Numerical Experiments Research Articles

Elliptic equations in weak oscillatory thin domains: Beyond periodicity with boundary‐concentrated reaction terms

In this paper, we analyze the limit behavior of a family of solutions of the Laplace operator with homogeneous Neumann boundary conditions, set in a two‐dimensional thin domain that presents weak oscillations on both boundaries and with terms concentrated in a narrow oscillating neighborhood of the top boundary. The aim of this problem is to study the behavior of the solutions as the thin domain presents oscillatory behaviors beyond the classical periodic assumptions,including scenarios such as quasi‐periodic or almost‐periodic oscillations. We then prove that the family of solutions converges to the solution of a one‐dimensional limit equation capturing the geometry and oscillatory behavior of boundary of the domain and the narrow strip where the concentration terms take place. In addition, we include a series of numerical experiments illustrating the theoretical results obtained in the quasi‐periodic context.

The nearest graph Laplacian in Frobenius norm

AbstractWe address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush–Kuhn–Tucker points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V‐FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms.

Improving detection accuracy of extreme-few-pixel Shack-Hartmann wavefront sensor based on tilt modulation

Conventional high spatial resolution Shack-Hartmann wavefront sensors have difficulty realizing high-precision wavefront detection under extreme-few-pixel conditions. A method of Shack-Hartmann wavefront sensor based on tilt modulation is proposed. The proposed method first increases the amount of sub-spot image information. Then, the corresponding wavefront detection accuracy can be improved by tuning parameters such as modulation amplitude, modulation angle, and modulation number. Finally, the detection accuracy for different parameter settings is fitted statistically. As observed in numerical simulations and experiments, the tilt modulation-based method effectively improves detection accuracy. Moreover, there are obvious advantages over the conventional Shack-Hartmann wavefront sensors in 4 × 4 sub-aperture pixels and a short focal length of the microlens array, which improves detection accuracy by about 84% in the experiments. Realization of the need for high spatial resolution and high detection accuracy of Shack-Hartmann wavefront sensors.

An Algorithm for Ship Detection in Complex Observation Scenarios Based on Mooring Buoys

Marine anchor buoys, as fixed-point profile observation platforms, are highly susceptible to the threat of ship collisions. Installing cameras on buoys can effectively monitor and collect evidence from ships. However, when using a camera to capture images, it is often affected by the continuous shaking of buoys and rainy and foggy weather, resulting in problems such as blurred images and rain and fog occlusion. To address these problems, this paper proposes an improved YOLOv8 algorithm. Firstly, the polarized self-attention (PSA) mechanism is introduced to preserve the high-resolution features of the original deep convolutional neural network and solve the problem of image spatial resolution degradation caused by shaking. Secondly, by introducing the multi-head self-attention (MHSA) mechanism in the neck network, the interference of rain and fog background is weakened, and the feature fusion ability of the network is improved. Finally, in the head network, this model combines additional small object detection heads to improve the accuracy of small object detection. Additionally, to enhance the algorithm’s adaptability to camera detection scenarios, this paper simulates scenarios, including shaking blur, rain, and foggy conditions. In the end, numerous comparative experiments on a self-made dataset show that the algorithm proposed in this study achieved 94.2% mAP50 and 73.2% mAP50:95 in various complex environments, which is superior to other advanced object detection algorithms.

Dynamic antifouling strategies of soft corals: Disrupting the flow field by tentacles

As permanently attached organism, soft corals can prevent fouling organisms from attaching to them through its anti-fouling mechanisms. In this paper, the antifouling mechanism of soft coral tentacles was revealed through the combination of numerical simulation and laboratory experiment. The results indicated that the interaction between tentacle structures and fluid medium affects the distribution and adhesion of bacteria. At the microscopic level, the boundary layer of low normal flow velocity forms a "water film" near the tentacles, preventing bacteria from approaching the tentacle surface. From the macro perspective, coral tentacles can disrupt the flow field through vortex streets, disrupt water flow stability, and make it difficult for bacteria to attach. The new discovery of bacterial movement and attachment behavior in the flow around a circular cylinder provides theoretical guidance for the design of antifouling surfaces.

A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests on (1) discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and (2) discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses. Numerical experiments complete the theoretical results.

UAdam: Unified Adam-Type Algorithmic Framework for Nonconvex Optimization.

Adam-type algorithms have become a preferred choice for optimization in the deep learning setting; however, despite their success, their convergence is still not well understood. To this end, we introduce a unified framework for Adam-type algorithms, termed UAdam. It is equipped with a general form of the second-order moment, which makes it possible to include Adam and its existing and future variants as special cases, such as NAdam, AMSGrad, AdaBound, AdaFom, and Adan. The approach is supported by a rigorous convergence analysis of UAdam in the general nonconvex stochastic setting, showing that UAdam converges to the neighborhood of stationary points with a rate of O(1/T). Furthermore, the size of the neighborhood decreases as the parameter β1 increases. Importantly, our analysis only requires the first-order momentum factor to be close enough to 1, without any restrictions on the second-order momentum factor. Theoretical results also reveal the convergence conditions of vanilla Adam, together with the selection of appropriate hyperparameters. This provides a theoretical guarantee for the analysis, applications, and further developments of the whole general class of Adam-type algorithms. Finally, several numerical experiments are provided to support our theoretical findings.

Optimal error estimates of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L2- and H1-norms are derived for the problem with initial data u0∈H01(Ω)∩H2(Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H1-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L∞ error estimate is obtained for 2D problems that too with the initial condition is in H01(Ω)∩H2(Ω). All results proved in this paper are valid uniformly as α→1−, where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

Approximation of the Hilbert transform on the half–line

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

On the properties of multivariate isotropic Random fields on the Ball

We consider multivariate isotropic random fields on the ball Bd. We first study their regularity properties in terms of Sobolev spaces. We further derive conditions guaranteeing the Hölder continuity of their covariance kernels and we prove the existence of sample Hölder continuous modifications for Gaussian random fields. Furthermore, we measure the error of truncated approximations of the corresponding series' representations. Moreover our developments are supported by numerical experiments. The majority of our results are new for multivariate random fields indexed over other domains, too. We express some of them for the case of the sphere.

Distributed Frank-Wolfe Solver for Stochastic Optimization With Coupled Inequality Constraints.

Distributed stochastic optimization (DSO) with local set constraints and coupled inequality constraints over a multiagent network is considered in this article. Usually, such problems are tackled by projected primal-dual methods, which require expensive projection operations when set constraints are complicated. In this context, this article focuses on the Frank-Wolfe (FW) framework, which provides computational simplicity by avoiding expensive projection operations, for solving DSO with local set and coupled inequality constraints. By combining recursive momentum and weighted averaging, this article proposes a distributed stochastic FW primal-dual algorithm (DSFWPD), which is the first stochastic FW solver for DSO problems with coupled constraints. The proposed algorithm achieves zero constraint violation on average with a sublinear decay of the optimality gap over a directed and time-varying network. The efficacy of DSFWPD is demonstrated by several numerical experiments.

Flux globalization-based well-balanced path-conservative central-upwind scheme for two-dimensional two-layer thermal rotating shallow water equations

We develop a flux globalization-based well-balanced path-conservative central-upwind scheme on Cartesian meshes for the two-dimensional (2-D) two-layer thermal rotating shallow water equations. The scheme is well-balanced in the sense that it can exactly preserve a variety of physically relevant steady states. In the 2-D case, preserving general “moving-water” steady states is difficult, and to the best of our knowledge, none of existing schemes can achieve this ultimate goal. The proposed scheme can exactly preserve the x- and y-directional jets in the rotational frame as well as certain genuinely 2-D equilibria. Numerical experiments demonstrate the performance of the proposed scheme in computationally non-trivial situations: in the presence of shocks, dry areas, non-trivial topographies, including discontinuous ones, and in the case of hyperbolicity loss. The scheme works equally well in both the f-plane and beta-plane frameworks.

Tolling scheme for traffic flow networks: modelling and control of positive systems

This article addresses modelling and control of traffic flow networks for congestion mitigation. We first model the traffic flow of an urban area and derive a state-space model composed of multiple compartments. Each compartment expresses the dynamics of the traffic demand that depends on drivers' decision-making on route choices influenced by tolling. Two practical properties are found in the proposed state space model: one is the positivity of the model, and the other is the conservation law in the number of vehicles. Based on the positivity of the model, we next propose the optimal tolling design aiming at the peak demand mitigation, which is expressed by nonlinear programming. We transform the nonlinear programming into a linear one, which is numerically tractable. Finally, we show that the proposed tolling design mitigates the peak demand in designated compartments in a numerical experiment and present a demonstration of the proposed control system.

Topology optimization of blazed gratings under conical incidence

A topology optimization method is presented and applied to a blazed diffraction grating in reflection under conical incidence. This type of grating is meant to disperse the incident light on one particular diffraction order, and this property is fundamental in spectroscopy. Conventionally, a blazed metallic grating is made of a sawtooth profile designed to work with the ±1st diffraction order in reflection. In this paper, we question this intuitive triangular pattern and look for optimal opto-geometric characteristics using topology optimization based on finite element modelling of Maxwell’s equations. In practical contexts, the grating geometry is mono-periodic, but it is enlightened by a 3D plane wave with a wave vector outside of the plane of invariance. Consequently, this study deals with the resolution of direct and inverse problems using the finite element method in this intermediate state between 2D and 3D: the so-called conical incidence. A multi-wavelength objective is used in order to obtain a broadband blazed effect. Finally, several numerical experiments are detailed. Our numerical results show that it is possible to reach a 98% diffraction efficiency on the −1st diffraction order if the optimization is performed on a single wavelength, and that the reflection integrated over the [400,1500] nm wavelength range can be 29% higher in absolute terms, 56% in relative terms, than that of the sawtooth blazed grating when using a multi-wavelength optimization criterion (from 52% to 81%).

Periodic spiking pulse formation in broadband nonlinear optoelectronic oscillator

An approach to generating periodic spiking pulses in a broadband opto-electronic oscillator (OEO) without external injection is proposed and demonstrated. Through biasing the electro-optic Mach-Zehnder modulator (MZM) in the OEO cavity at its nonlinear working point, spiking pulses are excited by the self-generated chaos under the combined action of the loop filter response time and the nonlinear excitation effect. An interaction force is introduced between pulses. As a result, the pulse number in a single roundtrip duration and the temporal distribution of the generated spiking pulses can be tuned by adjusting the loop gain and the direct-current (DC) bias voltage of the MZM. Both numerical simulation and experiment are carried out to demonstrate the spiking pulse generation mechanism, where ultra-short spiking pulse trains with even and uneven temporal distribution are generated.

Adaptive neural event-dependent intermittent fault-tolerant control of reaction–diffusion multi-agent systems

This article employs an event-dependent intermittent fault-tolerant control approach to address the adaptive consensus of reaction–diffusion multi-agent systems under directed topologies in both leaderless and leader-following networks. Initially, an event-dependent intermittent control mechanism is proposed to determine the work/rest time, relying on three partitions of the non-negative real region and the formulated Lyapunov function. Subsequently, under the introduced control mechanism, an adaptive consensus protocol is established, incorporating the updating law for adaptive coupling strength and adaptive weight estimation. Utilizing the neural network estimation theorem for fault estimation, the article presents explicit criteria to ensure the attainment of leaderless or leader-following consensus in reaction–diffusion multi-agent systems. Finally, two numerical simulation experiments are conducted to validate the robustness and effectiveness of the derived theorem.

Fast hybrid method for solving variational inequalities beyond monotonicity in real Banach spaces

In this work, we introduce a new inertial hybrid algorithm that combines the subgradient extragradient, Halpern, and S-iteration methods. We provide an approximate solution to variational inequality and common fixed point problem of relatively non-expansive mappings in real Banach spaces, where our cost operator is Lipschitz continuous and quasimonotone, and our step size is self-adaptive, non-monotone, and non-reliant on any line-search principle. We provide two strong convergence results with (and without) recourse to the monotonicity property, and further conduct several relevant numerical experiments to reveal the efficiency of our method over others in the literature.

Simulate the elastic wavefields in media with an irregular surface topography based on staggered grid finite difference

Abstract Wave equation forward modeling is a useful method to study the propagation regulation of seismic wavefields. Finite difference (FD) is one of the most extensively employed numerical approaches for computing wavefields in earthquake and exploration seismology. However, the FD approach relying on regular grids often struggles with calculating wavefields in regions featuring surface topographies. The elastic wave equation can more accurately describe the propagation of seismic wavefields in elastic media compared to the acoustic wave equation. We introduce a new FD scheme to calculate the elastic wavefields in an isotropic model with a surface topography. The novel approach can use a conventional staggered grid FD(SGFD) approach based on regular grids. A new elastic model with a horizontal surface is first obtained from the nearby surface's elastic properties and the undulating terrain elevation. We subsequently employ a topography-related strategy to eliminate the effects of surface topographies on the seismic wavefields in models with irregular surface topographies. The merits of our proposed scheme lie in its ability to stable numerically compute wavefields in models with irregular surface topographies without altering the conventional SGFD relying on regular grids. To validate the effectiveness and practicality of our method, we utilize elastic models featuring complex surface topographies. Numerical experiments demonstrate that our approach efficiently calculates elastic wavefields in isotropic media with irregular topographies based on conventional SGFD.

Neural interval‐censored survival regression with feature selection

AbstractSurvival analysis is a fundamental area of focus in biomedical research, particularly in the context of personalized medicine. This prominence is due to the increasing prevalence of large and high‐dimensional datasets, such as omics and medical image data. However, the literature on nonlinear regression algorithms and variable selection techniques for interval‐censoring is either limited or nonexistent, particularly in the context of neural networks. Our objective is to introduce a novel predictive framework tailored for interval‐censored regression tasks, rooted in Accelerated Failure Time (AFT) models. Our strategy comprises two key components: (i) a variable selection phase leveraging recent advances on sparse neural network architectures; (ii) a regression model targeting prediction of the interval‐censored response. To assess the performance of our novel algorithm, we conducted a comprehensive evaluation through both numerical experiments and real‐world applications that encompass scenarios related to diabetes and physical activity. Our results outperform traditional AFT algorithms, particularly in scenarios featuring nonlinear relationships.

Uncertainty modeling and propagation for groundwater flow: a comparative study of surrogates

We compare sparse grid stochastic collocation and Gaussian process emulation as surrogates for the parameter-to-observation map of a groundwater flow problem related to the Waste Isolation Pilot Plant in Carlsbad, NM. The goal is the computation of the probability distribution of a contaminant particle travel time resulting from uncertain knowledge about the transmissivity field. The latter is modelled as a lognormal random field which is fitted by restricted maximum likelihood estimation and universal kriging to observational data as well as geological information including site-specific trend regression functions obtained from technical documentation. The resulting random transmissivity field leads to a random groundwater flow and particle transport problem which is solved realization-wise using a mixed finite element discretization. Computational surrogates, once constructed, allow sampling the quantities of interest in the uncertainty analysis at substantially reduced computational cost. Special emphasis is placed on explaining the differences between the two surrogates in terms of computational realization and interpretation of the results. Numerical experiments are given for illustration.