Terms Of Lagrange Multipliers Research Articles

Highly conservative Allen–Cahn-type multi-phase-field model and evaluation of its accuracy

Abstract In the engineering field, it is necessary to construct a numerical model that can reproduce multiphase flows containing three or more phases with high accuracy. In our previous study, by extending the conservative Allen–Cahn (CAC) model, which is computationally considerably more efficient than the conventional Cahn–Hilliard (CH) model, to the multiphase flow problem with three or more phases, we developed the conservative Allen–Cahn type multi-phase-field (CAC–MPF) model. In this study, we newly construct the improved CAC–MPF model by modifying the Lagrange multiplier term of the previous CAC–MPF model to a conservative form. The accuracy of the improved CAC–MPF model is evaluated through a comparison of five models: three CAC–MPF models and two CH–MPF models. The results indicate that the improved CAC–MPF model can accurately and efficiently perform simulations of multiphase flows with three or more phases while maintaining the same level of volume conservation as the CH model. We expect that the improved CAC–MPF model will be applied to various engineering problems with multiphase flows with high accuracy. Graphic abstract

A Linear, Second-Order, and Unconditionally Energy-Stable Method for the L2-Gradient Flow-Based Phase-Field Crystal Equation

To solve the L2-gradient flow-based phase-field crystal equation accurately and efficiently, we present a linear, second-order, and unconditionally energy-stable method. We first truncate the quartic function in the Swift–Hohenberg energy functional. We also put the truncated function in the expansive part of the energy and add an extra term to have a linear convex splitting. Then, we apply the linear convex splitting to both the L2-gradient flow and the nonlocal Lagrange multiplier terms and combine it with the second-order SSP-IMEX-RK method. We prove that the proposed method is mass-conservative and unconditionally energy-stable. Numerical experiments including standard tests in the classical H−1-gradient flow-based phase-field crystal equation support that the proposed method is second-order accurate in time, mass conservative, and unconditionally energy-stable.

Unconditionally Maximum Bound Principle Preserving Linear Schemes for the Conservative Allen–Cahn Equation with Nonlocal Constraint

In comparison with the Cahn–Hilliard equation, the classic Allen-Cahn equation satisfies the maximum bound principle (MBP) but fails to conserve the mass along the time. In this paper, we consider the MBP and corresponding numerical schemes for the modified Allen–Cahn equation, which is formed by introducing a nonlocal Lagrange multiplier term to enforce the mass conservation. We first study sufficient conditions on the nonlinear potentials under which the MBP holds and provide some concrete examples of nonlinear functions. Then we propose first and second order stabilized exponential time differencing schemes for time integration, which are linear schemes and unconditionally preserve the MBP in the time discrete level. Convergence of these schemes is analyzed as well as their energy stability. Various two and three dimensional numerical experiments are also carried out to validate the theoretical results and demonstrate the performance of the proposed schemes.

Variational interior and interface transmission conditions: multidomain mixed Darcy/Stokes control problems

Multidomain mixed pairs of coupled stationary constrained boundary value problems, are formulated variationally in frameworks of reflexive Banach spaces as macro-hybrid systems, and analyzed. Interior synchronizing and interface coupling transmission constraints are modeled in terms of Lagrange multipliers as solutions of dual variational subpotential maximal monotone inclusions. Existence and uniqueness results are established via resolvent fixed-point characterizations of the corresponding primal and dual macro-hybridized problems. Multimedia mixed mechanical subsurface-surface Darcy/Stokes incompressible flow coupled pairs with intrinsic distributed control constraints exemplify the general macro-hybrid mixed systems. Coupling dual-primal and primal-dual interface macro-hybrid mixed variational conditions are lastly developed and applied to the Darcy/Stokes multimedia pair model.

Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia.

SOLUTION SETS OF SECOND-ORDER CONE LINEAR FRACTIONAL OPTIMIZATION PROBLEMS

We characterize the solution set for a second-order cone linear fractional optimization problem (P). We present sequential Lagrange multiplier characterizations of the solution set for the problem (P) in terms of sequential Lagrange multipliers of a known solution of (P).

Normalized Gradient Flow with Lagrange Multiplier for Computing Ground States of Bose--Einstein Condensates

The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evolution method is one of the most popular techniques for computing the ground states of Bose--Einstein condensates (BECs). In this paper, we revisit the time discretizations for the GFDN and its generalization to the multicomponent BECs. Several widely used time discretizations are demonstrated to be not accurate for computing the ground state solution in the general case, especially for the multicomponent BECs with two or more constraints even for the most accepted linearized backward Euler schemes. More precisely, these schemes usually converge to a solution with an error depending on the time step size. To accurately and efficiently compute the ground state solution of BECs, we propose the gradient flow with Lagrange multiplier (GFLM) method which can be viewed as the modified GFDN by introducing the explicit Lagrange multiplier terms or an approximation of the continuous normalized gradient flow. Through analysis and numerical computation, we clarify that, in order to accurately compute the ground state solution, the GFDN method must be discretized in very special ways in time such as the linearized backward Euler scheme, while the GFLM can be discretized by various schemes and works for multicomponent BEC with multiple constraints. We propose a simple backward-forward Euler scheme to discretize the GFLM, where only a linear system with constant coefficients needs to be solved at each time step. Taking the spin-1 BEC as an example, we demonstrate how the traditional GFDN approach fails in computing the ground state solution of general multicomponent BECs accurately. The proposed GFLM with backward-forward Euler discretization is shown to be accurate and efficient. Extensive numerical results are provided to verify our claims.

Decoupled, non-iterative, and unconditionally energy stable large time stepping method for the three-phase Cahn-Hilliard phase-field model

In this paper, we consider numerical approximations for a three-phase phase-field model, where three fourth-order Cahn-Hilliard equations are nonlinearly coupled together through a Lagrange multiplier term and a sixth-order polynomial bulk potential. By combining the recently developed SAV approach with the linear stabilization technique, we arrive at a novel stabilized-SAV scheme. At each time step, the scheme requires solving only four linear biharmonic equations with constant coefficients, making it the first, to the best of the author's knowledge, totally decoupled, second-order accurate, linear, and unconditionally energy stable scheme for the model. We further prove the unconditional energy stability rigorously and demonstrate the stability and the accuracy of the scheme numerically through the comparisons with the non-stabilized SAV scheme for simulating numerous benchmark numerical examples in 2D and 3D.

Deformations of the moduli space and superpotential flows in 3D SUSY QCD

We study the moduli space of three-dimensional $\mathcal{N}=2$ SQCD with $SU(N)$ gauge group and $F<N$ massless flavors. In the case of an $SU(2)$ theory with a single massless flavor, we explicitly calculate the quantum constraint $YM=1$ and generalize the calculation to models with arbitrary $N$ and $F=N-1$ flavors. In theories with $F<N-1$ flavors, we find that analogous constraints exist in locally defined coordinate charts of the moduli space. The existence of such constraints allows us to show that the Coulomb branch superpotential generated by single monopole effects is equivalent to the superpotential generated by multi-monopole contributions on the mixed Higgs-Coulomb branch. As a check for our result, we implement the local constraints as Lagrange multiplier terms in the superpotential and verify that deformations of a theory by a large holomorphic mass term for the matter fields results in a flow of the superpotential from the $F$-flavor model to the superpotential of an $(F-1)$-flavor model.

Pansharpening using data-centric optimization approach

ABSTRACT Earth’s observation satellites provide simultaneously both multispectral (XS) and panchromatic (pan) images but XS image has a lower spatial resolution when compared to pan image. Pansharpening is a pixel-level fusion technique resulting in a high-resolution multispectral image in terms of both spatial and spectral resolution. The problem lies in maintaining the spectral characteristics of each channel of the XS image when pan image is used to estimate the high spatial XS image. Many techniques have been proposed to address the problem. A popular method involves a sensor-based approach where correlation among the XS channels and correlation between the pan and spectral channels are incorporated. In this paper, we take a wholesome approach based on the reflectance data irrespective of the sensor physics. A linear regression model is formulated between the XS channel and the panchromatic data. We formulate an optimization problem in terms of Lagrange multiplier to maximise the spectral consistency of the fused data with respect to the original XS data, and to minimise the error in variance between the reference data and the computed data. We validate and compare our method with IHS and Brovey methods based on evaluation metrics such as Chi-square test and the R 2 test. The implementation is done and presented using IKONOS satellite data.

Characterizing the solution set of convex optimization problems without convexity of constraints

Some characterizations of solution sets of a convex optimization problem with a convex feasible set described by tangentially convex constraints are given. The results are expressed in terms of convex subdifferentials, tangential subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian-type function and establish a constant pseudo Lagrangian-type property for the solution set. This property is still valid in the case of a pseudoconvex locally Lipschitz objective function, and then used to derive Lagrange multiplier-based characterizations of the solution set. Some examples are given to illustrate the significances of our theoretical results.

New Mixed Elements for Maxwell Equations

New inf-sup stable mixed elements are proposed and analyzed for solving the Maxwell equations in terms of electric field and Lagrange multiplier. Nodal-continuous Lagrange elements of any order on simplexes in two- and three-dimensional spaces can be used for the electric field. The multiplier is compatibly approximated always by the discontinuous piecewise constant elements. A general theory of stability and error estimates is developed; when applied to the eigenvalue problem, we show that the proposed mixed elements provide spectral-correct, spurious-free approximations. Essentially optimal error bounds (only up to an arbitrarily small constant) are obtained for eigenvalues and for both singular and smooth solutions. Numerical experiments are performed to illustrate the theoretical results.

DATA FUSION OF PANCHROMATIC AND MULTISPECTRAL IMAGES BASED ON OPTIMIZATION USING LAGRANGE MULTIPLIER

Abstract. The remote sensing satellites provide complementary images of different resolutions which need to be integrated using the techniques of image fusion. In this paper, image fusion using the IKONOS satellite data is discussed. Unlike other models which are based on sensor model, our approach is data centric including the effects of the sensor as well as the reflectance characteristics of the imaged object. A linear relationship is built between the panchromatic channel and the multispectral channel data. We then formulate a minimisation function in terms of Lagrange multiplier to optimally maximise the spectral consistency and minimise the error in variance. The variances of the downsampled multispectral channels are observed and compared with the original multispectral data. A chi-square goodness of fit test is performed to evaluate the data computed based on our algorithm. Simulation results are presented using the IKONOS 1m resolution panchromatic and 4 m resolution multispectral data.

Informational theory of relativity

Assuming the minimal time to send a bit of information in the Einstein clock synchronization of the two clocks located at different positions, we introduce the extended metric to the information space. This modification of relativity changes the red shift formula keeping the geodesic equation intact. Extending the gauge symmetry hidden in the metric to the 5-dimensional general invariance, we start with the Einstein-Hilbert action in the 5-dimensional space-time. After the 4+1 decomposition, we obtain the effective action which includes the Einstein-Hilbert action for gravity, the Maxwell-like action for the velocity field and the Lagrange multiplier term which ensures the normalization of the time-like velocity field. As an application, we investigate a solution of the field equations in the case that a 4-dimensional part of the extended metric is spherically symmetric, which exhibits Schwarzschild-like space-time but with the minimal radius. As a discussion we present a possible informational model of synchronization process which is inherently stochastic. The model enables us to interpret the information quantity as a new spatial coordinate.

Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints

<p style='text-indent:20px;'>In this paper, we consider an uncertain convex optimization problem with a robust convex feasible set described by locally Lipschitz constraints. Using robust optimization approach, we give some new characterizations of robust solution sets of the problem. Such characterizations are expressed in terms of convex subdifferentails, Clarke subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian function and establish constant pseudo Lagrangian-type property for the robust solution set. We then used to derive Lagrange multiplier-based characterizations of robust solution set. By means of linear scalarization, the results are applied to derive characterizations of weakly and properly robust efficient solution sets of convex multi-objective optimization problems with data uncertainty. Some examples are given to illustrate the significance of the results.

Delay Minimal Policies in Energy Harvesting Communication Systems

We characterize delay minimal power scheduling policies in energy harvesting communication systems. We consider a continuous-time system, where the delay experienced by each bit is given by the time spent by the bit in the queue waiting to be transmitted to its receiver. We first consider a single-user channel, where the transmitter has a finite-sized battery to save its harvested energy. Data arrives during the course of communication and are saved in a finite data buffer as well. We find the optimal power policy that minimizes the average delay experienced by the bits subject to energy and data causality constraints. We characterize the optimal solution in terms of Lagrange multipliers, and calculate their values in a recursive manner. We show that, different from the existing literature, the optimum transmission power is not constant between the energy and data arrival events; the transmission power starts high, decreases linearly, and potentially reaches zero between energy and data arrivals. Intuitively, untransmitted bits experience cumulative delay due to the bits to be transmitted ahead of them, and hence the reason for transmission power starting high and decreasing over time. Next, we study a multiuser version of this problem, namely, a two-user broadcast channel, and characterize the optimal transmission policies that minimize the sum delay. For this setting, we consider the case, where the transmitter has an infinite-sized battery, and that all data packets intended for the receivers are available at the beginning of the communication session. We characterize the optimal solution in terms of Lagrange multipliers, and present an iterative solution that calculates their values. Our results show that in the optimal policy, both users may not be served simultaneously all the time; there may be times, where only one of the two users is served alone. We also show that the optimal policy may have gaps in transmission in between energy arrivals, where none of the users is served, echoing the results of the single-user setting.

A Mixed $H^1$-Conforming Finite Element Method for Solving Maxwell's Equations with Non-$H^1$ Solution

In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is intr...

Charge redistribution in QM:QM ONIOM model systems: a constrained density functional theory approach

ABSTRACTThe ONIOM hybrid method has found considerable success in QM:QM studies designed to approximate a high level of theory at a significantly reduced cost. This cost reduction is achieved by treating only a small model system with the target level of theory and the rest of the system with a low, inexpensive, level of theory. However, the choice of an appropriate model system is a limiting factor in ONIOM calculations and effects such as charge redistribution across the model system boundary must be considered as a source of error. In an effort to increase the general applicability of the ONIOM model, a method to treat the charge redistribution effect is developed using constrained density functional theory (CDFT) to constrain the charge experienced by the model system in the full calculation to the link atoms in the truncated model system calculations. Two separate CDFT-ONIOM schemes are developed and tested on a set of 20 reactions with eight combinations of levels of theory. It is shown that a scheme using a scaled Lagrange multiplier term obtained from the low-level CDFT model calculation outperforms ONIOM at each combination of levels of theory from 32% to 70%.

Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

Triply periodic minimal surface using a modified Allen–Cahn equation

In this paper, we propose a fast and efficient model for triply periodic minimal surface. The proposed model is based on the Allen–Cahn equation with a Lagrange multiplier term. The Allen–Cahn equation has the motion of mean curvature. And the Lagrange multiplier term corresponding to the constant volume constraint also relates to the average of mean curvature. By combining two terms, the mean curvature will be constant everywhere on the surface at the equilibrium condition. The proposed numerical method with the second-order accuracy of time and space exhibits excellent stability. In addition, the resulting discrete system is solved by a fast numerical method such as a multigrid method. Various numerical experiments are performed to demonstrate the accuracy and robustness of the proposed method.