Successive Approximation Procedure Research Articles

Stability of Solutions to a Caginalp Phase-Field Type Equations

This paper is concerned with the study of the asymptotic behavior of a generalization of the Caginalp phase-field model subject to homogeneous Neumann boundary conditions and regular potentials involving two temperatures. This work follows on from a paper in which the well-posedness of the problem, the dissipativity of the system, and the existence of global and exponential attractors were demonstrated. In addition, a study on the semi-infinite cylinder was also carried out. Indeed, if it is true that the existence of a global attractor makes it possible to predict the asymptotic behavior of solutions on a bounded domain, it does not say that these solutions converge. After having shown the existence of the global attractor, it is therefore important to look at the convergence of the solutions over time. There are several methods for determining the asymptotic behavior of the solutions of a differential system. We can mention the one that consists of transforming the given differential equations into integral equations and then applying the classical Picard successive approximation procedure to them. This work is devoted to the study of the convergence of solutions to steady states, adapting a well-known result concerning Lojasiewicz-Simon’s inequality.

A High-Accuracy RC Time Constant Auto-Tuning Scheme for Integrated Continuous-Time Filters.

The reliability of the resistor-capacitor (RC) time constant of a continuous-time (CT) filter has long been an obstacle with integrated circuits. Due to process and temperature variations in complementary metal-oxide semiconductor (CMOS) technology, the absolute value of the RC time constant may vary over ±50%, which is a big issue for many integrated continuous-time analog circuits. This study proposes an on-chip RC time constant auto-tuning scheme. The proposed scheme is based on the discrete master-slave auto-tuning concept. Considering the limitations in conventional works, a higher tuning accuracy is achieved by adopting two techniques: firstly, parasitic capacitance cancelation is proposed to eliminate the effects caused by parasitic capacitance; secondly, symmetric comparison is introduced to minimize the influence of the DC offset of the comparator. A successive approximation procedure is applied to improve the tuning speed. The proposed auto-tuning scheme has been validated in 55 nm CMOS technology with a fourth-order active-RC low-pass filter under PVT variations and 60 mV input offset voltage. The average tuning error is 2.21%, and the maximum error is 3.67%. The tuning error of the proposed scheme is considerably lower than the conventional scheme.

Data Rate Maximization in RIS-Assisted D2D Communication with Transceiver Hardware Impairments

We maximize the transmit rate of device-to-device (D2D) in a reconfigurable intelligent surface (RIS) assisted D2D communication system by satisfying the unit-modulus constraints of reflectin elements, the transmit power limit of base station (BS) and the transmitter in a D2D pair. Since it is a non-convex optimization problem, the block coordinate descent (BCD) technique is adopted to decouple this problem into three subproblems. Then, the non-convex subproblems are approximated into convex problems by using successive convex approximation (SCA) and penalty convex-concave procedure (CCP) techniques. Finally, the optimal solution of original problem is obtained by iteratively optimizing the subproblems. Simulation results reveal the validity of the algorithm that we proposed to solve the optimization problem and illustrate the effectiveness of RIS to improve the transmit rate of the D2D pair even with hardware impairments.

Intelligent Reflecting Surface Enhanced Secure Transmission Against Both Jamming and Eavesdropping Attacks

Both the jammer and the eavesdropper pose severe threat to wireless communications due to the broadcast nature of wireless channels. In this paper, an intelligent reflecting surface (IRS) assisted secure communication system is considered, where a base station (BS) wishes to reliably convey information to a user, in the presence of both a jammer and an eavesdropper whose transmit informations are not completely known. Specifically, with the imperfect third-party node's channel state information (CSI) and no knowledge of the jammer's transmit beamforming, we aim to maximize the system achievable rate by jointly designing the BS's transmit beamforming and the IRS's reflect beamforming, while limiting the information leakage to the potential eavesdropper. Due to the non-convexity and intractability of the original problem induced by the incompleted information, we utilize the auxiliary variables, Cauchy-Schwarz inequality, and General Sign-Definiteness transformation to convert the original optimization problem into a tractable convex optimization problem, and then obtain the high-quality optimal solution by using the successive convex approximation and penalty convex concave procedure. Numerical simulations demonstrate the superiority of our proposed optimization algorithm compared with existing approaches, and also reveal the impact of key parameters on the achievable system performance.

A Proof on Equivalence of Stagewise Newton and Dreyfus's Successive Approximation Procedures

Dreyfus in 1966 presented a second-order stagewise successive approximation procedure for computationally solving discrete-stage optimal control problems. The purpose of this paper is to provide a proof that Dreyfus's procedure generates a Newton step at each successive approximation. Hence, his 1966 procedure is the earliest stagewise Newton method in the literature. In the proof, we exploit an important property that Dreyfus's formulation readily adapts to an efficient second-order backpropagation that evaluates the Hessian matrix (as well as the gradient vector) in a stagewise fashion. After the proof, we also discuss the issues of practical Newton methods for solving a linear-quadratic problem with a globalization strategy.

Modification of kinetic theory of granular flow for frictional spheres, Part I: Two-fluid model derivation and numerical implementation

We derive a kinetic theory of granular flow (KTGF) for frictional spheres in dense systems, including rotation, sliding and sticking collisions. We use the Chapman–Enskog solution procedure of successive approximations, where the single-particle velocity distribution is assumed to be nearly Maxwellian both translationally and rotationally, as assumed by McCoy et al. (1966). An expression for the first-order particle velocity distribution function is derived, which includes the effects of particle rotation and friction. Using a simple moment method, balance equations for mass, momentum and energy are derived with closure equations for viscosities, and thermal conductivities and collisional energy dissipation rates of angular and translational kinetic energy. Because the internal angular momentum changes are coupled to the flow field, the stress tensor contains anti-symmetric components which are associated with a rotational viscosity. In the resulting closure equations, the rheological properties of the particles are explicitly described in terms of the friction coefficient. The model has been incorporated into our in-house two-fluid model (TFM) code for the modeling of dense gas–solid fluidized beds. For verification, a comparison of the present model in the limit of zero friction with the original (frictionless) KTGF model is carried out. Simulation results of both models agree well. In the next part, simulation results obtained with the new model will be compared with experimental data and discrete particle model (DPM) simulations.

Modified wave operators without loss of regularity for some long range Hartree equations. II

We continue the study of the theory of scattering for some long range Hartree equations with potential $|x|^{-\gamma}$, performed in a previous paper, denoted as I, in the range $1/2 < \gamma < 1$. Here we extend the results to the range $1/3 < \gamma < 1/2$. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem without loss of regularity between the asymptotic state and the solution, as in I, but in contrast to I, we are no longer able to cover the entire subcriticality range of regularity of the solutions. The method is an extension of that of I, using a better approximate asymptotic form of the solutions obtained as the next step of a natural procedure of successive approximations.

Modified Wave Operators Without Loss of Regularity for Some Long-Range Hartree Equations: I

We continue the study of the theory of scattering for some long range Hartree equations with potential |x|^-gamma, performed in a previous paper, denoted as I, in the range 1/2 < gamma < 1. Here we extend the results to the range 1/3 < gamma < 1/2. More precisely, we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem without loss of regularity between the asymptotic state and the solution, as in I, but in contrast to I, we are no longer able to cover the entire subcritical range of regularity of the solutions. The method is an extension of that of I, using a better asymptotic form of the solutions, obtained as the next step of a natural procedure of successive approximations.

Zeta: Zero Energy Transmissions Algebra for Wireless Sensor Networks

In this paper we describe ZETA-NR, a mathematical framework developed for multihop forwarding of packets. ZETA-NR provides proofs for optimal selection of the next relay. The communication model is based on medium access control protocols and energy models. The framework applies either distances between nodes or transmit power levels. We show possibilities to integrate multihop services into routing protocols using a successive approximation procedure. Simulation results indicate reduced energy for communication and improved throughput.

Contact problems for rough elastic solids being in nearly full contact or regarded as such for further elaboration

Problems on nearly full contact between elastic bodies with nominally flat rough boundaries are considered, so that only relatively small crack-like no-contact areas are present. Such a case is of essential applied interest and has been considerably less examined than the case of comparatively small contact areas, when only relatively sparsely located "contact spots" appear. This case may really appear under sufficiently large loads and appropriate roughness and compliance conditions (e.g. when metallic and polymeric materials contact) or it may be regarded as a zero- approximation, when roughness is essentially multi-scale and a procedure of successive approximations (iterations) is applied to solve corresponding contact problems. The latter situation is mainly considered below for plane strain, and, in a lesser degree, for 3D conditions, assuming no shear interaction. Particularly, the case is considered, when there is a coarse-scale Λ and a fine-scale λ (λ<<Λ) of roughness extent. In zero approximation, only coarse-scale roughness is taken into account, assuming that it is presented by a correspondingly large solitary socket. Fine-scale roughness is considered in first approximation only based on the Greenwood-Williamson model [1]. Similarly, the situations, where bending of a layer or a multi-layer structure plays dominant role are considered.

Development of a multi-fluid model for poly-disperse dense gas–solid fluidised beds, Part I: Model derivation and numerical implementation

We have derived a new set of closure equations for the rheologic properties of a dense gas–solid fluidised bed consisting of a multi-component mixture of slightly inelastic spheres, using the Chapman–Enskog solution procedure of successive approximations, where the particle velocity distribution of all particle species is assumed to be nearly Maxwellian around the particle mixture velocity with the particle mixture granular temperature. In this theory, differences in the mean velocities (i.e. particle segregation) and granular temperatures of the particle species result from higher order perturbation functions. Special attention is paid to assure thermodynamic consistency between the radial distribution function and the chemical potential of a hard-sphere particle specie appearing in the diffusion driving force when applying the revised Enskog theory, which is often overlooked. In the resulting closure equations, the rheologic properties of the particle mixture are explicitly described in terms of the particle mixture velocity and granular temperature, and the diffusion velocity and granular temperature of the individual particle phases can be computed from the mixture properties, which is a major advantage with respect to the numerical implementation. A new numerical solution strategy has been devised, which is an extension of the well-known SIMPLE algorithm and takes the compressibility of the solids phase directly into account, which allows for much larger time steps (about a factor of 10 larger). In Part 2 the simulation results obtained with the new model are compared with experimental data and discrete particle model (DPM) simulations.

Some axisymmetric problems for an elastic inhomogeneous thick-walled tube

An equilibrium differential equation for an axisymmetric problem is reduced to an integrable form under the assumption that the shear modulus is continuously differentiable and Poisson’s ratio is constant. A procedure of successive approximations is proposed for the case of a compressible material, and the Lame problem is solved exactly for the case of an incompressible material. A piecewise continuous variation of the Lame parameter as a function of radius is considered. Several examples of determining the stress-tensor components are given for various cases of inhomogeneity.

Successive approximation procedure of optimal tracking control for nonlinear similar composite systems

The optimal tracking control (OTC) problem for a class of affine nonlinear composite systems with similar structure is considered. By using a modeling technique, the nonlinear similar composite system is first transformed into some quasi-decoupled subsystems. Then the high-order, strongly coupled, nonlinear two-point boundary value (TPBV) problem is transformed into a sequence of linear decoupled TPBV problems through a successive approximation procedure. The obtained OTC law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence. A suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term.

A non-linear wear-contact problem for a Winkler foundation with an increasing contact area

A wear-contact problem for a Winkler foundation is considered in the case when the rate of wear depends nonlinearly on the contact pressure and the contact area increases. The corresponding integral and differential equations are obtained. A successive approximation procedure is proposed which enables an exact solution of the problem in the space of continuous functions to be found. The property of non-negativity of the contact pressure when it has non-negative initial values is established. It is shown, using a qualitative analysis and calculations, that the non-linearity of the wear law can have a considerable effect on the behaviour of the contact pressure during the wear process.

Refinement of the engineering method used to calculate foundation settlements and tilts of high-rise buildings

A procedure for calculation of bed settlement beneath a slab foundation, which was previously formulated in conformity with schematic representations of regulatory documents and which takes into account the shape of the foundation, the load distribution, and inhomogeneity of the bed is presented. For simulation of the load on the bed of a high-rise building, a diagram of its deformation beneath the lower surface of a “rigid plate” is proposed. A successive-approximation procedure based on use of equilibrium equations of the slab is developed for calculation of the settlement. Determination of the settlements and tilts of the foundation of a specific high-rise building are discussed as an example.

Arriving on Time

This research proposes a procedure for identifying dynamic routing policies in stochastic transportation networks. It addresses the problem of maximizing the probability of arriving on time. Given a current location (node), the goal is to identify the next node to visit so that the probability of arriving at the destination by time t or sooner is maximized, given the probability density functions for the link travel times. The Bellman principle of optimality is applied to formulate the mathematical model of this problem. The unknown functions describing the maximum probability of arriving on time are estimated accurately for a few sample networks by using the Picard method of successive approximations. The maximum probabilities can be evaluated without enumerating the network paths. The Laplace transform and its numerical inversion are introduced to reduce the computational cost of evaluating the convolution integrals that result from the successive approximation procedure.

Calculation of volume fraction of phase growing by a diffusion-type law

The problem of determining the volume fraction of a phase is considered for the case where the rate of nucleus growth is a decreasing function of its radius. The solution is obtained within the framework of the geometrical-probabilistic method suggested earlier. The procedure of successive approximations is described, which allows one to determine the volume fraction of a phase with the required accuracy. The errors arising in the calculation of the volume fraction of a phase from the Kolmogorov formula are estimated analytically. As an example of numerical estimates, the case of the diffusion growth mechanism is considered. It is shown that in the three-dimensional space, this error lies within 0.01 irrespective of the initial parameters of the problem.

Field-induced disturbance of the double layer electro-neutrality and non-linear electrophoresis

The influence of applied electric field on the field-induced variation of the electrolyte concentration (concentration polarization) disturbs the electro-neutrality of the system, represented by a dispersed particle and its electric double layer in electrolyte solution. The manifestation of this electro-neutrality disturbance in the non-linear electrophoresis was considered in the framework of a procedure of successive approximations in powers of the applied field strength. Analytic expressions describing the component of the electrophoretic velocity proportional to the cubic power of the applied field strength (cubic electrophoresis) were obtained for arbitrary values of the surface conductivity (of Dukhin number). The model restrictions are spherical non-conducting particle with homogeneous surface and thin double layer.

Two-Dimensional Bragg-Ewald's Dynamical Diffraction and Spontaneous Gratings: Qualitative Nonlinear Optics Treatment

A stimulated light scattering in thin waveguide film, filled by point Kerr-type dipoles, under inclined pumping beam incidence is examined. The iterative procedure of successive approximations is applied to define qualitatively the space spectrum of waveguide mode scattered field at the condition of its self-consistency with dipole oscillations. A comparison with experimental data is discussed briefly.

Riccati Dichotomic Basis Method for Solving Hypersensitive Optimal Control Problems

The solution of a completely hyper-sensitive Hamiltonian boundary-value problem (HBVP) arising in optimal control can be accurately approximated by concatenating an initial boundary-layer segment, an equilibrium segment, and a terminal boundary-layer segment. The approximations in the boundary-layers are essentially similar; only the direction of time is different. Consequently, it is sufficient to focus on the initial boundary-layer. In this paper it is shown that by using a dichotomic basis, the Hamiltonian vector field can be decomposed into its stable and unstable components. Furthermore, it is shown that a dichotomic basis can be constructed from by solving a Riccati differential equation. Using this result, a successive approximation procedure arises from which from which the approximate solution in the initial boundary-layer can be computed. The successive approximation procedure is illustrated on a problem in supersonic aircraft trajectory optimization and its range of applicability is discussed.