Recursive approach for constructing the q = 1/2 maximum entropy distribution from redundant data

A recursive approach for computing the q = 1/2 nonextensive maximum entropy distribution of the previously introduced formalism for data subset selection is proposed. Such an approach is based on an iterative biorthogonalization technique, which allows for the incorporation of the Lagrange multipliers that determine the distribution to the workings of the algorithm devised for selecting relevant data subsets. This technique circumvents the necessity of inverting operators and yields a recursive procedure to appropriately modify the Lagrange multipliers so as to account for each new constraint

Error analysis of variable degree mixed methods for elliptic problems via hybridization

Abstract. A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree RaviartThomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new variable degree Raviart-Thomas method, but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates. The error analysis is achieved by using a variational characterization of the Lagrange multipliers wherein the other unknowns do not appear. This approach can be applied to other hybridized mixed methods as well

Short-term Hydro-thermal Coordination By Lagrangian Relaxation: A New Algorithm for the Solution of the Dual Problem

For decades, researchers have been studying the unit commitment problem in electrical power generation. To solve this complex, large scale and constrained optimization (primal) problem in a direct manner is not a feasible approach, which is where Lagrangian relaxation comes in as the answer. The dual Lagrangian problem translates a relaxed problem approach, that indirectly leads to solutions of the original (primal) problem. In the coordination problem, a decomposition takes place where the global solution is achieved by coordinating between the respective subproblems solutions. This dual problem is solved iteratively, and Lagrange multipliers are updated between each iteration using subgradient methods. To tackle, time-consuming tuning tasks  or user related experience, a new adaptative algorithm, is proposed to better adjust the step-size used to update Lagrange multipliers, i.e., avoid the need to pre-select  a set of parameters. A results comparison against a traditionally employed step-size update mechanism, showed that the adaptive algorithm manages to obtain improved performances in terms of the targeted primal problem. Keywords: Hydro-Thermal coordination, Lagrangian relaxation, Lagrangian dual problem, Lagrange multipliers, Subgradient method

A Simplified Approach to Optimally Controlled Quantum Dynamics

A new formalism for the optimal control of quantum mechanical physical observables is presented. This approach is based on an analogous classical control technique reported previously[J. Botina, H. Rabitz and N. Rahman, J. chem. Phys. Vol. 102, pag. 226 (1995)]. Quantum Lagrange multiplier functions are used to preserve a chosen subset of the observable dynamics of interest. As a result, a corresponding small set of Lagrange multipliers needs to be calculated and they are only a function of time. This is a considerable simplification over traditional quantum optimal control theory[S. shi and H. Rabitz, comp. Phys. Comm. Vol. 63, pag. 71 (1991)]. The success of the new approach is based on taking advantage of the multiplicity of solutions to virtually any problem of quantum control to meet a physical objective. A family of such simplified formulations is introduced and numerically tested. Results are presented for these algorithms and compared with previous reported work on a model problem for selective unimolecular reaction induced by an external optical electric field.Comment: Revtex, 29 pages (incl. figures

Lagrange constraints for transient finite element surface contact

International audienceA new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included

Tsallis' entropy maximization procedure revisited

The proper way of averaging is an important question with regards to Tsallis' Thermostatistics. Three different procedures have been thus far employed in the pertinent literature. The third one, i.e., the Tsallis-Mendes-Plastino (TMP) normalization procedure, exhibits clear advantages with respect to earlier ones. In this work, we advance a distinct (from the TMP-one) way of handling the Lagrange multipliers involved in the extremization process that leads to Tsallis' statistical operator. It is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the Tsallis-Mendes-Plastino formalism.Comment: 17 pages, no figure

Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM

In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. We explore the capabilities of a recovery technique based on an enhanced MLS fitting, which directly provides continuous interpolated fields, to obtain estimates of the error in energy norm as an alternative to the superconvergent patch recovery (SPR). Boundary equilibrium is enforced using a nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose a nearly exact satisfaction of the internal equilibrium equation. The numerical results show the high accuracy of the proposed error estimator