Dual Lagrangian Research Articles

Dual Descent Augmented Lagrangian Method and Alternating Direction Method of Multipliers

Dual Descent Augmented Lagrangian Method and Alternating Direction Method of Multipliers

Duality symmetry of quantum electrodynamics and magnetic charges

The duality symmetry between electricity and magnetism hidden in classical Maxwell equations suggests the existence of dual charges, which have usually been interpreted as magnetic charges and have not been observed in experiments. In quantum electrodynamics (QED), the electric and magnetic fields are unified into a single gauge field, which makes this symmetry inconspicuous. Here, we revisit the duality symmetry of QED by introducing a dual gauge field and a dual symmetric Lagrangian. Within the framework of gauge-field theory, we show that the electric–magnetic duality symmetry cannot give any new conservation law. By checking the charge–charge interaction and the quantum Lorentz-force equation, we find that the introduced dual charges are electric charges, not magnetic charges. Notably, we show that genuine magnetic charges are incompatible with the traditional QED gauge-field framework, as the interaction between a magnetic charge and an electric charge cannot be mediated via the exchange of gauge photons.

A two steps Lagrangian–Eulerian numerical model for the simulation of explosive welding of three dissimilar materials joints

Explosion welding (EXW) is a solid-state joining process used to produce lap joints out of metal plates of dissimilar materials. During the process, a controlled explosive detonation results in a pressure wave pushing one of the plates to be welded, called flyer, against the other with high velocity. The high pressure and temperature generated, because of the impact energy decaying into heat, create the conditions for solid bonding phenomenon to take place. Due to the complexity of experimental tests, numerical simulation is considered a fundamental design tool for the process. Different approaches are found in literature to simulate the process. In this paper, a dual step Lagrangian–Eulerian approach is proposed to evaluate the effect of different explosives on the final quality of the weld. The numerical model was developed using the commercial software ABAQUS. A three dissimilar materials joint is considered, made of two outer plates of AA5083 aluminum alloy and A516 steel, respectively, and an intermediate layer made of AA1050 aluminum alloy. The model was first validated by comparing the wavy profile of the intermediate layer after the weld, and then used to evaluate the effect of different explosive by analyzing the distribution of the main process parameters as well as the presence of voids.

Dual formulations of vortex strings in a supersymmetric Abelian Higgs model

We discuss dual formulations of vortex strings (magnetic flux tubes) in the four-dimensional ${\cal N} =1$ supersymmetric Abelian Higgs model with the Fayet--Iliopoulos term in the superspace formalism. The Lagrangian of the model is dualized into a Lagrangian of the $BF$-type described by a chiral spinor gauge superfield including a 2-form gauge field. The dual Lagrangian is further dualized into a Lagrangian given by a chiral spinor superfield including a massive 2-form field. In both of the dual formulations, we obtain a superfield into which the vortex strings and their superpartners are embedded. We show the dual Lagrangians in terms of a superspace and a component formalism. In these dual Lagrangians, we explicitly show that the vortex strings of the original model are described by a string current electrically coupled with the 2-form gauge field or the massive 2-form field.

Deconstructing Odorant Identity via Primacy in Dual Networks.

In the olfactory system, odor percepts retain their identity despite substantial variations in concentration, timing, and background. We study a novel strategy for encoding intensity-invariant stimulus identity that is based on representing relative rather than absolute values of stimulus features. For example, in what is known as the primacy coding model, odorant identities are represented by the conditions that some odorant receptors are activated more strongly than others. Because, in this scheme, odorant identity depends only on the relative amplitudes of olfactory receptor responses, identity is invariant to changes in both intensity and monotonic nonlinear transformations of its neuronal responses. Here we show that sparse vectors representing odorant mixtures can be recovered in a compressed sensing framework via elastic net loss minimization. In the primacy model, this minimization is performed under the constraint that some receptors respond to a given odorant more strongly than others. Using duality transformation, we show that this constrained optimization problem can be solved by a neural network whose Lyapunov function represents the dual Lagrangian and whose neural responses represent the Lagrange coefficients of primacy and other constraints. The connectivity in such a dual network resembles known features of connectivity in olfactory circuits. We thus propose that networks in the piriform cortex implement dual computations to compute odorant identity with the sparse activities of individual neurons representing Lagrange coefficients. More generally, we propose that sparse neuronal firing rates may represent Lagrange multipliers, which we call the dual brain hypothesis. We show such a formulation is well suited to solve problems with multiple interacting relative constraints.

Fast Algorithms for Acoustic Impulse Response Shaping

Impulse response shaping (IRS) is a prefiltering technique for modifying the characteristics of a linear channel to achieve desirable characteristics. Acoustic impulse response shaping is used to apply partial equalization to reduce the effects of reverberation on audio signals propagating inside a room and is thus used for listening room compensation. To be capable of responding rapidly to time varying room acoustics, a shaping algorithm must be computationally fast; this is difficult to achieve since the filters to be designed may have thousands of coefficients. This paper derives variants of the Dual Augmented Lagrangian Method, presents their application to the design of the shaping filters, and validates their efficacy through experimental studies. In addition, the formulations have been modified to design shaping filters that are robust to change in microphone positions.

Optical helicity and Hertz vectors

We study the conserved quantity associated with the dual symmetry of the Maxwell equations, called the optical helicity, by means of transverse Hertz vectors. In the presence of charges, its evolution yields the integral of E⋅B which is the anomaly term for chiral fermions. We also discuss the helicity change in condensed matter systems where topological magnetoelectric effect emerges. An alternative expression of the optical helicity is also found. Lastly, a dual symmetric Hertz Lagrangian is constructed and its conserved charge is derived.

Efficient image reconstruction for fluorescence molecular tomography via linear regression approximation scheme with dual augmented Lagrangian method

With the development of non-contact fluorescence molecular tomography (FMT) imaging system, multi-fluorescence projections data can be obtained to improve the quality of reconstruction images. However, it remains a challenging issue to obtain fast and accurate reconstruction of the fluorescent probe distribution due to the large computational burden and the ill-posed nature of the inverse problem. In this work, we present an innovative method associating dual augmented lagrangian method (DALM) with a linear regression approximation (LRA) strategy to locate the fluorescence probe, which guarantees the accuracy, efficiency, and robustness for FMT reconstruction. Numerical experiments based on a heterogeneous phantom are performed to validate the feasibility of the proposed method. The results demonstrate that the proposed method can achieve accurate target localization, and satisfactory computational efficiency. Furthermore, this approach is robust even under quite ill-posed condition.

Dual polarization radar Lagrangian parameters: a statistics-based probabilistic nowcasting model

The aim of this study is to present a statistics-based Lagrangian nowcasting model to predict intense rainfall convective events based on dual polarization radar parameters. The data employed in this study are from X-band radar collected during the CHUVA-Vale campaign from November 2011 to March 2012 in southeast Brazil. The model was designed to catch the important physical characteristics of storms, such as the presence of supercooled water above 0 °C isotherm, vertical ice crystals in high levels, graupel development in the mixed-phase layer and storm vertical growth, using polarimetric radar in the mixed-phase layer. These parameters are based on different polarimetric radar quantities in the mixed phase, such as negative differential reflectivity (Z DR) and specific differential phase (K DP), low correlation coefficient (ρ hv) and high reflectivity Z h values. Storms were tracked to allow the Lagrangian temporal derivation. The model is based on the estimation of the proportion of radar echo volume in the mixed phase that is likely to be associated with intense storm hydrometeors. Thirteen parameters are used in this probabilistic nowcasting model, which is able to predict the potential for future storm development. The model distinguishes two different categories of storms, intense and non-intense rain cell events by determining how many parameters reach the “intense” storm threshold.

Dual QCD, confinement potential and thermal effects

Based on the topological structure of non-Abelian gauge theories, a dual QCD gauge formulation has been developed in terms of magnetic symmetry, which manifests the topological structure of the symmetry group in a non-trivial way. The dynamical configuration of the resulting dual QCD vacuum and its flux tube configuration have been investigated for analyzing the non-perturbative features of QCD. Utilizing the dual QCD Lagrangian in the dynamically broken phase of magnetic symmetry and applying Zwanziger formalism, the non-perturbative gluon propagator has been derived and used to extract the quark confining potential for both quenched and full QCD. The quenched confining mechanism is responsible for linear confinement and points towards the permanent confinement of the colored quarks inside the hadrons. In full QCD due to light-quark polarization the quark-antiquark potential automatically screens signaling the instability in the flux tube at large inter-quark distances and such screening increases with the increase of infrared cutoff. Using the partition function approach alongwith the mean-field treatment for the QCD monopole field the thermal response of the QCD vacuum has been analyzed by deriving the finite-temperature form of quark confining potential. A continuous vanishing of the associated string tension has been observed in the vicinity of critical temperature, which, in turn, leads to the restoration of magnetic symmetry in the domain of high temperatures and signals the onset of second-order deconfinement phase transition.

On the Quantum Mechanical Treatment of the Bateman-Morse-Feshbach Damped Oscillator with Variable Mass

The harmonic oscillator with time-dependent (indefinite and variable) mass subject to the force proportional to velocity is studied by extending Bateman’s dual Lagrangian and Hamiltonian formalism. To study the quantum analog of such a dissipative system, the Batemann-Morse-Feshback classical Hamiltonian of the damped harmonic oscillator with varying (time-dependent) mass is canonically quantized. In order to discuss the stability of the quantum dissipative system due to the influence of varying mass and the dissipative force, we derived a formula for the vacuum state of the dissipative system with the help of quantum field theoretical framework. It is shown that the formula based on this simple model could be used to study the influence of dissipation such as the instability due to the dissipative force and/or the variable mass. It is understood that the change in the oscillator mass corresponds to a control parameter in quantum dissipative systems.

A globally and quadratically convergent primal–dual augmented Lagrangian algorithm for equality constrained optimization

We present a primal–dual augmented Lagrangian method to solve an equality constrained minimization problem. This is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing an augmented Lagrangian function. An important aspect of this approach is that, by a choice of suitable updating rules of parameters, the algorithm reduces to a regularized Newton method applied to a sequence of optimality systems. The global convergence is proved under mild assumptions. An asymptotic analysis is also presented and quadratic convergence is proved under standard regularity assumptions. Some numerical results show that the method is very efficient and robust.

S-duality for D3-brane in NS-NS and R-R backgrounds

We construct the low-energy effective field theory for a D3-brane in constant R-R 2-form potential background as the S-dual theory of a D3-brane in NS-NS B-field background. Despite the non-Abelian algebra of the noncommutative U(1) gauge symmetry, the electromagnetic duality transformation can be carried out to all orders, and the dual Lagrangian is given in a compact form. The gauge algebra is found to be a mixture of a deformed area-preserving diffeomorphism and the usual U(1) gauge symmetry.

Performance evaluation of popular l1-minimization algorithms in the context of Compressed Sensing

Compressed sensing (CS) is a data acquisition technique that is gaining popularity because of the fact that the reconstruction of the original signal is possible even if it was sampled at a sub-Nyquist rate. In contrast to the traditional sampling method, in CS we take a few measurements from the signal and the original signal can then be reconstructed from these measurements by using an optimization technique called l1-minimization. Computer engineers and mathematician have been equally fascinated by this latest trend in digital signal processing. In this work we perform an evaluation of different l1-minimization algorithms for their performance in reconstructing the signal in the context of CS. The algorithms that have been evaluated are PALM (Primal Augmented Lagrangian Multiplier method), DALM (Dual Augmented Lagrangian Multiplier method) and ISTA (Iterative Soft Thresholding Algorithm). The evaluation is done based on three parameters which are execution time, PSNR and RMSE.

An alternating direction method for second-order conic programming

An alternating direction dual augmented Lagrangian method for second-order cone programming (SOCP) problems is proposed. In the algorithm, at each iteration it first minimizes the dual augmented Lagrangian function with respect to the dual variables, and then with respect to the dual slack variables while keeping the other two variables fixed, and then finally it updates the Lagrange multipliers. Convergence result is given. Numerical results demonstrate that our method is fast and efficient, especially for the large-scale second-order cone programming.

Learning a common substructure of multiple graphical Gaussian models

Properties of data are frequently seen to vary depending on the sampled situations, which usually change along a time evolution or owing to environmental effects. One way to analyze such data is to find invariances, or representative features kept constant over changes. The aim of this paper is to identify one such feature, namely interactions or dependencies among variables that are common across multiple datasets collected under different conditions. To that end, we propose a common substructure learning (CSSL) framework based on a graphical Gaussian model. We further present a simple learning algorithm based on the Dual Augmented Lagrangian and the Alternating Direction Method of Multipliers. We confirm the performance of CSSL over other existing techniques in finding unchanging dependency structures in multiple datasets through numerical simulations on synthetic data and through a real world application to anomaly detection in automobile sensors.

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

We consider the one-dimensional motion of a particle immersed in a potential field U ( x ) under the influence of a frictional (dissipative) force linear in velocity ( − γ x ̇ ) and a time-dependent external force ( K ( t ) ). The dissipative system subject to these forces is discussed by introducing the extended Bateman’s system, which is described by the Lagrangian: ℒ = m x ̇ y ̇ − U ( x + 1 2 y ) + U ( x − 1 2 y ) + γ 2 ( x y ̇ − y x ̇ ) − x K ( t ) + y K ( t ) , which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting U ( x ± y / 2 ) = 1 2 k ( x ± y / 2 ) 2 specifically for a dual extended damped–amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman’s Hamiltonian ℋ . The Heisenberg equations of motion utilizing the quantized Hamiltonian ℋ ̂ surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K ( t ) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped–amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.

Super-Linear Convergence of Dual Augmented Lagrangian Algorithm for Sparsity Regularized Estimation

We analyze the convergence behaviour of a recently proposed algorithm for regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is based on a new interpretation of DAL as a pr...

Topological interactions of non-Abelian vortices with quasiparticles in high density QCD

Non-Abelian vortices are topologically stable objects in the color-flavor locked (CFL) phase of dense QCD. We derive a dual Lagrangian starting with the Ginzburg-Landau effective Lagrangian for the CFL phase, and obtain topological interactions of non-Abelian vortices with quasiparticles such as $U(1)_B$ Nambu-Goldstone bosons (phonons) and massive gluons. We find that the phonons couple to the translational zero modes of the vortices while the gluons couple to their orientational zero modes in the internal space.

Alternating direction augmented Lagrangian methods for semidefinite programming

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our algorithms are robust and very efficient due to their ability or exploit special structures, such as sparsity and constraint orthogonality in these problems.