Lagrange Multiplier Equations Research Articles

Numerical simulation for fractional optimal control problems via euler wavelets

Abstract This paper suggests employing Euler wavelets to provide a precise and effective computational approach for certain fractional optimal control problems. The primary objective of the approach is to transform the fractional optimal control problem defined by the dynamical system and performance index into systems of algebraic equations, which then can be readily solved using matrix techniques. Since Euler polynomials are employed to build Euler wavelets and since Euler polynomials have fewer terms than most of commonly used other polynomials to build wavelets, employing them for the numerical approach results in sparser operational matrices. The speed of the recommended numerical method is improved due to the fewer terms in Euler wavelet operational matrices. We obtain the corresponding systems of algebraic equations for state variable, control variable, and Lagrange multipliers (used to determine the essential conditions of optimality) by incorporating operational matrices of Euler wavelets. Subsequently, those systems of algebraic equations are solved to obtain the numerical solution. The suggested method’s efficiency is demonstrated using a few typical examples. The suggested procedure is accurate and efficient, according to the results.

Lagrange multiplier and variational equations in mechanics

Lagrange multiplier and variational equations in mechanics

Euclidean distance degree of zero-set of two polynomials

In this note, we recall the study of the Euclidean distance degree of an algebraic set X which is the zero-point set of a polynomial (see [BSW]). Specifically, consider a hypersurface defined by a general polynomial f with its support and contains the origin i.e 0 support of f. In the paper [BSW], the authors study about the Euclidean distance degree (EDD) and found that the EDD of this hypersurface is approximately by the mixed volume (MV) of some Newton polytopes. The main purpose of this note is to study the case that the manifold is defined by two polynomials . We show that the Euclidean distance degree is equal to the solution of the Lagrange multiplier equation. Furthermore, we also find out that the EDD of this variety is not greater than the mixed volume of Newton polytopes of the associated Lagrange multiplier equations.

An effective operator splitting scheme for two-dimensional conservative nonlocal Allen–Cahn equation

An effective operator splitting scheme for solving the nonlocal Allen–Cahn equation with a Lagrange multiplier is studied. Firstly, based on the operator splitting method, the original equation is discretized into a nonlinear equation, a nonlocal equation and a Lagrange multiplier equation, respectively. Then, the nonlinear equation is solved analytically, the nonlocal equation is discretized using Crank–Nicolson format, and the Lagrange multiplier equation is solved explicitly. The mass conservation and convergence analysis of the numerical algorithm are analyzed theoretically. Numerical experiments are presented to confirm the validity of the proposed method, including the convergence, energy decline and mass conservation.

Euclidean Distance Degree and Mixed Volume

We initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface f=0 defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of f=0 equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.

Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.

Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion

Location estimation plays a crucial role in Location-Based Services (LBSs) with satisfactory user experience. The Wireless Local Area Network (WLAN) localization approach is preferred as a cost-efficient solution to indoor localization on account of the widely-deployed WLAN infrastructures. In this paper, we propose a new WLAN Received Signal Strength (RSS)-based indoor localization approach using the semi-supervised manifold alignment with dimension expansion. In concrete terms, we first construct an innovative objective function based on the augmented physical coordinates and the corresponding WLAN RSS measurements. Second, the closed-form solution to the objective function is derived out according to the Lagrange multiplier equation, which results in the manifold in physical coordinate space. Third, the target location is estimated by matching the transformed newly-collected RSS against the manifold. The localization performance with noise perturbation is analyzed upon the constructed objective function, and meanwhile, the closed-form solution to the objective function with respect to multiple types of measurements is also derived out for the sake of leveraging all of the potential measurements for indoor localization. The extensive testing results show that the proposed approach performs well in localization accuracy even at low calibration load, and its performance can be further improved by using multiple types of measurements for localization.

A novel design method based on flow pattern construction for flow passage with low flow drag and pressure drop

A novel design method for flow passage with low flow drag and pressure drop was developed based on flow pattern construction. The specific flow pattern with minimum viscous dissipation rate represents low pressure drop in the flow passage. Thus, the fluid flow requires to be optimized as a variational problem subject to the minimum viscous dissipation rate principle, which is also considered as flow pattern construction. By using calculus of variations, the governing equations for the construction of the specific flow pattern were derived from the functional extremum problem represented by the Lagrange multiplier equation. After that, a computational fluid dynamics model was set up and applied in the solution procedure of the governing equations to generate the geometry of the flow passages based on the flow pattern construction. As the illustrative examples to validate the proposed design method, the geometries of flow passages were optimized in terms of pressure drop in a wide range of Re numbers.

Study on Dynamics Model of Macro/Micro Mechanical Arm of the New Workover

Macro/micro mechanical arm is an important component of workover mechanical system. According to its composition characteristics and working principle, the arm was simplified to a plane system that all components were working in the same plane. Based on rigid body kinematics and dynamics theories, the kinematics and dynamics mathematical models of arm lifting progress were established by using constraint equations, rate equation, acceleration equations, virtual principle work, lagrange multiplier and differential-algebraic mixed equations of motion. It provides theoretical basis for kinematics and dynamics analysis of the macro/micro mechanical arm in lifting process.

Closed form solutions for water-filling problems in optimization and game frameworks

We study power control in optimization and game frameworks. In the optimization framework there is a single decision maker who assigns network resources and in the game framework users share the network resources according to Nash equilibrium. The solution of these problems is based on so-called water-filling technique, which in turn uses bisection method for solution of non-linear equations for Lagrange multipliers. Here we provide a closed form solution to the water-filling problem, which allows us to solve it in a finite number of operations. Also, we produce a closed form solution for the Nash equilibrium in symmetric Gaussian interference game with an arbitrary number of users. Even though the game is symmetric, there is an intrinsic hierarchical structure induced by the quantity of the resources available to the users. We use this hierarchical structure to perform a successive reduction of the game. In addition to its mathematical beauty, the explicit solution allows one to study limiting cases when the crosstalk coefficient is either small or large. We provide an alternative simple proof of the convergence of the Iterative Water Filling Algorithm. Furthermore, it turns out that the convergence of Iterative Water Filling Algorithm slows down when the crosstalk coefficient is large. Using the closed form solution, we can avoid this problem. Finally, we compare the non-cooperative approach with the cooperative approach and show that the non-cooperative approach results in a more fair resource distribution.

Conservation properties of the bridging domain method for coupled molecular/continuum dynamics

AbstractAn overlaid domain decomposition method, called the bridging domain method, for the coupled simulation of molecular/continuum systems is analyzed. In this method, compatibility between the atomistic and the continuum subdomains is enforced by Lagrange multipliers. Two forms are considered: (1) a consistent constraint form, which employs the exact non‐diagonal Lagrange multiplier equations and thus enforces compatibility exactly, but is computationally demanding and (2) a diagonalized constraint form, which is computationally more efficient. It is shown that the consistent constraint form conserves linear momentum, angular momentum and energy. The diagonalized constraint form dissipates energy. The diagonalized form is shown to be more effective in suppressing spurious reflections at the interface. Copyright © 2008 John Wiley & Sons, Ltd.

Desensitizing models using covariance matrix transforms or counter‐balanced distortions

AbstractThis paper presents a generalization of the Lagrange multiplier equation for a regression subject to constraints. It introduces two methods for desensitizing models to anticipated spectral artifacts such as baseline variations, wavelength shift, or trace contaminants. For models derived from a covariance matrix such as multiple linear regression (MLR) and principal components regression (PCR) models, the first method shows how a covariance matrix can be desensitized to an artifact spectrum, v, by adding σ2 v ⊗ v to it. For models not derived from a covariance matrix, such as partial least squares (PLS) or neural network (NN) models, the second method shows how distorted copies of the original spectra can be prepared in a counter‐balanced manner to achieve desensitization. Unlike earlier methods that added random distortions to spectra, these new methods never introduce any accidental correlations between the added distortions and the Y‐block. The degree of desensitization is controlled by a parameter, σ, for each artifact from zero (no desensitization) to infinity (complete desensitization, which is the Lagrange multiplier limit). Unlike Lagrange multipliers, these methods permit partial desensitization so we can individually vary the degree of desensitization to each artifact, which is important when desensitization to one artifact inhibits desensitization to another. Copyright © 2005 John Wiley & Sons, Ltd.

The method of determining internal forces at any cross section of links in mechanisms

In the paper it is introduced a method of determining internal forces at any cross section of the links of mechanisms. As known, so far it is used the method of D'Alembert, which consists of two steps, the determination of the acceleration states of links and the establishment of the equilibrium equations for the set of forces including the forces of inertia and the internal forces at the cross section. A. I. Lurie proposed a method of analytical mechanics for this problem. Its concept is to make a new system called the released one by cutting the link at a cross section under consideration and adding some coordinates. Only one condition putting restriction on the released system is the additional coordinates must equal zero. Under this restriction the new created system is coincided to the original one. This restriction is equivalent to put the mechanical constraints, whose reaction forces are the components of internal forces at the cross section under consideration. It is necessary emphasize that the Lurie's method is convenient only for opened loops, but is not applied for closed ones. Moreover, the Lagrange's multiplier equations applied by A. I. Lurie are unsuitable. In this paper it is presented the generalized Lurie's method, which is applied for the opened and closed loops by using the Principle of Compatibility.

A constrained variational approach for energy derivatives in Fock-space multireference coupled-cluster theory.

In this paper, we present a formulation based on constrained variational approach to enable efficient computation of energy derivatives using Fock-space multireference coupled-cluster theory. Adopting conventional normal ordered exponential with Bloch projection approach, we present a method of deriving equations when general incomplete model spaces are used. Essential simplifications arise when effective Hamiltonian definition becomes explicit as in the case of complete model spaces or some special quasicomplete model spaces. We apply the method to derive explicit generic expressions upto third-order energy derivatives for [0,1], [1,0], and [1,1] Fock-space sectors. Specific diagrammatic expressions for zeroth-order Lagrange multiplier equations for [0,1], [1,0], and [1,1] sectors are presented.

A Schwarz Preconditioner for a Hybridized Mixed Method

AbstractIn this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest-order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to precondition even the higher-order cases of these methods.

Data Sharpening for Nonparametric Inference Subject to Constraints

Data sharpening involves perturbing the data to improve the performance of a statistical method. The versions of it that have been proposed in the past have been for bias reduction in curve estimation, and the amount of perturbation of each datum has been determined by an explicit formula. This article suggests a distance-based form of data sharpening, in which the sum of the distances that data are moved is minimized subject to a constraint imposed on an estimator. The constraint could be one that leads to bias reduction, or to variance or variability reduction, or to a curve estimator being monotone or unimodal. In contrast to earlier versions of the method, in the form presented in this article the amount and extent of sharpening is determined implicitly by a formula that is typically given as the solution of a Lagrange-multiplier equation. Sometimes the solution can be found by Newton–Raphson iteration, although when qualitative constraints are imposed it usually requires quadratic programming or a related method.

Basic equations, theory and principle of computational stock market (III)—basic theories

By basic equations, two basic theories are presented: 1. Theory of stock' s value υ* (t) = υ*(0) exp(ar*2 t); 2. Theory of conservation of stock' s energy. Let stock' s energy ϕ be defined as a quadratic function of stock' s price υ and its derivative\(\dot v,\phi = {\rm A}v^2 + Bv\dot v + C\dot v^2 + Dv\), under the constraint of basic equation, the problem was reduced to a problem of constrained optimization along optimal path. Using Lagrange multiplier and Euler equation of variation method, it can be proved that ϕ keeps conservation for any υ,\(\dot v\). The application of these equations and theories on judgement and analysis of tendency of stock market are given, and the judgement is checked to be correct by the recorded tendency of Shenzhen and Shanghai stock markets.

ON MODELLING VARIABLE STRUCTURE DYNAMICS OF HYBRID PARAMETER MULTIPLE BODY SYSTEMS

Recently a methodology for closed-form modelling of non-holonomic hybrid parameter multiple body systems (HPMBS) was presented. The tool provides a means to model non-holonomic HPMBS with sets of minimal closed-form hybrid differential equations; these equations are minimal in the sense that there are no Lagrange multiplier equations required. Also provided by the method is a means to generate closed-form models of the momentum change imparted on a HPMBS when the system undergoes impulsive loads or configuration space changes. In this paper, these techniques will be merged to form an algorithm for modelling HPMBS under the influence of changes in the configuration space due to the variable presence of, or lack of, constraints. It should be noted that restitution coefficients are not needed to describe momentum transferred within the system when the configuration changes. Included in this work is the explicit minimal closed-form model of a flexible two-link planar manipulator interacting with a work-piece and the numerical simulation of this system under various conditions. The process of modelling this system will elucidate the algorithm which is applicable to higher order systems.

Parameter estimation to improve accuracy of mechanical system simulations using rigid body codes

A procedure is proposed to improve the accuracy of responses obtained from rigid body codes used to simulate mechanisms, or spring loaded mechanisms that act like structures. In the method, dynamic parameters of a constrained planar mechanical system were estimated from the independent kinematic responses and measured external forces of the system. The Lagrange equation was used to develop the equations of motion of the system and the linear equations with respect to the dynamic parameters being estimated. The generalized coordinate partitioning technique using the Lagrange multiplier and constraint equations of the system was applied to reduce the dimension of the constrained dynamic system. The rank deficient linear system was solved by SVD (singular value decomposition). ADAMS simultions for three different models revealed good correlation between the actual and the estimated parameters, by comparison between the simulated system response from the actual parameters and from the estimated parameters.

One-dimensional modelling of contact impact problem in guided transport vehicle crash

It is impractical to rely solely on prototype testing for the global analysis of structures in impact. Numerical simulation is also necessary. We propose using a lumped-mass model, linked by springs and dampers. In certain crash configurations, links between bodies may be fully compressed. Difficulties encountered in solving these types of problems include the treatment of contact velocity reversals in impact. We state the contact equations between bodies using the Lagrange multiplier equations, where we introduce a restitution coefficient which takes body compressibility and energy losses into account. Then, we propose a new algorithm, and show in a one-dimensional example its advantages, over a penalty method. Finally, we apply this method to a 30 km/h frontal impact of a railway carriage.