and cost for the evaluation of the mortar integrals to compute the coupling operators. In this paper, we identify bottlenecks in parallel data layout and domain decomposition that hinder an efficient evaluation of the mortar integrals. We then propose a set of computational strategies to restore optimal parallel communication and scalability for the core kernels devoted to the evaluation of mortar terms. We exemplarily study the proposed algorithmic components in the context of three-dimensional large-deformation contact mechanics, both for cases with fixed and dynamically varying interface topology, yet these concepts can naturally and easily be transferred to other mortar applications, e.g. classical meshtying problems. To restore parallel scalability, we employ overlapping domain decompositions of the interface discretization independent from the underlying volumes and then tackle parallel communication for the mortar evaluation by a geometrically motivated reduction of ghosting data. Using three-dimensional contact examples, we demonstrate strong and weak scalability of the proposed algorithms up to 480 parallel processes as well as study and discuss improvements in parallel communication related to mortar finite element methods. For the first time, dynamic load balancing is applied to mortar contact problems with evolving contact zones, such that the computational work is well balanced among all parallel processors independent of the current state of the simulation.

In this paper, both semidiscrete and fully discrete finite element methods are analyzed for the penalized two-dimensional unsteady Navier-Stokes equations with nonsmooth initial data. First order backward Euler method is applied for the time discretization, whereas conforming finite element method is used for the spatial discretization. Optimal $L^2$ error estimates for the semidiscrete as well as the fully discrete approximations of the velocity and of the pressure are derived for realistically assumed conditions on the data. The main ingredient in the proof is the appropriate exploitation of the inverse of the penalized Stokes operator, negative norm estimates and time weighted estimates. Numerical examples are discussed at the end which conform our theoretical results.

This paper presents a hybrid numerical method for linear collisional kinetic equations with diffusive scaling. The aim of the method is to reduce the computational cost of kinetic equations by taking advantage of the lower dimensionality of the asymptotic fluid model while reducing the error induced by the latter approach. It relies on two criteria motivated by a pertubative approach to obtain a dynamic domain decomposition. The first criterion quantifies how far from a local equilibrium in velocity the distribution function of particles is. The second one depends only on the macroscopic quantities that are available on the whole computing domain. Interface conditions are dealt with using a micro-macro decomposition and the method is significantly more efficient than a standard full kinetic approach. Some properties of the hybrid method are also investigated, such as the conservation of mass.

Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modelled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon's method or as the minimum variance method. Numerical examples illustrate our theoretical findings.

In this paper, we propose Nesterov Accelerated Shuffling Gradient (NASG), a new algorithm for the convex finite-sum minimization problems. Our method integrates the traditional Nesterov's acceleration momentum with different shuffling sampling schemes. We show that our algorithm has an improved rate of $\mathcal{O}(1/T)$ using unified shuffling schemes, where $T$ is the number of epochs. This rate is better than that of any other shuffling gradient methods in convex regime. Our convergence analysis does not require an assumption on bounded domain or a bounded gradient condition. For randomized shuffling schemes, we improve the convergence bound further. When employing some initial condition, we show that our method converges faster near the small neighborhood of the solution. Numerical simulations demonstrate the efficiency of our algorithm.

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, discretization, etc. in 1D and 2D. Our model outperforms state-of-the-art numerical solvers in the low resolution regime in terms of speed and accuracy.

An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction--diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $H^1$ norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.

Analog to Digital Converters (ADCs) are a major contributor to the energy consumption on the receiver side of millimeter-wave multiple-input multiple-output (MIMO) systems with large antenna arrays. Consequently, there has been significant interest in using low-resolution ADCs along with hybrid beam-forming at MIMO receivers for energy efficiency. However, decreasing the ADC resolution results in performance loss -- in terms of achievable rates -- due to increased quantization error. In this work, we study the application of practically implementable nonlinear analog operations, prior to sampling and quantization at the ADCs, as a way to mitigate the aforementioned rate-loss. A receiver architecture consisting of linear analog combiners, implementable nonlinear analog operators, and one-bit threshold ADCs is designed. The fundamental information theoretic performance limits of the resulting communication system, in terms of achievable rates, are investigated under various assumptions on the set of implementable nonlinear analog functions. In order to justify the feasibility of the nonlinear operations in the proposed receiver architecture, an analog circuit is introduced, and circuit simulations exhibiting the generation of the desired nonlinear analog operations are provided.

In this work, we present a modification of explicit Runge-Kutta temporal integration schemes that guarantees the preservation of any locally-defined quasiconvex set of bounds for the solution. These schemes operate on the basis of a bijective mapping between an admissible set of solutions and the real domain to strictly enforce bounds. Within this framework, we show that it is possible to recover a wide range of methods independently of the spatial discretization, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. Furthermore, these schemes are proven to recover the order of accuracy of the underlying Runge-Kutta method upon which they are built. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. We show the utility of this approach in numerical experiments using a pseudospectral spatial discretization without any explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.

Computations of incompressible flows with velocity boundary conditions require solution of a Poisson equation for pressure with all Neumann boundary conditions. Discretization of such a Poisson equation results in a rank-deficient matrix of coefficients. When a non-conservative discretization method such as finite difference, finite element, or spectral scheme is used, such a matrix also generates an inconsistency which makes the residuals in the iterative solution to saturate at a threshold level that depends on the spatial resolution and order of the discretization scheme. In this paper, we examine inconsistency for a high-order meshless discretization scheme suitable for solving the equations on a complex domain. The high order meshless method uses polyharmonic spline radial basis functions (PHS-RBF) with appended polynomials to interpolate scattered data and constructs the discrete equations by collocation. The PHS-RBF provides the flexibility to vary the order of discretization by increasing the degree of the appended polynomial. In this study, we examine the convergence of the inconsistency for different spatial resolutions and for different degrees of the appended polynomials by solving the Poisson equation for a manufactured solution as well as the Navier-Stokes equations for several fluid flows. We observe that the inconsistency decreases faster than the error in the final solution, and eventually becomes vanishing small at sufficient spatial resolution. The rate of convergence of the inconsistency is observed to be similar or better than the rate of convergence of the discretization errors. This beneficial observation makes it unnecessary to regularize the Poisson equation by fixing either the mean pressure or pressure at an arbitrary point. A simple point solver such as the SOR is seen to be well-convergent, although it can be further accelerated using multilevel methods.