Lagrangian Parameters Research Articles

Halo Bias in the Peak Model: A First-principles Nonparametric Approach

The Press–Schechter (PS) and excursion set (ES) models of structure formation fail in reproducing the halo bias found in simulations, while the ES-peaks' formalism built in the peak model reproduces it only at high masses and does not address in a fully satisfactory manner peak nesting, and the mass and time of ellipsoidal collapse of triaxial peaks in the Gaussian-smoothed density field. Here, we apply the confluent system of peak trajectories formalism fixing all these issues from first principles and with no free parameters to infer the Lagrangian local peak bias parameters, which adopt very simple analytic expressions similar to those found in the PS and ES models. The predicted Eulerian linear halo bias recovers the results of simulations. More specifically, we show that the only small departure observed at intermediate and low masses can be due to the spurious halo splitting and grouping caused by the spherical overdensity halo-finding algorithm used in simulations.

Hp-Multigrid Preconditioner for a Divergence-Conforming HDG Scheme for the Incompressible Flow Problems

In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^3$$\\end{document}.

Topological optimization in 3D-magnetostatics: development of adjoint methods using the equations of magnetic moments

PurposeThe purpose of this paper is mainly to develop the adjoint method within the method of magnetic moment (MMM) and thus, to provide an efficient new way to solve topology optimization problems in magnetostatic to design 3D-magnetic circuits.Design/methodology/approachFirst, the MMM is recalled and the optimization design problem is reformulated as a partial derivative equation-constrained optimization problem where the constraint is the Maxwell equation in magnetostatic. From the Karush–Khun–Tucker optimality conditions, a new problem is derived which depends on a Lagrangian parameter. This problem is called the adjoint problem and the Lagrangian parameter is called the adjoint parameter. Thus, solving the direct and the adjoint problems, the values of the objective function as well as its gradient can be efficiently obtained. To obtain a topology optimization code, a semi isotropic material with penalization (SIMP) relaxed-penalization approach associated with an optimization based on gradient descent steps has been developed and used.FindingsIn this paper, the authors provide theoretical results which make it possible to compute the gradient via the continuous adjoint of the MMMs. A code was developed and it was validated by comparing it with a finite difference method. Thus, a topology optimization code associating this adjoint based gradient computations and SIMP penalization technique was developed and its efficiency was shown by solving a 3D design problem in magnetostatic.Research limitations/implicationsThis research is limited to the design of systems in magnetostatic using the linearity of the materials. The simple examples, the authors provided, are just done to validate our theoretical results and some extensions of our topology optimization code have to be done to solve more interesting design cases.Originality/valueThe problem of design is a 3D magnetic circuit. The 2D optimization problems are well known and several methods of resolution have been introduced, but rare are the problems using the adjoint method in 3D. Moreover, the association with the MMMs has never been treated yet. The authors show in this paper that this association could provide gains in CPU time.

Simultaneous optimal control of directional missed discovery rates in data stream diagnosis

High-dimensional data streams are ubiquitous in modern manufacturing, due to their ability to provide valuable information about the industrial system’s performance on a real-time basis. If a shift occurs in a production process, fault diagnosis based on the data streams is of critical importance for identifying the root cause. Existing methods have largely focused on controlling the total missed discovery rate without distinguishing missed signals for positive versus negative components of the shift vector. In practice, however, losses incurred from the two directional shifts can differ substantially, so it is desirable to constrain the proportions of missed signals for positive and negative components at two distinctive levels. In this article, we propose a fault classification procedure that controls the two proportions separately. By formulating the problem as Lagrangian multiplier optimization, we show that the proposed procedure is optimal in the sense that it minimizes the expected number of false discoveries. We also suggest an iterative adjustment algorithm that converges to the optimal Lagrangian parameters. The asymptotic optimality for the data-driven version of our procedure is also established. Theoretical justification and numerical comparison with state-of-the-art methods show that the proposed procedure works well in applications.

Opinion Models, Election Data, and Political Theory.

A unifying setup for opinion models originating in statistical physics and stochastic opinion dynamics are developed and used to analyze election data. The results are interpreted in the light of political theory. We investigate the connection between Potts (Curie-Weiss) models and stochastic opinion models in the view of the Boltzmann distribution and stochastic Glauber dynamics. We particularly find that the q-voter model can be considered as a natural extension of the Zealot model, which is adapted by Lagrangian parameters. We also discuss weak and strong effects (also called extensive and nonextensive) continuum limits for the models. The results are used to compare the Curie-Weiss model, two q-voter models (weak and strong effects), and a reinforcement model (weak effects) in explaining electoral outcomes in four western democracies (United States, Great Britain, France, and Germany). We find that particularly the weak effects models are able to fit the data (Kolmogorov-Smirnov test) where the weak effects reinforcement model performs best (AIC). Additionally, we show how the institutional structure shapes the process of opinion formation. By focusing on the dynamics of opinion formation preceding the act of voting, the models discussed in this paper give insights both into the empirical explanation of elections as such, as well as important aspects of the theory of democracy. Therefore, this paper shows the usefulness of an interdisciplinary approach in studying real world political outcomes by using mathematical models.

λ-Domain Rate Control via Wavelet-Based Residual Neural Network for VVC HDR Intra Coding.

High dynamic range (HDR) video offers a more realistic visual experience than standard dynamic range (SDR) video, while introducing new challenges to both compression and transmission. Rate control is an effective technology to overcome these challenges, and ensure optimal HDR video delivery. However, the rate control algorithm in the latest video coding standard, versatile video coding (VVC), is tailored to SDR videos, and does not produce well coding results when encoding HDR videos. To address this problem, a data-driven λ-domain rate control algorithm is proposed for VVC HDR intra frames in this paper. First, the coding characteristics of HDR intra coding are analyzed, and a piecewise R-λ model is proposed to accurately determine the correlation between the rate (R) and the Lagrange parameter λ for HDR intra frames. Then, to optimize bit allocation at the coding tree unit (CTU)-level, a wavelet-based residual neural network (WRNN) is developed to accurately predict the parameters of the piecewise R-λ model for each CTU. Third, a large-scale HDR dataset is established for training WRNN, which facilitates the applications of deep learning in HDR intra coding. Extensive experimental results show that our proposed HDR intra frame rate control algorithm achieves superior coding results than the state-of-the-art algorithms. The source code of this work will be released at https://github.com/TJU-Videocoding/WRNN.git.

Flavour and Higgs physics in Z2-symmetric 2HD models near the decoupling limit

With no evidence of any exotic particle detected so far beyond the Standard Model, the new physics may lie above the presently accessible energies at colliders and, at low-energies, can be accounted for via an effective description. The interplay of flavour and Higgs physics data allows setting stringent bounds on the parameters of the effective Lagrangian. In this paper, we focus on Z2-symmetric two Higgs doublet models near the decoupling limit: the corresponding effective description relies on only a few parameters, thus predicting many interesting correlations between observables that work as tests of the theory. We present the results of a global fit to the existing data, updating and extending over the past literature. We comment on the triple Higgs coupling as a probe of an extended scalar sector and on the recent CDF II measurement of the W-mass.

The disparity between optimal and practical Lagrangian multiplier estimation in video encoders

With video streaming making up 80% of the global internet bandwidth, the need to deliver high-quality video at low bitrate, combined with the high complexity of modern codecs, has led to the idea of a per-clip optimisation approach in transcoding. In this paper, we revisit the Lagrangian multiplier parameter, which is at the core of rate-distortion optimisation. Currently, video encoders use prediction models to set this parameter but these models are agnostic to the video at hand. We explore the gains that could be achieved using aper-clipdirect-search optimisation of the Lagrangian multiplier parameter. We evaluate this optimisation framework on a much larger corpus of videos than that has been attempted by previous research. Our results show that per-clip optimisation of the Lagrangian multiplier leads to BD-Rate average improvements of 1.87% for x265 across a 10 k clip corpus of modern videos, and up to 25% in a single clip. Average improvements of 0.69% are reported for libaom-av1 on a subset of 100 clips. However, we show that a per-clip, per-frame-type optimisation ofλfor libaom-av1 can increase these average gains to 2.5% and up to 14.9% on a single clip. Our optimisation scheme requires about 50–250 additional encodes per-clip but we show that significant speed-up can be made using proxy videos in the optimisation. These computational gains (of up to ×200) incur a slight loss to BD-Rate improvement because the optimisation is conducted at lower resolutions. Overall, this paper highlights the value of re-examining the estimation of the Lagrangian multiplier in modern codecs as there are significant gains still available without changing the tools used in the standards.

Cornering large-Nc QCD with positivity bounds

The simple analytic structure of meson scattering amplitudes in the large-Nc limit, combined with positivity of the spectral density, provides precise predictions on low-energy observables. Building upon previous studies, we explore the allowed regions of chiral Lagrangian parameters and meson couplings to pions. We reveal a structure of kinks at all orders in the chiral expansion and develop analytical tools to show that kinks always correspond to amplitudes with a single light pole. We build (scalar- and vector-less) deformations of the Lovelace-Shapiro and Coon UV-complete amplitudes, and show that they lie close to the boundaries. Moreover, constraints from crossing-symmetry imply that meson couplings to pions become smaller as their spin increases, providing an explanation for the success of Vector Meson Dominance and holographic QCD. We study how these conclusions depend on assumptions about the high-energy behavior of amplitudes. Finally, we emphasize the complementarity between our results and Lattice computations in the exploration of large-Nc QCD.

Distributed Optimization Subject to Inseparable Coupled Constraints: A Case Study on Plant-Wide Ethylene Process

Plant-wide optimization plays a vital role in improving the overall performance of large-scale industrial processes. Considering the modeling complexity and convergence difficulty of centralized plant-wide optimization, this paper proposes a distributed framework by decomposing the global optimization problem into a set of subproblems, where multiple local units interact with each other between nodes. According to the proposed framework, plant-wide optimization problem can be effectively solved by distributed optimization. To eliminate the limitations of existing distributed algorithms, we introduce constraint node to describe the inseparable coupled constraints between nodes. By combining Lagrange duality and parameter projection, the proposed algorithm can solve optimization problems with multiple constraints. Taking ethylene production process as an example, the global energy consumption optimization is guaranteed without the whole-process mechanism model. Numerical simulation and industrial experiment results demonstrate that the proposed algorithm can reduce the energy consumption of the entire ethylene process with fewer computation time.

A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint

Starting from a reformulation of the mass balance law based on the Bilby–Kröner–Lee (BKL) decomposition of the deformation gradient tensor, we study some peculiar mechanical aspects of growth in a monophasic continuum by regarding the reformulated mass balance equation as a non-holonomic and rheonomic constraint. Such constraint restricts the admissible rates of the growth tensor, i.e., one of the two factors of the BKL decomposition, to comply with a growth law provided phenomenologically. For our purposes, we put the constraint in Pfaffian form, and treat time as a fictitious, additional Lagrangian parameter, subjected to the condition that its rate must be unitary. Then, by taking some suggestions from the literature, we assume the existence of generalized forces conjugated with the virtual variations of the growth tensor, and we write a constrained version of the Principle of Virtual Work (PVW) that leads to a mixed boundary value problem whose unknowns are the motion, the growth tensor, and the Lagrange multipliers of the considered theory. This allows to extrapolate a physical interpretation of the role that the growth-conjugated forces play on the components of the growth tensor, especially on the distortional ones, i.e., those that are not directly related to the variation of mass of the body. The core message of our work is conceptual: we show that the growth laws usually encountered in the literature, which are prescribed phenomenologically, but may be difficult to justify theoretically, can be put in the framework of the PVW by regarding them as constraints. Moreover, we retrieve more particularized frameworks of growth available in the literature, while being able to switch to a theory of growth of grade one, such as a Cahn–Hilliard model of growth.

Mixed Eulerian–Lagrangian modeling of sheet metal roll forming

We propose a nonlinear shell finite element model to simulate sheet metal roll forming, a continuous forming process to produce endless metal profiles. A mixed Eulerian–Lagrangian kinematic description is employed to overcome the drawbacks of the common Lagrangian parametrization. The finite element mesh is detached from the particle motion in axial direction and, thus, facilitates a two-step solution procedure to capture the continuous forming process: First, an equilibrium is sought with the account for contact and plastic flow. Secondly, the material transport is taken into account, which amounts to the integration of an advection problem for the plastic variables. The continuum plasticity model with through-the-thickness integration for the stress resultants guarantees a precise resolution of the forming process in each cross section of the Kirchhoff–Love shell. A series of simulations is carried out to ascertain the convergence of the numerical scheme, to highlight the impact of characteristic parameters and to establish a correspondence to a reference computation with the commercial software Abaqus in a simplified static setting. A physical experiment is devised on an actual roll forming mill to assess the quality of the current computational model.

Inequality constrained stochastic nonlinear optimization via active-set sequential quadratic programming

We study nonlinear optimization problems with a stochastic objective and deterministic equality and inequality constraints, which emerge in numerous applications including finance, manufacturing, power systems and, recently, deep neural networks. We propose an active-set stochastic sequential quadratic programming (StoSQP) algorithm that utilizes a differentiable exact augmented Lagrangian as the merit function. The algorithm adaptively selects the penalty parameters of the augmented Lagrangian, and performs a stochastic line search to decide the stepsize. The global convergence is established: for any initialization, the KKT residuals converge to zero almost surely. Our algorithm and analysis further develop the prior work of Na et al. (Math Program, 2022. https://doi.org/10.1007/s10107-022-01846-z). Specifically, we allow nonlinear inequality constraints without requiring the strict complementary condition; refine some of designs in Na et al. (2022) such as the feasibility error condition and the monotonically increasing sample size; strengthen the global convergence guarantee; and improve the sample complexity on the objective Hessian. We demonstrate the performance of the designed algorithm on a subset of nonlinear problems collected in CUTEst test set and on constrained logistic regression problems.

Convergence analysis of consensus-ADMM for general QCQP

We analyze the convergence properties of the consensus-alternating direction method of multipliers (ADMM) for solving general quadratically constrained quadratic programs. We prove that the augmented Lagrangian function value is monotonically non-increasing as long as the augmented Lagrangian parameter is chosen to be sufficiently large. Simulation results show that the augmented Lagrangian function is bounded from below when the matrix in the quadratic term of the objective function is positive definite. In such a case, the consensus-ADMM is convergent.

Neutrino mixing and leptogenesis in a L_e-L_mu -L_tau model

We present a simple extension of the Standard Model with three right-handed neutrinos in a SUSY framework, with an additional hbox {U}(1)_text {F} abelian flavor symmetry with a non standard leptonic charge L_e-L_mu -L_tau for lepton doublets and arbitrary right-handed charges. Our model predicts an inverted neutrino mass hierarchy and it is able to reproduce the experimental values of the mixing angles of the PMNS matrix and of the r=Delta m_text {sun}^2/Delta m_text {atm}^2 ratio, with only a moderate fine tuning of the Lagrangian free parameters. The baryon asymmetry of the Universe is generated via thermal leptogenesis through CP-violating decays of the heavy right-handed neutrinos. We present a detailed numerical solution of the relevant Boltzmann equation, accounting for the impact of the distribution of the asymmetry in the various lepton flavors.

Standard model at 200 GeV

The Standard Model can be defined quantitatively by running parameters in a mass-independent renormalization scheme at a fixed reference scale. We provide a set of simple interpolation formulas that give the fundamental Lagrangian parameters in the $\overline{\mathrm{MS}}$ scheme at a renormalization scale of 200 GeV, safely above the top-quark mass and suitable for matching to candidate new physics models at very high mass scales using renormalization group equations. These interpolation formulas take as inputs the on-shell experimental quantities and use the best available calculations in the pure $\overline{\mathrm{MS}}$ scheme. They also serve as an accounting of the parametric uncertainties for the short-distance Standard Model Lagrangian. We also include an interpolating formula for the $W$-boson mass. This paper is based on results obtained with the publicly available computer program smdr.

Adaptive search space decomposition method for pre- and post-buckling analyses of space truss structures

The paper proposes a novel adaptive search space decomposition method and a novel gradient-free optimization-based formulation for the pre- and post-buckling analyses of space truss structures. Space trusses are often employed in structural engineering to build large steel constructions, such as bridges and domes, whose structural response is characterized by large displacements. Therefore, these structures are vulnerable to progressive collapses due to local or global buckling effects, leading to sudden failures. The method proposed in this paper allows the analysis of the load-equilibrium path of truss structures to permanent and variable loading, including stable and unstable equilibrium stages and explicitly considering geometric nonlinearities. The goal of this work is to determine these equilibrium stages via optimization of the Lagrangian kinematic parameters of the system, determining the global equilibrium. However, this optimization problem is non-trivial due to the undefined parameter domain and the sensitivity and interaction among the Lagrangian parameters. Therefore, we propose to formulate this problem as a nonlinear, multimodal, unconstrained, continuous optimization problem and develop a novel adaptive search space decomposition method, which progressively and adaptively re-defines the search domain (hypersphere) to evaluate the equilibrium of the system using a gradient-free optimization algorithm. We tackle three benchmark problems and evaluate a medium-sized test representing a real structural problem in this paper. The results are compared to those available in the literature regarding displacement–load curves and deformed configurations. The accuracy and robustness of the adopted methodology show a high potential for gradient-free algorithms to analyze space truss structures.

Application of the Lagrange Equation for Intelligent Sensor Vibration Control for Power Network Monitoring

Objective. In order to control the vibration of the beam structure more effectively and improve the safety and availability of the beam structure, an application study of the Lagrange equation for vibration control of smart sensors for power grid monitoring is proposed. The vibration of the beam structure, the displacement of the beam structure under the excitation of seismic acceleration, the response analytical electrical formula, and the displacement response formula of the beam structure under the action of the kinematic force are deduced. The optimal parameters of the beam-TMDI system are given, and the parameter sensitivity analysis is carried out. Then, the control effect of the TMDI system is studied by numerical analysis, and the vibration reduction effect of the TMDI system and the tuned mass damper (TMD) system is compared. Experimental results show that when the mass ratio μ of the TMDI system and the TMD system are both set to a fixed value of 0.005, and the parameter β of the TMDI system is set to 0, namely β = b = 0 , at this time, the TMDI system degenerates into a TMD system. The TMD natural frequency is 14.179 rad/s and the damping ratio is 0.0432 by the DH optimization method, while the TMD natural frequency is 14.1812 rad/s, and the damping ratio is 0.0436 by the augmented Lagrangian optimization algorithm. Conclusion. The vibration displacement response spectrum of a beam structure obtained by the frequency domain method can effectively reflect electricity in the displacement response of a beam structure. The parameters that minimize the vibration response of the beam structure can be accurately obtained by using the augmented Lagrangian parameter optimization method. The sensitivity of the TMDI system is controlled by the inertial device, and the inertial device has a significant impact on its robustness. The vibration reduction performance of the TMDI system is obviously better than in the conventional TMD systems.

A method of circle center fitting for water wall cladding

In the process of automatic laser cladding of the water wall, the determination of the cladding path is a vital step. As the cladding progresses, the central angle of the measurable pipe gradually decreases, and the pipe will be slightly deformed due to thermal stress. A method for obtaining the center of the pipe is proposed to guarantee a high fitting accuracy under the condition of small arc. The principle of the method is to introduce Lagrangian constraint parameters λx, λy to the circle center obtained by a fitting curve. Then the gradient descent method is performed on the parameters to find the optimal solution for the center of the section circle. For the problem of poor fitting accuracy for small arc, the simulation and experimental results show that when the central angle is 10 degrees, the center of the fitting circle's error is 0.2326 mm. It possesses a higher fitting circle center accuracy for water wall cladding.

Beyond Jarlskog: 699 invariants for CP violation in SMEFT

As SMEFT is a framework of growing importance to analyze high-energy data, understanding its parameter space is crucial. The latter is commonly split into CP-even and CP-odd parts, but this classification is obscured by the fact that CP violation is actually a collective effect that is best captured by considering flavor-invariant combinations of Lagrangian parameters. First we show that fermion rephasing invariance imposes that several coefficients associated to dimension-six operators can never interfere with operators of dimension ≤ 4 and thus cannot appear in any physical observable at mathcal{O} 1/Λ2. For those that can, instead, we establish a one-to-one correspondence with CP-odd flavor invariants, all linear with respect to SMEFT coefficients. We explicitly present complete lists of such linear CP-odd invariants, and carefully examine their relationship to CP breaking throughout the parameter space of coefficients of dimension ≤ 4. Requiring that these invariants all vanish, together with the Jarlskog invariant, the strong-CP phase, and the 6 CP-violating dimension-6 bosonic operators, provides 699(+1 + 1 + 6) conditions for CP conservation to hold in any observable at leading order, mathcal{O} (1/Λ2).