A topology optimization algorithm based on topological derivative and level-set function for designing phononic crystals

PurposeThis work presents a topology optimization methodology for designing microarchitectures of phononic crystals. The objective is to get microstructures having, as a consequence of wave propagation phenomena in these media, bandgaps between two specified bands. An additional target is to enlarge the range of frequencies of these bandgaps.Design/methodology/approachThe resulting optimization problem is solved employing an augmented Lagrangian technique based on the proximal point methods. The main primal variable of the Lagrangian function is the characteristic function determining the spatial geometrical arrangement of different phases within the unit cell of the phononic crystal. This characteristic function is defined in terms of a level-set function. Descent directions of the Lagrangian function are evaluated by using the topological derivatives of the eigenvalues obtained through the dispersion relation of the phononic crystal.FindingsThe description of the optimization algorithm is emphasized, and its intrinsic properties to attain adequate phononic crystal topologies are discussed. Particular attention is addressed to validate the analytical expressions of the topological derivative. Application examples for several cases are presented, and the numerical performance of the optimization algorithm for attaining the corresponding solutions is discussed.Originality/valueThe original contribution results in the description and numerical assessment of a topology optimization algorithm using the joint concepts of the level-set function and topological derivative to design phononic crystals.

Feynman’s propagator in Schwinger’s picture of Quantum Mechanics

A novel derivation of Feynman’s sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac–Feynman–Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function [Formula: see text] on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian [Formula: see text] allows us to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman’s original derivation of the propagator for a point particle described by a classical Lagrangian L.

A Dynamical Alternating Direction Multiplier Method for Two-Block Optimization Problems

In this paper, we propose a dynamic alternating direction method of multipliers for two-block separable optimization problems. The well-known classical ADMM can be obtained after the time discretization of the dynamical system. Under suitable condition, we prove that the trajectory asymptotically converges to a saddle point of the Lagrangian function of the problems. When the coefficient matrices in the constraint are identiy matrices, we prove the worst-case O(1/t) convergence rate in ergodic sense.

Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators

In previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards.

Adaptive multimedia data hiding and watermarking

Watermarking is a technique of hiding a message about a work of media within that work itself in· the purpose of protecting the digital information against illegal duplication and manipulation. The objectives of this study are to analyze the robustness and distortion performance of watermarking system and to explore watermarking schemes which balance the robustness-distortion tradeoff optimally. In this thesis, We present a detector algorithm to adaptively extract spread spectrum watermark by filtering the watermarked images with Wiener filter. Two optimization algorithms for quantization watermarking are proposed. First one optimizes uniform quantization based look-up table embedding which minimizes watermarking distortion. Secondly, we analyze the robustness-distortion tradeoff and formulate the robustness-distortion tradeoff into a Lagrangian function. Hence optimal quantizers for watermarking subject to given robustness or fidelity constraint are achieved.

Adaptive multimedia data hiding and watermarking

Watermarking is a technique of hiding a message about a work of media within that work itself in· the purpose of protecting the digital information against illegal duplication and manipulation. The objectives of this study are to analyze the robustness and distortion performance of watermarking system and to explore watermarking schemes which balance the robustness-distortion tradeoff optimally. In this thesis, We present a detector algorithm to adaptively extract spread spectrum watermark by filtering the watermarked images with Wiener filter. Two optimization algorithms for quantization watermarking are proposed. First one optimizes uniform quantization based look-up table embedding which minimizes watermarking distortion. Secondly, we analyze the robustness-distortion tradeoff and formulate the robustness-distortion tradeoff into a Lagrangian function. Hence optimal quantizers for watermarking subject to given robustness or fidelity constraint are achieved.

Lagrangian Formulation of Lorenz and Chen Systems

This paper presents the formulation of Lagrangian function for Lorenz, Modified Lorenz and Chen systems using Lagrangian functions depending on fractional derivatives of differentiable functions, and the estimation of the conserved quantity associated with the respective systems.

On semidifferentiable interval-valued programming problems

In this paper, we consider the semidifferentiable case of an interval-valued minimization problem and establish sufficient optimality conditions and Wolfe type as well as Mond–Weir type duality theorems under semilocal E-preinvex functions. Furthermore, we present saddle-point optimality criteria to relate an optimal solution of the semidifferentiable interval-valued programming problem and a saddle point of the Lagrangian function.

A Neurodynamic Optimization Approach for L1 Minimization with Application to Compressed Image Reconstruction

This paper presents a neurodynamic optimization approach for l1 minimization based on an augmented Lagrangian function. By using the threshold function in locally competitive algorithm (LCA), subgradient at a nondifferential point is equivalently replaced with the difference of the neuronal state and its mapping. The efficacy of the proposed approach is substantiated by reconstructing three compressed images.

Improved Lagrangian-PPA based prediction correction method for linearly constrained convex optimization

<p style='text-indent:20px;'>This paper presents an improved Lagrangian-PPA based prediction correction method to solve linearly constrained convex optimization problem. At each iteration, the predictor is achieved by minimizing the proximal Lagrangian function with respect to the primal and dual variables. These optimization subproblems involved either admit analytical solutions or can be solved by a fast algorithm. The new update is generated by using the information of the current iterate and the predictor, as well as an appropriately chosen stepsize. Compared with the existing PPA based method, the parameters are relaxed. We also establish the convergence and convergence rate of the proposed method. Finally, numerical experiments are conducted to show the efficiency of our Lagrangian-PPA based prediction correction method.</p>