Showing papers on "Linear approximation published in 2017"

Arbitrary order 2D virtual elements for polygonal meshes: part II, inelastic problem

Abstract: The present work deals with the formulation of a virtual element method for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II (Artioli et al. in Comput Mech, 2017) the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a computer code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical results are also compared with the one recovered using the classical finite element method.

Linear power-flow models in multiphase distribution networks

Abstract: This paper considers multiphase unbalanced distribution systems and develops approximate AC power-flow models wherein voltages, line currents, and powers at the point of common coupling are linearly related to the nodal net power injections. The linearization approach is grounded on a fixed-point interpretation of the AC power-flow equations, and it is applicable to distribution systems featuring a variety of connections for loads and generation units (e.g., wye and ungrounded delta, as well as line-to-line and line-to-grounded-neutral at the secondary of the distribution transformers). The approximate linear models can be naturally leveraged to facilitate the development of computationally-affordable optimization and control applications — from applications for advanced distribution management systems to online and distributed optimization routines. The approximation accuracy of the proposed method is evaluated on different feeders.

Optimal Reactive Power Dispatch With Accurately Modeled Discrete Control Devices: A Successive Linear Approximation Approach

Abstract: In this paper, a novel solution to the optimal reactive power dispatch (ORPD) problem is proposed. The nonlinearity of the power flow equations is handled by a new successive linear approximation approach. For the voltage magnitude terms, a mathematical transformation that improves the accuracy and facilitates the linear modeling of shunt capacitors is used. Without loss of accuracy, the load tap changers and shunt capacitors are both modeled by linear constraints using discrete variables, which facilitates the linearly constrained mixed-integer formulation of the proposed ORPD model. An efficient iterative solving algorithm is introduced. The obtained solution strictly satisfies the power flow equations. Case studies on several IEEE benchmark systems show that the proposed algorithm can efficiently provide near-optimal solutions with the error of the objective functions of less than 0.1%. Compared with several commercial solvers, the proposed method shows distinct advantages in terms of both robustness and efficiency. Moreover, based on the round-off results, a heuristic method that reduces the optimization ranges of the discrete control variables is proposed. This method can further improve the computational efficiency with small losses in accuracy.

Solving OPF using linear approximations: fundamental analysis and numerical demonstration

Abstract: Due to the unique advantages in computational robustness and convergence, the linear approximation approach is and will remain to be an important method to solve the optimal power flow (OPF) problem, especially for industrial applications. The DC power flow method, which is currently used in the majority of power industries, is the representative. Many studies extend the DC power flow method by including voltage magnitude, reactive power, and losses. This study provides a detailed analysis and breakdown investigation of existing linear approximations of the OPF problem. The formulation and accuracy of existing linear approximations are compared. Taking advantage of the decoupled formulation of linear approximations, the property of power flow equations is illustrated from a new perspective. Why reactive power flow equations are hard to linearise is explained theoretically. The numerical performance of existing linear approximations is demonstrated in IEEE and Polish test systems. Evidence from the theoretical analysis and numerical studies shows that the accuracy of linear approximations could be substantially improved using a mathematical transformation of the non-linear voltage magnitude term. This finding provides a new research direction for solving the OPF problem using linear approximations.

Superconductor in a weak static gravitational field

Abstract: We provide the detailed calculation of a general form for Maxwell and London equations that takes into account gravitational corrections in linear approximation. We determine the possible alteration of a static gravitational field in a superconductor making use of the time-dependent Ginzburg–Landau equations, providing also an analytic solution in the weak field condition. Finally, we compare the behavior of a high- $$T_\text {c}$$ superconductor with a classical low- $$T_\text {c}$$ superconductor, analyzing the values of the parameters that can enhance the reduction of the gravitational field.

An algorithm for the construction of substitution box for block ciphers based on projective general linear group

Abstract: The aim of this work is to synthesize 8*8 substitution boxes (S-boxes) for block ciphers. The confusion creating potential of an S-box depends on its construction technique. In the first step, we have applied the algebraic action of the projective general linear group PGL(2,GF(28)) on Galois field GF(28). In step 2 we have used the permutations of the symmetric group S256 to construct new kind of S-boxes. To explain the proposed extension scheme, we have given an example and constructed one new S-box. The strength of the extended S-box is computed, and an insight is given to calculate the confusion-creating potency. To analyze the security of the S-box some popular algebraic and statistical attacks are performed as well. The proposed S-box has been analyzed by bit independent criterion, linear approximation probability test, non-linearity test, strict avalanche criterion, differential approximation probability test, and majority logic criterion. A comparison of the proposed S-box with existing S-boxes sho...

Improved bridge constructs for stochastic differential equations

Abstract: We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an Ito stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler---Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time process remains intractable, necessitating the use of computationally intensive methods such as Markov chain Monte Carlo. Designing an efficient proposal mechanism which can be applied to a noisy and partially observed system that exhibits nonlinear dynamics is a challenging problem, and is the focus of this paper. By partitioning the process into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications.

Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks

Abstract: We present an approach for the verification of feed-forward neural networks in which all nodes have a piece-wise linear activation function. Such networks are often used in deep learning and have been shown to be hard to verify for modern satisfiability modulo theory (SMT) and integer linear programming (ILP) solvers. The starting point of our approach is the addition of a global linear approximation of the overall network behavior to the verification problem that helps with SMT-like reasoning over the network behavior. We present a specialized verification algorithm that employs this approximation in a search process in which it infers additional node phases for the non-linear nodes in the network from partial node phase assignments, similar to unit propagation in classical SAT solving. We also show how to infer additional conflict clauses and safe node fixtures from the results of the analysis steps performed during the search. The resulting approach is evaluated on collision avoidance and handwritten digit recognition case studies.

Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation

Abstract: The concepts of local controllability and observability of nonlinear discrete-time systems with the Caputo-, Riemann–Liouville-and Grunwald–Letnikov-type h-difference fractional order operators are studied. The Implicit Function Theorem is used in order to show that the nonlinear systems are locally observable or controllable in a finite number of steps if their linear approximations are observable or controllable, respectively, in the same number of steps.

Superconductor in a weak static gravitational field

Abstract: We provide the detailed calculation of a general form for Maxwell and London equations that takes into account gravitational corrections in linear approximation. We determine the possible alteration of a static gravitational field in a superconductor making use of the time-dependent Ginzburg-Landau equations, providing also an analytic solution in the weak field condition. Finally, we compare the behavior of a high-Tc superconductor with a classical low-Tc superconductor, analyzing the values of the parameters that can enhance the reduction of the gravitational field.

An enhanced linear Kalman filter EnLKF algorithm for parameter estimation of nonlinear rational models

Abstract: In this study, an enhanced Kalman Filter formulation for linear in the parameters models with inherent correlated errors is proposed to build up a new framework for nonlinear rational model parameter estimation The mechanism of linear Kalman filter LKF with point data processing is adopted to develop a new recursive algorithm The novelty of the enhanced linear Kalman filter EnLKF in short and distinguished from extended Kalman filter EKF is that it is not formulated from the routes of extended Kalman Filters to approximate nonlinear models by linear approximation around operating points through Taylor expansion and also it includes LKF as its subset while linear models have no correlated errors in regressor terms No matter linear or nonlinear models in representing a system from measured data, it is very common to have correlated errors between measurement noise and regression terms, the EnLKF provides a general solution for unbiased model parameter estimation without extra cost to convert model structure The associated convergence is analysed to provide a quantitative indicator for applications and reference for further research Three simulated examples are selected to bench-test the performance of the algorithm In addition, the style of conducting numerical simulation studies provides a user-friendly step by step procedure for the readers/users with interest in their ad hoc applications It should be noted that this approach is fundamentally different from those using linearisation to approximate nonlinear models and then conduct state/parameter estimate

BEM numerical simulation of coupled heat, air and moisture flow through a multilayered porous solid

Abstract: The problem of unsteady coupled moisture, air and heat energy transport through a porous solid is studied numerically using singular boundary integral representation of the governing equations. The governing transport equations are written and solved for the continuous driving potentials, i.e. relative humidity, temperature and air pressure. The boundary and interface conditions are discussed. The integral equations are discretized using mixed-boundary elements and a multidomain method also known as the macro-elements technique. The numerical model uses quadratic approximation over space and linear approximation over time for all field functions, which provides highly accurate numerical results. Three test benchmark examples (moisture uptake in a semi-infinite region, air transfer through a lightweight wall, and moisture redistribution inside a multilayered wall with capillary-active interior insulation), were analyzed to show the applicability and accuracy of the simulation model developed.

Reserve constrained dynamic economic dispatch with valve-point effect: A two-stage mixed integer linear programming approach

Abstract: This paper proposes a deterministic two-stage mixed integer linear programming (TSMILP) approach to solve the reserve constrained dynamic economic dispatch (DED) problem considering valve-point effect (VPE). In stage one, the nonsmooth cost function and the transmission loss are piecewise linearized and consequently the DED problem is formulated as a mixed integer linear programming (MILP) problem, which can be solved by commercial solvers. In stage two, based on the solution obtained in stage one, a range compression technique is proposed to make a further exploitation in the subspace of the whole solution domain. Due to the linear approximation of the transmission loss, the solution obtained in stage two dose not strictly satisfies the power balance constraint. Hence, a forward procedure is employed to eliminate the error. The simulation results on four test systems show that TSMILP makes satisfactory performances, in comparison with the existing methods.

Successive Linear Approximation Methods for Leak Detection in Water Distribution Systems

Abstract: In many modern water networks, an emerging trend is to measure pressure at various points in the network for operational reasons. Because leaks typically induce a signature on pressure, the...

Improved Parameter Estimates for Correlation and Capacity Deviates in Linear Cryptanalysis

Abstract: Statistical attacks form an important class of attacks against block ciphers. By analyzing the distribution of the statistics involved in the attack, cryptanalysts aim at providing a good estimate of the data complexity of the attack. Recently multiple papers have drawn attention to how to improve the accuracy of the estimated success probability of linear key-recovery attacks. In particular, the effect of the key on the distribution of the sample correlation and capacity has been investigated and new statistical models developed. The major problem that remains open is how to obtain accurate estimates of the mean and variance of the correlation and capacity. In this paper, we start by presenting a solution for a linear approximation which has a linear hull comprising a number of strong linear characteristics. Then we generalize this approach to multiple and multidimensional linear cryptanalysis and derive estimates of the variance of the test statistic. Our simplest estimate can be computed given the number of the strong linear approximations involved in the offline analysis and the resulting estimate of the capacity. The results tested experimentally on SMALLPRESENT-[4] show the accuracy of the estimated variance is significantly improved. As an application we give more realistic estimates of the success probability of the multidimensional linear attack of Cho on 26 rounds of PRESENT.

Quantitative model of diffuse speckle contrast analysis for flow measurement.

Abstract: Diffuse speckle contrast analysis (DSCA) is a noninvasive optical technique capable of monitoring deep tissue blood flow. However, a detailed study of the speckle contrast model for DSCA has yet to be presented. We deduced the theoretical relationship between speckle contrast and exposure time and further simplified it to a linear approximation model. The feasibility of this linear model was validated by the liquid phantoms which demonstrated that the slope of this linear approximation was able to rapidly determine the Brownian diffusion coefficient of the turbid media at multiple distances using multiexposure speckle imaging. Furthermore, we have theoretically quantified the influence of optical property on the measurements of the Brownian diffusion coefficient which was a consequence of the fact that the slope of this linear approximation was demonstrated to be equal to the inverse of correlation time of the speckle.

Quadratically Constrained Quadratic-Programming Based Control of Legged Robots Subject to Nonlinear Friction Cone and Switching Constraints

Abstract: A hierarchical control architecture is presented for energy-efficient control of robots subject to variety of linear/nonlinear inequality constraints such as Coulomb friction cones, switching unilateral contacts, actuator saturation limits, and yet minimizing the power losses in the joint actuators. The control formulation can incorporate the nonlinear friction cone constraints into the control without recourse to the common linear approximation of the constraints or introduction of slack variables. A performance metric is introduced that allows trading-off the multiple constraints when otherwise finding an optimal solution is not feasible. Moreover, the projection-based controller does not require the minimal-order dynamics model and hence allows switching contacts that are particularly appealing for legged or walking robots. The fundamental properties of constrained inertia matrix derived are similar to those of general inertia matrix of the system, and subsequently these properties are greatly exploited for control design purposes. The problem of task space control with minimum (point-wise) power dissipation subject to all physical constraints is transcribed into a quadratically constrained quadratic programming that can be solved by barrier methods. Experimental results are appended to comparatively demonstrate the efficiency and performance of the optimal controller.

Assessment of learning tomography using Mie theory

Abstract: In Optical diffraction tomography, the multiply scattered field is a nonlinear function of the refractive index of the object. The Rytov method is a linear approximation of the forward model, and is commonly used to reconstruct images. Recently, we introduced a reconstruction method based on the Beam Propagation Method (BPM) that takes the nonlinearity into account. We refer to this method as Learning Tomography (LT). In this paper, we carry out simulations in order to assess the performance of LT over the linear iterative method. Each algorithm has been rigorously assessed for spherical objects, with synthetic data generated using the Mie theory. By varying the RI contrast and the size of the objects, we show that the LT reconstruction is more accurate and robust than the reconstruction based on the linear model. In addition, we show that LT is able to correct distortion that is evident in Rytov approximation due to limitations in phase unwrapping. More importantly, the capacity of LT in handling multiple scattering problem are demonstrated by simulations of multiple cylinders using the Mie theory and confirmed by experimental results of two spheres.

Sensitivity-Based Linear Approximation Method to Estimate Time-Dependent Origin–Destination Demand in Congested Networks:

Abstract: This paper presents a bi-level optimization problem to estimate offline time-dependent origin–destination (time-dependent O-D) demand on the basis of link flows and historical O-D matrices. The upper-level problem aimed to minimize the summation of errors in both traffic counts and O-D demand. Conventionally, O-D flows are linearly mapped to link flows with the assignment matrix proportions obtained from the dynamic traffic assignment, which is typically formulated as the lower-level problem. However, the linear relationship may be invalid when congestion builds up in the network, and a nonlinear relation between O-D flows and link flows may result. The nonlinearity may lead to a converged solution that is far from the global optimum. An accurate solution should be able to relax the linear assumption and to consider the effect of other O-D flows on the links’ traffic volumes. In this study, a solution method that relied on the sensitivity of assignment proportions to O-D flows was proposed and applied; it...

Analysis of the linear approximation of seismic inversions for various structural pairs

Abstract: Thanks to space-based photometry missions CoRoT and Kepler, we benefit from a wealth of seismic data for stars other than the sun. In the future, K2, Tess, and Plato will complement this data and provide observations in addition to those already at hand. The availability of this data leads to questions on how it is feasible to extend linear structural inversion techniques to stars other than the sun. Linked to this problem is the question of the validity of the linear assumption. In this study, we analyse its limitations with respect to changes of structural variables.We wish to provide a more extended theoretical background to structural linear inversions by doing a study of the validity of the linear assumption for various structural variables. First, we recall the origins of the linear assumption for structural stellar inversions and explain its importance for asteroseismic studies. We recall the impact of unknown structural quantities such as the mass and the radius of the star on structural inversion results. We explain how kernels for new structural variables can be derived using two methods, one suited to asteroseismic targets, the other to helioseismic targets. For this second method, we present a new structural pair, namely the (A, Y) structural kernels. The kernels are tested in various numerical experiments that enable us to evaluate the weaknesses of different pairs and the domains of validity of their respective linear regime. The numerical tests we carry out allow us to disentangle the impact of various uncertainties in stellar models on the verification of the linear integral relations. We show the importance of metallicity, the equation of state, extra-mixing, and inaccuracies in the microphysics. We discuss the limitations of the method of conjugated functions due to the lack of extremely precise determinations of masses or radii in the asteroseismology.

First Order Error Correction for Trimmed Quadrature in Isogeometric Analysis

Abstract: In this work, we develop a specialized quadrature rule for trimmed domains, where the trimming curve is given implicitly by a real-valued function on the whole domain. We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a predefined base case. We then extend the classical approach of linear approximation of the trimming curve by adding an error correction term based on a Taylor expansion of the blending between the linearized implicit trimming curve and the original one. This approach leads to an accurate method which improves the convergence of the quadrature error by one order compared to piecewise linear approximation of the trimming curve. It is at the same time efficient, since essentially the computation of one extra one-dimensional integral on each trimmed cell is required. Finally, the method is easy to implement, since it only involves one additional line integral and refrains from any point inversion or optimization operations. The convergence is analyzed theoretically and numerical experiments confirm that the accuracy is improved without compromising the computational complexity .

On the stability of the permanent rotations of a charged rigid body-gyrostat

Abstract: We consider the motion of a charged rigid body about a fixed point carrying a rotor that is attached along one of the principal axes of the body. This motion occurs under the action of the resultant of the uniform gravity field and the homogeneous magnetic field. The equations of motion are formulated, and they are presented by means of the Hamiltonian function in the framework of the Lie–Poisson system. These equations of motion have six equilibrium solutions. The sufficient conditions for instability for these equilibria are studied by utilizing the linear approximation method, while the sufficient conditions for stability are presented by means of the energy-Casimir method. For certain configuration of the body, the regions of Lyapunov stability and instability are determined in the plane of some parameters. Furthermore, we clarify that the regions of Lyapunov stability are a portion of the regions of linear stability.

Dielectric permeability tensor and linear waves in spin-1/2 quantum kinetics with non-trivial equilibrium spin-distribution functions

Abstract: A consideration of waves propagating parallel to the external magnetic field is presented. The dielectric permeability tensor is derived from the quantum kinetic equations with non-trivial equilibrium spin-distribution functions in the linear approximation on the amplitude of wave perturbations. It is possible to consider the equilibrium spin-distribution functions with nonzero z-projection proportional to the difference of the Fermi steps of electrons with the chosen spin direction, while x- and y-projections are equal to zero. It is called the trivial equilibrium spin-distribution functions. In the general case, x- and y-projections of the spin-distribution functions are nonzero which is called the non-trivial regime. A corresponding equilibrium solution is found in Andreev [Phys. Plasmas 23, 062103 (2016)]. The contribution of the nontrivial part of the spin-distribution function appears in the dielectric permeability tensor in the additive form. It is explicitly found here. A corresponding modificatio...