Variational Method Research Articles

A comparison of two nonlinear data assimilation methods

Abstract. Advanced numerical data assimilation (DA) methods, such as the four-dimensional variational (4DVAR) method, are elaborate and computationally expensive. Simpler methods exist that take time variability into account, providing the potential of accurate results with a reduced computational cost. Recently, two of these DA methods were proposed for a nonlinear ocean model, an implementation which is costly in time and expertise, developing the need to first evaluate a simpler comparison between these two nonlinear methods. The first method is diffusive back-and-forth nudging (D-BFN), which has previously been implemented in several complex models, most specifically an ocean model. The second is the concave–convex nonlinearity (CCN) method that has a straightforward implementation and promising results with a toy model. D-BFN is less costly than a traditional variational DA system but requires an iterative implementation of equations that integrate the nonlinear model forward and backward in time, whereas CCN only requires integration of the nonlinear model forward in time. This paper investigates if the CCN algorithm can provide competitive results with the already tested D-BFN within simple chaotic models. Results show that observation density and/or frequency, as well as the length of the experiment window, significantly impacts the results for CCN, whereas D-BFN is fairly robust to sparser observations, predominately in time.

Determinant- and derivative-free quantum Monte Carlo within the stochastic representation of wavefunctions

Abstract Describing the ground states of continuous, real-space quantum many-body systems, like atoms and molecules, is a significant computational challenge with applications throughout the physical sciences. Recent progress was made by variational methods based on machine learning (ML) ansatzes. However, since these approaches are based on energy minimization, ansatzes must be twice differentiable. This (a) precludes the use of many powerful classes of ML models; and (b) makes the enforcement of bosonic, fermionic, and other symmetries costly. Furthermore, (c) the optimization procedure is often unstable unless it is done by imaginary time propagation, which is often impractically expensive in modern ML models with many parameters. The stochastic representation of wavefunctions (SRW), introduced in (Atanasova et al 2023 Nat. Commun. 14 3601), is a recent approach to overcoming (c). SRW enables imaginary time propagation at scale, and makes some headway towards the solution of problem (b), but remains limited by problem (a). Here, we argue that combining SRW with path integral techniques leads to a new formulation that overcomes all three problems simultaneously. As a demonstration, we apply the approach to generalized ‘Hooke’s atoms’: interacting particles in harmonic wells. We benchmark our results against state-of-the-art data where possible, and use it to investigate the crossover between the Fermi liquid and the Wigner molecule within closed-shell systems. Our results shed new light on the competition between interaction-driven symmetry breaking and kinetic-energy-driven delocalization.

Existence and concentration of positive solutions to generalized Chern-Simons-Schrödinger system with critical exponential growth

We are concerned with a class of generalized Chern-Simons-Schrödinger systems{−Δu+λV(x)u+A0u+∑j=12Aj2u=f(u),∂1A2−∂2A1=−12|u|2,∂1A1+∂2A2=0,∂1A0=A2|u|2,∂2A0=−A1|u|2, where λ>0 denotes a sufficiently large parameter, V:R2→R admits a potential well Ω≜intV−1(0) and the nonlinearity f fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. Under some suitable assumptions on V and f, based on variational method together with some new technical analysis, we are able to get the existence of positive solutions for some large λ>0, and the asymptotic behavior of the obtained solutions is also investigated when λ→+∞.

A novel engineered sandwich composite with ceramic-coated graphite fibre-reinforced face sheets and an auxetic core for enhanced broadband electromagnetic absorption: design using multiscale computational analysis

Abstract A novel sandwich composite is designed for enhanced broadband absorption using a multiscale computational analysis. The sandwich panel is configured with BaTiO3 coated graphite fiber reinforced polymer (C-GFRP) composite, while BaTiO3 based auxetic material is used as the core material. A computationally efficient in-house tool based on the variational asymptotic method is used to homogenise the electromagnetic properties and evaluate the reflection loss characteristics of the proposed sandwich structure using the scattering matrices. A detailed parametric study of 300 analyses is conducted to identify the optimal sandwich panel configurations for achieving broadband reflection loss under the normal incidence of transverse electric (TE) and transverse magnetic (TM) polarised waves. Two different configurations have been obtained, satisfying the requirements of broadband reflection loss in the X-band frequency range. Configuration (a) Vf = 15% without any ceramic coating, and configuration (b) Vf = 20% with ceramic coating volume fraction (Cf) of 70%. Configuration (a) and (b) maintained the desired reflection loss of greater than -10 dB up to an incidence angle of 40◦ and 60◦, respectively. With the demonstrated capability of covering the entire broadband frequency range, the proposed configurations can serve as baselines to explore novel ceramic-based engineered composite architectures for broadband absorption applications.

Robust estimation of structural orientation parameters and 2D/3D local anisotropic Tikhonov regularization

Understanding the orientation of geologic structures is crucial for analyzing the complexity of the earths’ subsurface. For instance, information about geologic structure orientation can be incorporated into local anisotropic regularization methods as a valuable tool to stabilize the solution of inverse problems and produce geologically plausible solutions. We introduce a new variational method that uses the alternating direction method of multipliers within an alternating minimization scheme to jointly estimate orientation and model parameters in 2D and 3D inverse problems. Specifically, our approach adaptively integrates recovered information about structural orientation, enhancing the effectiveness of anisotropic Tikhonov regularization in recovering geophysical parameters. The paper also discusses the automatic tuning of algorithmic parameters to maximize the new method’s performance. Our algorithm is tested across diverse 2D and 3D examples, including structure-oriented denoising and trace interpolation. The results indicate that the algorithm is robust in solving the considered large and challenging problems, alongside efficiently estimating the associated tilt field in 2D cases and the dip, strike, and tilt fields in 3D cases. Synthetic and field examples show that our anisotropic regularization method produces a model with enhanced resolution and provides a more accurate representation of the true structures.

Bridge modal identification method based on adaptive VMD-HT algorithm

ABSTRACT Bridge engineering structures play a crucial role in transportation. Traditional methods of road and bridge closures for inspection and evaluation, while effective, increase bridge inspection costs and disrupt normal traffic flow. Therefore, a rapid detection and evaluation method for bridge structural modal parameters in the operational state is more suitable for practical engineering applications. However, under real operating conditions, bridge vibration signals typically contain a large amount of noise and interference, and modal identification must overcome issues of non-stationarity and non-linearity in the vibration signals caused by vehicle loads and environmental changes to ensure the accuracy of the identification. This study replaces the traditional Empirical Mode Decomposition algorithm in the Hilbert-Huang Transform with the Variational Mode Decomposition algorithm. It proposes a new method that combines the VMD algorithm with the Hilbert Transform, referred to as the Variational Mode Decomposition-Hilbert Transform method. Compared to the EMD algorithm, it offers significant improvements in suppressing mode mixing and endpoint effects. A vehicle-bridge coupling model was constructed, and the bridge’s natural modal information was successfully extracted from the vibration response signals. The research results indicate that the improved algorithm offers higher accuracy and faster computational speed compared to traditional identification algorithms.

Existence results for p-Kirchhoff type problems on closed manifolds

This paper is to establish the existence, uniqueness, and compactness of positive solutions for the p-Kirchhoff type problems on a closed Riemannian manifold (M, g) of dimension n ≥ 3. Both subcritical and critical cases are considered under certain conditions by variational methods. Our results generalize several previous works from the special case of p = 2 to the general case of 1 < p < n.

Examining the influence of electric field on the photoionization cross section and spin polaronic shift in a semimagnetic double quantum well

In the current research, we performed a numerical investigation of a shallow donor impurity confined in a diluted magnetic semiconductor (DMS) double quantum well (DQW) made of a Cd1−xwMnxwTe/Cd1−xbMnxbTe under the applied electric field in the z-direction. The optoelectronic properties were carried out using the effective-mass approximation and the variational method. The binding energy and photoionization cross section were examined by considering different factors such as, well width, barrier width, impurity position, and Manganese mole fraction. Additionally, the spin polaronic shift corrections to the donor binding energy, caused by the strong exchange interaction between the magnetic moment of the Mn2+ the confined carrier spin, have been examined under the aforementioned effects.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Analysis of degenerate [formula omitted]-Laplacian elliptic equations involving Hardy terms: Existence and numbers of solutions

This article investigates the existence of solutions to quasilinear degenerate elliptic equation with Hardy singular coefficient, in which the weighted function ω(x) is unbounded (singular), then we cannot use the classical space W01,p(Ω), so we have to find another space W01,p(ω,Ω) to deal with the difficulties caused by singularities or degeneracies. New criteria for the existence of at least one and at least two generalized solutions are established via variational methods and critical point theorems provided that the nonlinearity satisfies appropriate hypotheses.

Existence results for some nonlinear elliptic systems on graphs

In this paper, several nonlinear elliptic systems are investigated on graphs. One type of the Sobolev embedding theorem and a new version of the strong maximum principle are established. Then, by using the variational method, the existence of different types of solutions to some elliptic systems is confirmed. Such problems extend the existence results on closed Riemann surface to graphs and extend the existence results for one single equation on graphs Grigor'yan et al. (2016) [13] to nonlinear elliptic systems on graphs. Such problems can also be viewed as one type of discrete version of the elliptic systems on Euclidean space and Riemannian manifold.

Bayesian self-calibration and imaging in very long baseline interferometry

Context. Self-calibration methods with the CLEAN algorithm have been widely employed in very long baseline interferometry (VLBI) data processing in order to correct antenna-based amplitude and phase corruptions present in the data. However, human interaction during the conventional CLEAN self-calibration process can impose a strong effective prior, which in turn may produce artifacts within the final image and hinder the reproducibility of final results. Aims. In this work, we aim to demonstrate a combined self-calibration and imaging method for VLBI data in a Bayesian inference framework. The method corrects for amplitude and phase gains for each antenna and polarization mode by inferring the temporal correlation of the gain solutions. Methods. We use Stokes I data of M87 taken with the Very Long Baseline Array (VLBA) at43 GHz, pre-calibrated using the rPICARD CASA-based pipeline. For antenna-based gain calibration and imaging, we use the Bayesian imaging software resolve. To estimate gain and image uncertainties, we use a variational inference method. Results. We obtain a high-resolution M87 Stokes I image at 43 GHz in conjunction with antenna-based gain solutions using our Bayesian self-calibration and imaging method. The core with counter-jet structure is better resolved, and extended jet emission is better described compared to the CLEAN reconstruction. Furthermore, uncertainty estimation of the image and antenna-based gains allows us to quantify the reliability of the result. Conclusions. Our Bayesian self-calibration and imaging method is able to reconstruct robust and reproducible Stokes I images and gain solutions with uncertainty estimation by taking into account the uncertainty information in the data.

Hydrodynamics of multicomponent vesicles: A phase-field approach

In this paper, we introduce a thermodynamically-consistent phase-field model to investigate the hydrodynamics of inextensible multicomponent vesicles in various fluid flows with inertial forces. Our model couples the fluid field, surface concentration field representing chemical species on the membrane, and vesicle dynamics, while enforcing global area and volume constraints through a Lagrange multiplier method. Specifically, we employ full Navier–Stokes equations for the fluid field, the Cahn–Hilliard equations for the species concentration on the membrane, a nonlinear advection–diffusion equation to describe the evolution of the vesicle membrane, and an additional equation to enforce local inextensibility. We utilize a residual-based variational multiscale method for the Navier–Stokes equations and a standard Galerkin finite element framework for the remaining equations. The PDEs are solved using an implicit, monolithic scheme based on the generalized-α time integration method. We extend previous models for homogeneous vesicles (Aland et al., 2014; Valizadeh and Rabczuk, 2022) to multicomponent vesicles, introducing Cahn–Hilliard equations while maintaining thermodynamic consistency. Additionally, we employ isogeometric analysis (IGA) for higher accuracy. We present a variety of two-dimensional numerical examples, including multicomponent vesicles in shear flow and Poiseuille flow with and without obstructions, using the resistive immersed surface method to handle obstructions. Furthermore, we provide three-dimensional simulations of multicomponent vesicles in Poiseuille flow.

A novel locally statistical active contour model for SAR image segmentation and faster algorithm

Abstract In this paper, a novel locally statistical active contour model (LACM) based on the Aubert-Aujol (AA) denoising model and variational level set method is proposed, which can be used for SAR image segmentation with intensity inhomogeneity. By simple transformation, a fast segmentation algorithm with a global optimization solver can be obtained. Experiments using some challenging synthetic images and Envisat SAR images demonstrate the superiority of our proposed models with respect to classic models. Further, we can find that the structure of the proposed algorithm is the same as that of the Transformer or RNN. We will study the nature of the above two types of algorithms to propose a more effective segmentation algorithm based on the energy model.

Enhancing performance of sandwich panel with three-dimensional orthogonal accordion cores

This study introduces a novel three-dimensional orthogonal accordion structure (3D-OAS) as the cellular core of sandwich panels, achieving multi-directional zero Poisson’s ratio through the orthogonal combination of two-dimensional accordion structures. To analyze its static characteristics effectively, a two-dimensional equivalent Reissner–Mindlin model (2D-ERM) was established utilizing the variational asymptotic method (VAM). The accuracy of 2D-ERM was confirmed by conducting three-point bending tests on 3D-printed specimens and analyzing the in-plane and out-of-plane deformation results of the 3D finite element model (3D-FEM). The comparison of global displacement contours and path-displacement curves between 3D-FEM and 2D-ERM showed a high level of agreement in predicting static deformation. The equivalent stiffness of SP-3D-OAS steadily increased as the inclined angle deviates by 90-degree, irrespective of whether it pertains to the convex or concave angle. Evaluation of deformability in sandwich panels with different cellular core forms revealed superior comprehensive performance in 3D-OAS, followed by 3D-YRS and 3D-XYAS, with a reduction of 16.41% and 17.35% in specific stiffness, respectively. Compared to the 3D-FEM, 2D-ERM significantly reduces computation time without compromising engineering accuracy. The research results provide a useful reference for optimal design of sandwich panels with multi-directional ZPR cellular core.

A variational method for the sheath potential of hypersonic leading edges with space-charge limitations

Electron transpiration cooling for the leading edges (LE) of hypersonic aircraft utilizes thermionic emission; however, space-charge effects limit the electron emission rate, potentially diminishing the efficiency of this cooling mechanism. We develop a variational weak form of the Poisson equation that describes the sheath potential and then numerically solve it using the finite element method. This formulation has two main benefits: (1) the space-charge limit condition can be incorporated as a constraint and (2) it allows for the analysis of three-dimensional geometries with complex boundary conditions. We demonstrate that the current emitted from the surface of an LE is generally a small fraction of the Child–Langmuir limit due to space charge. We then propose several methods to enhance the emitted current from the surface and to boost the cooling effect of thermionic emission. These include increasing the plasma density, applying a negative surface potential, and using fringe fields under suitable geometric conditions. For a LaB6 emitting LE, the total emitted current is shown to be minimal and independent of the temperature of a surface with floating potential. However, when a negative potential is applied and the surface is heated, the emitted current follows the Richardson–Dushman relationship up to a critical temperature, beyond which it remains constant. At an applied surface potential of −5 V, the critical temperature is around 1700 K.

NUMERICAL ASSESSMENT OF SOME SEMI-ANALYTICAL TECHNIQUES FOR SOLVING A FRACTIONAL-ORDER LEPTOSPIROSIS MODEL

This research aims to apply and compare two semi-analytical techniques, the Variational Iterative Method (VIM) and the New Iterative Method (NIM), for solving a pre-formulated mathematical model of Fractional-order Leptospirosis. Leptospirosis is a significant bacterial infection affecting humans and animals. By implementing the VIM and NIM algorithms, numerical experiments are conducted to solve the leptospirosis model. Comparing the obtained findings demonstrates that VIM and NIM are effective semi-analytical methods for solving systems of fractional differential equations. Notably, our study unveils a crucial dynamic in the disease's spread. The application of VIM and NIM offers a refined depiction of the biological dynamics, highlighting that the susceptible human population gradually decreases, the infectious human population declines, the recovered human population increases, and a significant rise in the infected vector population is observed over time. This nuanced portrayal of the disease's dynamics is crucial for understanding the intricate interplay of Leptospirosis among human and vector populations. The study's outcomes contribute valuable insights into the applicability and performance of the methods in solving the Fractional Leptospirosis model. Results indicate rapid convergence and comparable outcomes for both methods.

Застосування методу двобічних наближень до розв’язання першої крайової задачі для одновимірного рівняння теплопровідності з експоненціально нелінійним коефіцієнтом теплопровідності

The problem of heat conduction in nonlinear media reduces to solving boundary value problems for nonlinear heat conduction equations, where the coefficients of the equation or the function of the power of heat sources depend on temperature according to some law. Among the numerical methods for solving problems for nonlinear equations of mathematical physics, one can distinguish finite difference methods, finite element methods, variational and projection methods, as well as iterative methods. Among the latter group, the two-sided approximation method is particularly attractive due to its ability to provide a convenient estimate for the error of the approximate solution and to prove the existence of a solution to the original problem. The theory of linear partially ordered spaces was developed by L. V. Kantorovich in the second half of the 1930s. Further development of this theory is associated with the works of M. A. Krasnosel’skii, H. Amann, V. I. Opojtsev, N. S. Kurpel, B. A. Shuvar, A. I. Kolosov etc. The aim of this article is to develop a two-sided approximation method based on the use of Green's functions for solving the first boundary value problem for a one-dimensional nonlinear heat conduction equation and to investigate its performance when solving test problems. To achieve this goal, the unknown function was replaced, and the boundary value problem was reduced to a Hammerstein integral equation, which was considered as a nonlinear operator equation in a partially ordered Banach space. Conditions for the existence of a unique positive solution to the problem and conditions for two-sided convergence of successive approximations to it were obtained. The developed method was implemented in software and tested on solving test problems. The results of the computational experiment are illustrated with graphical and tabular information

On a Dirichlet problem related to anisotropic fluid flow in bidisperse porous media

This paper is concerned with the study of a Dirichlet boundary value problem for a system of two coupled anisotropic Darcy–Forchheimer–Brinkman equations on a bounded Lipschitz domain in . Using variational methods and fixed point techniques, we obtain a well‐posedness result in ‐based Sobolev spaces for sufficiently small data. As an application, we investigate numerically the lid‐driven flow problem in a square cavity saturated with a bidisperse porous medium and analyze the effect of various physical parameters on the fluid motion.

Simulation and Experimental Research of V-Crack Testing of Rail Surfaces Based on Laser Ultrasound

Rail surface cracks are widespread damage that can lead to uneven surfaces of railheads and affect traveling safety. Non-destructive testing is needed to inspect rails regularly to ensure the normal operation of railroads. This paper proposes a laser ultrasonic testing method combining variational mode decomposition and diffractive Rayleigh wave time-of-flight to detect tiny cracks on the rail surface quantitatively. The finite element method was combined with experiments to simulate and experimentally investigate cracks of different sizes numerically. In the numerical simulation, the location of the crack was determined by B-scan. Afterward, the interaction between various types of ultrasound and cracks was comparatively analyzed, and the crack size was quantitatively characterized using useful information from the ultrasound signals. The results show that the time-of-flight method can detect arbitrary cracks with low error. Therefore, the experimentally acquired ultrasound signals used the time difference between the diffracted Rayleigh wave and other ultrasound waves to detect the crack information quantitatively. The variational mode decomposition method was used to separate the ultrasonic signals and extract the best surface wave modes to improve the signal-to-noise ratio. The results show that the combination of variational mode decomposition and time-of-flight method can effectively detect the size of cracks.