Lagrangian Optimization Research Articles

Design of channel section for minimum water loss using Lagrange optimization and artificial neural networks

Evaporation and seepage are the most important forms of water loss in an open channel. Evaporation loss depends on the area of free surface while, seepage loss is function of channel geometry. The design that achieves minimum water loss is determined for trapezoidal and rectangular cross sections using Lagrangian method with Manning’s equation as a constraint. The water loss function which comprises evaporation and seepage loss was used. Matlab code is created to solve the equations of optimization by trial and error. An approach to determine the bed width and water depth from known discharge, side slope, longitudinal slope, manning’s coefficient, evaporation rate, and soil type is proposed. Design charts which determine bed width and water depth that ensure minimum water loss are presented. Moreover, artificial neural networks are used to get the best design for any value of evaporation rate. Using the created design charts and neural networks optimum design can be determined easily, accurately and quickly.

An ultra-reliable and low latency communications assisted modulation based non-orthogonal multiple access scheme

The technological demand drastically requires Internet of Things (IoT) execution. Certainly, it will bring a paradigm transformation from a local city to a smart city. Besides several other applications, Intelligent transportation system (ITS) will bring a dominant smart society shift. ITS will benefit growing traffic system safety, reliability, demand and security. 5G Ultra-Reliable and Low Latency Communications (URLLC) amalgamation is certain with ITS due to big data and traffic requirements. In this paper we demontrate our Modulation based Non Orthogonal Multiple Access (M-NOMA) technique support for Ultra-Reliable and Low Latencey Communication (URLLC) user case of 5G. Then we present it as a user case for ITS. M-NOMA guarantees reliability and low latency of the system due to its minimum symbol error rate (SER), reduced interference, better data rate and hence the Quality of service (QoS). In this paper, we derive SER for M-NOMA with Additive White Gaussian Noise (AWGN) and fading channel. We also optimized the derived SER, data rate with Lagrange optimization. The proposed system supports the low latency and high reliability requirements. Simulation result confirmed the added benefits offered by our M-NOMA scheme over the traditional schemes.

Hybrid RF/FSO Backhaul Networks With Statistical-QoS-Aware Buffer-Aided Relaying

We investigate adaptive transmission schemes over a hybrid radio frequency (RF)/free space optical (FSO) backhaul network for connecting macro-cell base station with small-cell base station through multiple parallel buffer-aided relay nodes. The proposed adaptive transmission schemes improve delay-throughput trade-off of backhaul network by exploiting the degrees-of-freedom over both RF and FSO links and capturing quality-of-service (QoS) requirements. We study two different system configurations, namely, single-carrier and multi-carrier hybrid RF/FSO systems. Our goal is to maximize the constant data arrival rate to the network such that the total queue occupancy in the network is bounded with certain acceptable queue-length bound violation probability. Towards this goal, we formulate novel optimization problems in order to maximize the end-to-end effective capacity over RF and FSO links of both system configurations. Efficient solutions are derived by employing the Lagrangian optimization technique. Some interesting insights are also revealed by considering special cases on link conditions and QoS requirements. Simulation results provide the following two observations: (i) the proposed adaptive transmission scheme outperforms the conventional switch-over hybrid RF/FSO transmission, especially in clear to moderately harsh weather conditions; and (ii) compared to several non-buffer-aided and buffer-aided relaying with non-adaptive transmission schemes, the proposed adaptive transmission scheme substantially improves the supportable data arrival for the hybrid RF/FSO backhaul network.

Sub-Frame Layer Rate Distortion Optimization for High Efficiency Video Coding

This paper proposes a sub-frame layer RDO algorithm to further improve coding performance of low-delay hierarchical prediction structure in High Efficiency Video Coding. In the proposed algorithm, each frame is segmented into two sub-frames according to motion intensity, and a sub-frame layer is taken as a basic optimizing unit. An inter-frame dependency model is proposed to quantify inter-frame dependency of a frame or a sub-frame. The proposed inter-frame dependency model is composed of two parts, Laplace distribution based direct inter-frame dependency model and distortion propagation chain. The direct inter-frame dependency model is used to evaluate dependency caused by direct reference, and the distortion propagation chain shows the relationship between direct reference and indirect reference. With quantified inter-frame dependency, Lagrange optimization is then used to solve the optimal $\lambda $ -QP pair for each sub-frame layer in frame layer 1 (L1) to frame layer 4 (L4). For frame layer 0 (L0), additional L0 frames are adaptively inserted according to inter-frame dependency of the previous L0 frame, and the optimal $\lambda $ -QP pair for each sub-frame in L0 is solved using Lagrange optimization. When peak signal to noise ratio is used as quality criterion, the proposed algorithm achieves 7.0% and 7.0% bit-rate reduction with lowdelay_P_main (LDP) and lowdelay_main (LDB) configurations respectively; when structural similarity is used as quality criterion, the proposed algorithm achieves 13.3% and 13.1% bit-rate reduction with LDP and LDB configurations respectively.

Rate-Energy Tradeoff for Wireless Simultaneous Information and Power Transfer in Full-Duplex and Half-Duplex Systems

In this paper, we study the rate-energy tradeoff for wireless simultaneous information and power transfer in full-duplex and half-duplex scenarios. To this end, the weighting function of energy efficiency and transmission rate, as rate-energy tradeoff metric is first introduced and the metric optimization problem is formulated. Applying Karush-Kuhn-Tucker (KKT) conditions for Lagrangian optimality and a series of mathematical approximations, the metric optimization problem can be simplified. The closed-form solution of the power ratio is obtained, building direct relationship between power ratio and the rate-energy tradeoff metric. By choosing power ratio, one can make the tradeoff between information rate and harvested power in a straightforward and efficient way. Using the method similar to the half duplex systems, the optimal power ratio can be obtained in the full duplex systems, so as to balance the information transmission rate and energy transmission efficiency. Simulation results validate that the information rate is non-increasing with harvested power in half-duplex systems and the tradeoff of information rate and harvested power can be simply made. In the full duplex systems, the power ratio solution of the rate-energy tradeoff metric optimization problem can be used as the approximate optimal solution of the optimization problem and the approximation error is negligible.

A trio of simple optimized axisymmetric kinematic dynamos in a sphere.

Planetary magnetic fields are generated by the motion of conductive fluid in the planet's interior. Complex flows are not required for dynamo action; simple flows have been shown to act as efficient kinematic dynamos, whose physical characteristics are more straightforward to study. Recently, Chen et al. (2018, J. Fluid Mech. 839, 1-32. (doi:10.1017/jfm.2017.924)) found the optimal, unconstrained kinematic dynamo in a sphere, which, despite being of theoretical importance, is of limited practical use. We extend their work by restricting the optimization to three simple two-mode axisymmetric flows based on the kinematic dynamos of Dudley & James (1989, Proc. R. Soc. Lond. A 425, 407-429. (doi:10.1098/rspa.1989.0112)). Using a Lagrangian optimization, we find the smallest critical magnetic Reynolds number for each flow type, measured using an enstrophy-based norm. A Galerkin method is used, in which the spectral coefficients of the fluid flow and magnetic field are updated in order to maximize the final magnetic energy. We consider the t 0 1 s 0 1, t 0 1 s 0 2 and t 0 2 s 0 2 flows and find enstrophy-based critical magnetic Reynolds numbers of 107.7, 142.4 and 125.5 (13.7, 19.6 and 16.4, respectively, with the energy-based definition). These are up to four times smaller than the original flows. These simple and efficient flows may be used as benchmarks in future studies.

Silly rubber

Simulating viscoelastic polymers and polymeric fluids requires a robust and accurate capture of elasticity and viscosity. The computation is known to become very challenging under large deformations and high viscosity. Drawing inspirations from return mapping based elastoplasticity treatment for granular materials, we present a finite strain integration scheme for general viscoelastic solids under arbitrarily large deformation and non-equilibrated flow. Our scheme is based on a predictor-corrector exponential mapping scheme on the principal strains from the deformation gradient, which closely resembles the conventional treatment for elastoplasticity and allows straightforward implementation into any existing constitutive models. We develop a new Material Point Method that is fully implicit on both elasticity and inelasticity using augmented Lagrangian optimization with various preconditioning strategies for highly efficient time integration. Our method not only handles viscoelasticity but also supports existing elastoplastic models including Drucker-Prager and von-Mises in a unified manner. We demonstrate the efficacy of our framework on various examples showing intricate and characteristic inelastic dynamics with competitive performance.

Optimal Signaling Schemes and Capacity of Non-Coherent Rician Fading Channels With Low-Resolution Output Quantization

Low-resolution analog-to-digital converter (ADC) has been considered as a promising solution to save power and cost in communication systems using high bandwidth and/or multiple RF chains. The goal of this work is to address the design of optimal signaling schemes and establish the capacity limit of Rician fading channels with low-resolution output quantization. This fading channel can be used to accurately model a wide range of wireless channels with the line-of-sight (LOS) components, including emerging mm-wave communications. The focus is on non-coherent fast fading channels where neither the transmitter nor the receiver knows the channel state information (CSI). By examining the continuity of the input-output mutual information, the existence of the optimal input signal is first validated. Then, considering the case of 1-bit ADC, we show that the optimal input is $\pi /2$ circularly symmetric. A necessary and sufficient condition for an input signal to be optimal, which is referred to as the Kuhn-Tucker condition (KTC), and Lagrangian optimization problem are then established. By exploiting the novel log-quadratic bounds on the Gaussian $Q$ -function, it is then demonstrated that for a given mass point’s amplitude, the corresponding rotated mass points through the phase of LOS component must form a square grid centered at zero. Furthermore, the amplitude of the mass points in the optimal distribution can take on only one value. As a result, the capacity-achieving input with 1-bit ADC is a rotated quadrature phase-shift keying (QPSK) constellation, and the rotation angle depends on the Rician factor. The characterization of the optimal input has also been extended to the case of multi-bit ADCs. Specifically, it is shown that for a $K$ -bit ADC, the optimal input is discrete having atmost $2^{2K}$ mass points. In both the cases of 1-bit and $K$ -bit ADCs, the channel capacities are established in closed-form.

Error Concealment-Aware Encoding for Robust Video Transmission

In this paper an error concealment (EC)-aware encoding scheme is proposed to improve the quality of decoded video in broadcast environments subject to transmission errors and data loss. The proposed scheme is based on a scalable coding approach where the best EC methods to be used at the decoder are optimally determined at the encoder and signalled to the decoder through supplemental enhancement information messages. Such optimal EC modes are found by simulating transmission losses, followed by a Lagrangian optimization of the signalling rate—EC distortion cost. A generalized saliency-weighted distortion is used and the residue between coded frames and their EC substitutes is encoded using a rate-controlled enhancement layer. When data loss occurs the signalling information is used by the decoder to improve the reconstruction quality. The simulation results show that the proposed method achieves consistent quality gains in comparison with other reference methods and previous works. Using only the EC mode signalling, i.e., without any residue transmitted in the enhancement layer, an average PSNR gain up to 2.95 dB is achieved, while using the full EC-aware scheme, i.e., including residue encoded in the enhancement layer, the proposed scheme outperforms other comparable methods, with a PSNR gain up to 3.79 dB.

Learning Complexity-Aware Cascades for Pedestrian Detection.

The problem of pedestrian detection is considered. The design of complexity-aware cascaded pedestrian detectors, combining features of very different complexities, is investigated. A new cascade design procedure is introduced, by formulating cascade learning as the Lagrangian optimization of a risk that accounts for both accuracy and complexity. A boosting algorithm, denoted as complexity aware cascade training (CompACT), is then derived to solve this optimization. CompACT cascades are shown to seek an optimal trade-off between accuracy and complexity by pushing features of higher complexity to the later cascade stages, where only a few difficult candidate patches remain to be classified. This enables the use of features of vastly different complexities in a single detector. In result, the feature pool can be expanded to features previously impractical for cascade design, such as the responses of a deep convolutional neural network (CNN). This is demonstrated through the design of pedestrian detectors with a pool of features whose complexities span orders of magnitude. The resulting cascade generalizes the combination of a CNN with an object proposal mechanism: rather than a pre-processing stage, CompACT cascades seamlessly integrate CNNs in their stages. This enables accurate detection at fairly fast speeds.

Virtual Network Functions Scheduling Under Delay-Weighted Pricing

We consider a cost minimization problem for High Volume Servers (HVS) equipped with Virtual Machines (VMs) to serve Virtual Network Functions (VNF) demands for resources. Given a scheduling period, a central scheduler decides which VM to use for each VNF demand. Each VM can be activated or disabled with different costs. Each VNF has a delay-weighted pricing function to indicate its completion time tolerance. We prove the NP-completeness of the problem and develop an algorithm based on Lagrangian relaxation and subgradient optimization to deal with this computational complexity. Finally, our numerical results demonstrate our algorithm’s effectiveness compared to two benchmarks.

Sensitivity of high-speed boundary-layer stability to base-flow distortion

The linear stability of high-speed boundary layers can be altered by distortions to the base velocity and temperature profiles. An analytic expression for the sensitivity is derived for parallel and spatially developing boundary layers, the latter using linear parabolized stability equations and their adjoint. Both the slow mode, S, and the fast mode, F, are investigated at Mach number 4.5. The mode S is more sensitive with respect to distortion in base velocity than in base temperature. The sensitivity is largest within the boundary layer away from the wall. Near the critical layer, where the phase speed of the mode equals the base streamwise velocity, the sensitivity to the base streamwise velocity is negative. For the mode F, there is a discontinuous jump in the sensitivity when the phase speed is below unity, and a critical layer is established. The sensitivity of the two modes increases with the Reynolds number, but there is a sudden drop and a jump in the sensitivities of the modes S and F, respectively, near the synchronization point where the phase speeds of the two modes are equal. Furthermore, the maximum uncertainty bounds are obtained for the distorted base state that maximizes the destabilization or stabilization of the modes by solving the Lagrangian optimization problem for the sensitivity. The sensitivity of the flow stability to surface heating is then studied, and changes in growth rate and the$N$-factor are evaluated. The formulation provides a clear physical interpretation of these changes, and establishes uncertainty bounds for stability predictions for a given level of uncertainty in wall temperature.

Three-dimensional fluid topology optimization for heat transfer

In this work, an in house topology optimization (TO) solver is developed to optimize a conjugate heat transfer problem: realizing more complex and efficient coolant systems by minimizing pressure losses and maximizing the heat transfer. The TO method consists in an idealized sedimentation process in which a design variable, in this case impermeability, is iteratively updated across the domain. The optimal solution is the solidified region uniquely defined by the final distribution of impermeability. Due to the geometrical complexity of the optimal solutions obtained, this design method is not always suitable for classic manufacturing methods (molding, stamping....) On the contrary, it can be thought as an approach to better and fully exploit the flexibility offered by additive manufacturing (AM), still often used on old and less efficient design techniques. In the present article, the proposed method is developed using a Lagrangian optimization approach to minimize stagnation pressure dissipation while maximizing heat transfer between fluid and solid region. An impermeability dependent thermal conductivity is included and a smoother operator is adopted to bound thermal diffusivity gradients across solid and fluid. Simulations are performed on a straight squared duct domain. The variability of the results is shown on the basis of different weights of the objective functions. The solver builds automatically three-dimensional structures enhancing the heat transfer level between the walls and the flow through the generation of pairs of counter rotating vortices. This is consistent to solution proposed in literature like v-shaped ribs, even if the geometry generated is more complex and more efficient. It is possible to define the desired level of heat transfer and losses and obtain the closest optimal solution. It is the first time that a conjugate heat transfer optimization problem, with these constraints, has been tackled with this approach for three-dimensional geometries.

Determination of the economical insulation thickness of building envelopes simultaneously in energy-saving renovation of existing residential buildings

ABSTRACTWhen the energy saving rate of existing residential buildings renovation is determined, the thermal performances of external walls, windows, and roof interact with each other. Therefore, it is necessary to study the determination of economical insulation thickness of building envelopes considering the interaction among building envelope performances. The objective function and its bound of envelope thermal performance optimization in the energy-saving renovation of existing residential buildings in severe cold and cold zones in China were established. It is the conditional extremum problem and can be solved through Lagrange’s method of multipliers to determine the economical insulation thickness of external walls and roofs simultaneously. The method is proved to be feasible by an existing residential building in Beijing. When the same window types are selected, the energy-saving renovation program of the building envelope determined by the Lagrangian optimization method can produce the minimum investment in insulation, minimum investment payback period, and the largest net present value (NPV) of the life cycle.

A dual–polarimetric approach to earthquake damage assessment

ABSTRACTIn this study, a novel physical approach is proposed to detect damages due to earthquakes using dual polarimetric (DP) coherent Synthetic Aperture Radar (SAR) imagery. An optimization method, aimed at enhancing scattering basis differences between measurements collected before and after the event, is designed exploiting Lagrange optimization of the difference between two polarimetric covariance matrices. A meaningful showcase is presented to demonstrate the soundness of the proposed approach that consists of processing Sentinel–1 C–band scenes related to 2016 Central Italy Earthquake. The proposed approach, which is contrasted with the conventional coherence based single– and dual–polarization approaches, results in the best sensitivity to damages.

A Two-Stage Approach for Robust HEVC Coding and Streaming

The increased compression ratios achieved by the High Efficiency Video Coding (HEVC) standard lead to reduced robustness of coded streams, with increased susceptibility to network errors and consequent video quality degradation. This paper proposes a method based on a two-stage approach to improve the error robustness of HEVC streaming, by reducing temporal error propagation in the case of frame loss. The prediction mismatch that occurs at the decoder after frame loss is reduced through the following two stages. First, at the encoding stage, the reference pictures are dynamically selected based on constraining conditions and Lagrangian optimization, which distributes the use of reference pictures, by reducing the number of prediction units that depend on a single reference. Second, at the streaming stage, a motion vector (MV) prioritization algorithm, based on spatial dependencies, selects an optimal subset of MVs to be transmitted, redundantly, as side information to reduce mismatched MV predictions at the decoder. The simulation results show that the proposed method significantly reduces the effect of temporal error propagation. Compared with the reference HEVC, the proposed reference picture selection method is able to improve the video quality at low-packet-loss rates (e.g., 1%) using the same bitrate, achieving quality gains up to 2.3 dB for 10% of packet loss ratio. It is shown, for instance, that the redundant MVs are able to boost the performance achieving quality gains of 3 dB when compared with the reference HEVC, at the cost using 4% increase in total bitrate.

Segmenting Oil Spills from Blurry Images Based on Alternating Direction Method of Multipliers

We exploit the alternating direction method of multipliers (ADMM) for developing an oil spill segmentation method, which effectively detects oil spill regions in blurry synthetic aperture radar (SAR) images. We commence by constructing energy functionals for SAR image deblurring and oil spill segmentation separately. We then integrate the two energy functionals into one overall energy functional subject to a linear mapping constraint that correlates the deblurred image and the segmentation indicator. The overall energy functional along with the linear constraint follows the form of ADMM and thus enables an effective augmented Lagrangian optimization. Furthermore, the iterative updates in the ADMM maintain information exchanges between the energy minimizations for SAR image deblurring and oil spill segmentation. Most existing blurry image segmentation strategies tend to consider deblurring and segmentation as two independent procedures with no interactions, and the operation of deblurring is thus not guided for obtaining an accurate segmentation. In contrast, we integrate deblurring and segmentation into one overall energy minimization framework with information exchanges between the two procedures. Therefore, the deblurring procedure is inclined to operate in favor of the more accurate oil spill segmentation. Experimental evaluations validate that our framework outperforms the separate deblurring and segmentation strategy for detecting oil spill regions in blurry SAR images.

Estimation of the parameters of the extended growth curve model under multivariate skew normal distribution

The Growth Curve Model (GCM) assumes the same shape of profiles for each group, where group means are assumed to be represented by polynomials of the same degree. The model, therefore, is inappropriate when analyzing data from studies involving groups with mean growth curves represented by different shapes. We consider the Extended Growth Curve Model (EGCM), which is a model that allows the group means to follow different degrees of polynomials over time. Existing inference on EGCM assumes multivariate normal errors and produces estimators that are not optimal, when used in the analysis of data with skewed distributions. In this paper, we consider the multivariate skew normal (MSN) distribution as the underlying distribution for the EGCM and provide estimators for its mean and covariance parameters. We adopted the Restricted Expectation–Maximization (REM) algorithm, which is based on the multivariate Newton–Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and the existing REM algorithm are only applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, that consequently allowed us to extend the REM algorithm to matrix estimators. The performance of the proposed estimators was examined using extensive simulations and a real data example was also considered to illustrate the application of our proposed estimators.

Optimal control of a separated boundary-layer flow over a bump

The optimal control of a globally unstable two-dimensional separated boundary layer over a bump is considered using augmented Lagrangian optimization procedures. The present strategy allows for controlling of the flow from a fully developed nonlinear state back to the steady state using a single actuator. The method makes use of a decomposition between the slow dynamics associated with the base flow modification and the fast dynamics, known as flapping, characterized by a large scale oscillation of the recirculation region. Starting from a steady state forced by a suction actuator located near the separation point, the base flow modification is shown to be controlled by a vanishing suction strategy. For weakly unstable flow regimes, this control law can be further optimized by means of direct–adjoint iterations of the nonlinear Navier–Stokes equations. In the absence of external noise, this novel approach proves to be capable of controlling the transient dynamics and the base flow modification simultaneously.

Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations

We develop error estimates for the semidiscrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by Gamba and Tharkabhushanam in [J. Comput. Phys., 228 (2009), pp. 2012--2036]. In addition we study the long time convergence of such semidiscrete solution to the equilibrium Maxwellian distribution that conserves the mass, momentum, and energy associated with the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely, conservation of mass, momentum, and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semidiscrete analysis on convergence of the proposed numerical method which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical approximation. The results of this work answer a long standing open problem posed by Cercignani, Illner, and Pulvirenti in [Mathematical Theory of Dilute Gases, Springer, New York, 1994, Chapter 12] about finding error estimates for a numerical scheme associated with the Boltzmann equation, as well as showing the semidiscrete numerical solution converges to the equilibrium Maxwellian distribution associated with the initial value problem.