Variational Approach Research Articles

A variational approach to deriving Hamiltonian functions under non-holonomic constraints — the case of the Gaussian thermostats

The Hamiltonian function (H), if it exists, provides a powerful platform for studying the dynamics of constrained many-particle systems. For systems under holonomic constraints, H can be derived from the principle of least action (PLA) formulated in terms of a suitable set of generalized coordinates. In molecular dynamics (MD) simulations, where the system temperature often prescribes a non-holonomic constraint, the challenge is to find the system’s Hamiltonian. The Gaussian Isokinetic (GIK) thermostat, the first deterministic physics-preserving thermostat in MD, was derived from the principle of least constraint (PLC) in the early 1980s: its H, interestingly, was discovered almost 15 years later. A mathematically rigorous connection of the GIK Hamiltonian with PLA, however, has been missing until now. In this article, we adopt a variational approach to establish the equivalence of PLC with PLA for a class of non-holonomically constrained many-particle systems, and present the general formulation of the Euler–Lagrange equations of motion through time rescaling. Making use of the particular forms of the Lagrange multiplier obtained from the principle of least constraint, we derive H for the GIK and the Gaussian Isoenergetic thermostats.

Stress and energy release rate analysis of cross-ply carbon fiber-reinforced plastic laminate with transverse cracks subjected to ultrasonic vibration using a variational approach

Application of carbon fiber reinforced plastics (CFRP) to structures that are subjected to increasingly high numbers of cyclic loads is expected to expand, so evaluation of fatigue properties up to the very-high cycle regime is essential to prove their long-term reliability. Conventional fatigue testing is not practical for such conditions owing to the excessive time required; ultrasonic fatigue testing, which can be performed at a frequency of 20 kHz, is an effective alternative. Stress distribution in ultrasonic fatigue testing differs from that in conventional fatigue testing because the specimen is resonated. The stress distribution of cross-ply laminates with transverse cracks subjected to resonance was calculated using a variational approach. The energy release rate was calculated based on the results of a previous study on transverse crack initiation. The results of the stress analysis are in good agreement with those of finite-element analysis. Quantitative evaluation of transverse crack initiation was also conducted.

L’approche variationnelle de la rupture : un exemple de collaboration fructueuse entre mécaniciens et mathématiciens

This article reviews some important steps in the development and application of the variational approach to fracture. It emphasizes the essential role played by the permanent dialogue between mechanicians and applied mathematicians in bringing about this new paradigm and ensuring its undeniable success.

Responding to consumer environmental awareness: A variational inequality approach to green manufacturing optimization

In response to increasing consumer environmental awareness, our study introduces a sustainable supply chain optimization model that stimulates competition among multiple green manufacturers within an uncertain market context. Each manufacturer crafts a selection of products showcasing various degrees of greenness, purposefully designed to meet the expanding demand of environmentally conscious consumers. This strategic model emphasizes profit maximization for green manufacturers, necessitating the definition of ideal greenness levels and production quantities for an array of product types. By employing variational inequality theory, we transmute the preliminary optimization model into an interconnected variational inequality problem. We propose an improved Euler algorithm, a proficient tool for attaining equilibrium solutions. Empirical scenarios inspired by the electric vehicle industry lend credibility to our model, indicating that the surge in consumer environmental awareness compels manufacturers to diversify their offerings and accentuate their products’ greenness levels. Intriguingly, the research suggests that escalating competition can incentivize manufacturers to elevate their products’ greenness levels, strategically countering potential profit erosion. In addition, the study reveals that intensifying consumers’ green utility sensitivity doesn’t invariably augment demand for manufacturers, even with significant improvements in product greenness. This paradox highlights the necessity for manufacturers to strategically adjust their production approaches in light of varying consumer preferences. Moreover, when the level of consumer environmental awareness achieves a certain level, the fierce competition might not always be harmful to some manufacturers and the industry if they can adopt effective strategies to improve green utility sensitivity.

Flat flow solution to the mean curvature flow with volume constraint

Abstract In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai, Seis and Spadaro [L. Mugnai, C. Seis and E. Spadaro, Global solutions to the volume-preserving mean-curvature flow, Calc. Var. Partial Differential Equations 55 2016, 1, Article ID 18]. This method is based on the gradient flow approach due to Almgren, Taylor and Wang [F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 1993, 2, 387–438] and Luckhaus and Sturzenhecker [S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 1995, 2, 253–271] and my aim is to replace the volume penalization with the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on second variation argument.

A novel variational approach for multiphoton microscopy image restoration: from PSF estimation to 3D deconvolution

In multi-photon microscopy (MPM), a recent in-vivo fluorescence microscopy system, the task of image restoration can be decomposed into two interlinked inverse problems: firstly, the characterization of the point spread function (PSF) and subsequently, the deconvolution (i.e. deblurring) to remove the PSF effect, and reduce noise. The acquired MPM image quality is critically affected by PSF blurring and intense noise. The PSF in MPM is highly spread in 3D and is not well characterized, presenting high variability with respect to the observed objects. This makes the restoration of MPM images challenging. Common PSF estimation methods in fluorescence microscopy, including MPM, involve capturing images of sub-resolution beads, followed by quantifying the resulting ellipsoidal 3D spot. In this work, we revisit this approach, coping with its inherent limitations in terms of accuracy and practicality. We estimate the PSF from the observation of relatively large beads (approximately 1 μm in diameter). This goes through the formulation and resolution of an original non-convex minimization problem, for which we propose a proximal alternating method along with convergence guarantees. Following the PSF estimation step, we then introduce an innovative strategy to deal with the high level multiplicative noise degrading the acquisitions. We rely on a heteroscedastic noise model for which we estimate the parameters. We then solve a constrained optimization problem to restore the image, accounting for the estimated PSF and noise, while allowing a minimal hyper-parameter tuning. Theoretical guarantees are given for the restoration algorithm. These algorithmic contributions lead to an end-to-end pipeline for 3D image restoration in MPM, that we share as a publicly available Python software. We demonstrate its effectiveness through several experiments on both simulated and real data.

Variational approach for time-space fractal Bogoyavlenskii equation

In this paper, we apply the variational approach for solving the time-space fractal Bogoyavlenskii equation. By the two-scale fractal complex transformation, the fractal modification of Bogoyarlenskii equation is equivalently rewritten as the original Bogoyavlenskii equation. By variational principle, different type solutions including fractal bright soliton solution, fractal bright-like soliton solution, fractal kinky-bright soliton solution and fractal periodic wave solution are provided, which are not touched in the existing literature. Numerical results are presented to show the propagation behaviours.

Two-dimensional modulation instability and soliton clusters in nonlocal media with competing cubic–quintic nonlinearities

Using linear-stability analysis, two-dimensional modulation instability (MI) of plane wave is studied in nonlocal media with competing cubic–quintic (CQ) nonlinearities, which shows that MI can be effectively eliminated by strong nonlocality and competing quintic nonlinearity. Furthermore, propagation properties of higher-order soliton clusters, i.e., Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) solitons are also investigated. Bifurcated solutions of these solitons are obtained analytically with variational approach. We also demonstrate in detail the propagation dynamics of the HG and LG solitons with split-step Fourier transform.

Qubit Efficient Quantum Algorithms for the Vehicle Routing Problem on Noisy Intermediate‐Scale Quantum Processors

AbstractThe vehicle routing problem with time windows (VRPTW) is a common optimization problem faced within the logistics industry. In this work, the use of a previously‐introduced qubit encoding scheme is explored to reduce the number of qubits, to evaluate the effectiveness of Noisy Intermediate‐Scale Quantum (NISQ) devices when applied to industry relevant optimization problems. A quantum variational approach is applied to a testbed of multiple VRPTW instances ranging from 11 to 3964 routes. These intances are formulated as quadratic unconstrained binary optimization (QUBO) problems based on realistic shipping scenarios. The results are compared with standard binary‐to‐qubit mappings after executing on simulators as well as various quantum hardware platforms, including IBMQ, AWS (Rigetti), and IonQ. These results are benchmarked against the classical solver, Gurobi. The approach can find approximate solutions to the VRPTW comparable to those obtained from quantum algorithms using the full encoding, despite the reduction in qubits required. These results suggest that using the encoding scheme to fit larger problem sizes into fewer qubits is a promising step in using NISQ devices to find approximate solutions for industry‐based optimization problems, although additional resources are still required to eke out the performance from larger problem sizes.

Schur multipliers of C⁎-algebras, group-invariant compactification and applications to amenability and percolation

Let Γ be a countable discrete group. Given any sequence (fn)n≥1 of ℓp-normalized functions (p∈[1,2)), consider the associated positive definite matrix coefficients 〈fn,ρ(⋅)fn〉 of the right regular representation ρ. We construct an orthogonal decomposition of the corresponding Schur multipliers on the reduced group C⁎-algebra or the uniform Roe algebra of Γ. We identify this decomposition explicitly via the limit points of the orbits (f˜n)n≥1 in the group-invariant compactification of the quotient space constructed by Varadhan and the first author in [14]. We apply this result and use positive-definiteness to provide two (quite different) characterizations of amenability of Γ – one via a variational approach and the other using group-invariant percolation on Cayley graphs constructed by Benjamini, Lyons, Peres and Schramm [1]. These results underline, from a new point of view to the best of our knowledge, the manner in which Schur multipliers capture geometric properties of the underlying group Γ.

A Variational Approach for Joint Image Recovery and Feature Extraction Based on Spatially Varying Generalised Gaussian Models

A Variational Approach for Joint Image Recovery and Feature Extraction Based on Spatially Varying Generalised Gaussian Models

Modeling the initiation and propagation of complex networks of cracks in reinforced concrete plates

The present paper proposes a new finite element formulation for the analysis of RC slabs, accounting for concrete cracking and reinforcement yielding. The proposed formulation, developed within the framework of Lumped Damage Mechanics, considers that all inelastic effects are localized in lines and relates all numerical parameters to the mechanical properties of the slab. The proposed variational approach is presented as a Nash noncooperative equilibrium point. It is verified, numerically, that the proposed FE leads to mesh-independent results. Numerical simulations of some experimental tests are also presented. The results in terms of collapse mechanism, cracking network, and force-deflection curves showed that the model has significant accuracy, even for non-symmetric free clamped and simple supported boundary conditions. It is also shown, through the confrontation with experimental results, that the model correctly characterizes the initiation and development of complex networks of cracks.

Breaking the limitations with sparse inputs by variational frameworks (BLIss) in terahertz super-resolution 3D reconstruction

Data acquisition, image processing, and image quality are the long-lasting issues for terahertz (THz) 3D reconstructed imaging. Existing methods are primarily designed for 2D scenarios, given the challenges associated with obtaining super-resolution (SR) data and the absence of an efficient SR 3D reconstruction framework in conventional computed tomography (CT). Here, we demonstrate BLIss, a new approach for THz SR 3D reconstruction with sparse 2D data input. BLIss seamlessly integrates conventional CT techniques and variational framework with the core of the adapted Euler-Elastica-based model. The quantitative 3D image evaluation metrics, including the standard deviation of Gaussian, mean curvatures, and the multi-scale structural similarity index measure (MS-SSIM), validate the superior smoothness and fidelity achieved with our variational framework approach compared with conventional THz CT modal. Beyond its contributions to advancing THz SR 3D reconstruction, BLIss demonstrates potential applicability in other imaging modalities, such as X-ray and MRI. This suggests extensive impacts on the broader field of imaging applications.

PMF-GRN: a variational inference approach to single-cell gene regulatory network inference using probabilistic matrix factorization.

Inferring gene regulatory networks (GRNs) from single-cell data is challenging due to heuristic limitations. Existing methods also lack estimates of uncertainty. Here we present Probabilistic Matrix Factorization for Gene Regulatory Network Inference (PMF-GRN). Using single-cell expression data, PMF-GRN infers latent factors capturing transcription factor activity and regulatory relationships. Using variational inference allows hyperparameter search for principled model selection and direct comparison to other generative models. We extensively test and benchmark our method using real single-cell datasets and synthetic data. We show that PMF-GRN infers GRNs more accurately than current state-of-the-art single-cell GRN inference methods, offering well-calibrated uncertainty estimates.

On a class of fractional p( x, y)−Kirchhoff type problems with indefinite weight

This paper is concerned with a class of fractional \(p(x,y)-\)Kirchhoff type problems with Dirichlet boundary data along with indefinite weight of the following form \begin{equation*} \left\lbrace\begin{array}{ll} M\left(\int_{Q}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dx\,dy\right)\\ (-\triangle_{p(x)})^s+|u(x)|^{q(x)-2}u(x) & \\ =\lambda V(x)|u(x)|^{r(x)-2}u(x)& \text{in }\Omega,\\ u=0, & \text{in }\mathbb{R}^N\Omega. \end{array}\right. \end{equation*} By means of direct variational approach and Ekeland’s variational principle, we investigate the existence of nontrivial weak solutions for the above problem in case of the competition between the growth rates of functions \(p\) and \(r\) involved in above problem, this fact is essential in describing the set of eigenvalues of this problem.

Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik–Schnirelman theory, and new estimates on the category of suitable configuration spaces.

A New 2D Formulation of Modified General Relativity

AbstractIt is well known that the Einstein–Hilbert action in two dimensions is topological and yields an identically vanishing Einstein tensor. Consequently one is faced with difficulties when formulating a non‐trivial gravity model. The authors present a new, intrinsically two‐dimensional, approach to this problem based on the Einstein action. This yields a well defined variational approach which results in new field equations that break diffeomorphism invariance. This proposed approach does not require the introduction of additional scalar fields, nor the use of conformal transformations. However, how including conformal counter terms leads to equivalent results is shown. In doing so, an explanation for why previous approaches worked can be provided. Solutions to the field equations are briefly discussed.

Angular traveling waves of the high‐dimensional Boussinesq equation

AbstractThis paper studies traveling waves with nonzero wave speed (angular traveling waves) of the high‐dimensional Boussinesq equation that have not been studied before. We analyze the properties of these waves and demonstrate that, unlike the unique stationary solution, they lack positivity, radial symmetry, and exponential decay. By employing variational and geometric approaches, along with perturbation theory, we establish the orbital (in)stability and strong instability of these traveling waves.

Variational approach for a technology acquisition strategy model in context of the Internet of Things

Variational approach for a technology acquisition strategy model in context of the Internet of Things

A new yield criterion based constitutive model for porous materials

In the present work, a new explicit expression of macroscopic yield criterion is derived for porous materials, such as soil, rock and cement-based materials. To consider the asymmetric behavior between tensile and compression loading, the matrix is assumed to obey to a Drucker–Prager local criterion. The influence of porosity and the effect of matrix properties are well taken into account by this criterion. By using the stress variational approach with a suitable statically admissible stress field, a new more accurate estimation with a simple expression has been obtained for the case of purely deviatoric loading. The exact solutions can be retrieved by this criterion in the case of hydrostatic loading. It also resolves the problem of discontinuity between tension and compression. The weaknesses of the main existing criteria have been overcome by this new yield criterion. By comparing with the finite element numerical results with a wide range values of porosity and frictional parameter of the matrix, the proposed macroscopic yield criterion has a great accuracy and improves the existing ones. Based on this yield criterion, a complete constitutive model is constructed and implemented. This micro-mechanics based model is then applied to describe the mechanical behavior of porous sandstone, and validated by the comparisons with experimental results at different confining pressures.