Extended Conjugate Gradient Method Research Articles

Optimal solution techniques for control problem of evolution equations

A lot of optimal solution techniques exist in literature and have been employed to solve optimization problems. However, in this work, two optimal solution techniques namely the Steepest Descent Method (SDM) and the Extended Conjugate Gradient Method (ECGM) were applied to two evolution equations to determine which best approximates the system to its analytic solution. Comparisons were also carried out between the SDM and the ECGM for optimal solution and convergence. In doing so, the Liberty Basic Programming Language (LBPL) was employed to perform various numerical iterations on the respective algorithms of the techniques. It was discovered that the SDM proved to be a better technique in obtaining an optimal solution than the ECGM.

Deep Unfolded Extended Conjugate Gradient Method for Massive MIMO Processing with Application to Reciprocity Calibration

In this paper, we consider deep unfolding the standard iterative conjugate gradient (CG) algorithm to solve a linear system of equations. Instead of being adjusted with known rules, the parameters are learned via backpropagation to yield the optimal results. However, the proposed unfolded CG (UCG) is extended wherein a scalar parameter is substituted by a matrix-parameter to augment the degrees of freedom per layer. Once the training is completed, the UCG has revealed to require far a smaller number of layers than the number of iterations needed using the standard iterative CG. It is also shown to be very robust to noise and outperforms the standard CG in low signal to noise ratio (SNR) region. A key merit of the proposed approach is the fact that no explicit training data is dedicated to the learning phase as the optimization process relies on the residual error which is not explicitly expressed as a function of the desired data. As an example, the proposed UCG is applied to solve the reciprocity calibration problem encountered in massive MIMO (Multiple-Input Multiple-Output) systems.

An Optimal Penalty Constant For Discrete Optimal Control Regulator Problems

Indiscriminate and imprecise use of penalty constants can lead to substantial computational problems and consequently erroneous conclusions and deductions. This paper view this, with utmost seriousness and thus establish an optimal penalty constant φ that optimizes the minimization of the cost functional while solving discrete optimal control problems using extended conjugate gradient method (ECGM). I employed some measures that examine optimal control of dynamic processes that can be described by Differential Algebraic Equations (DAEs) that entails integer restriction on some or all of the control functions. I established the construction of an optimal penalty constant which can be employed in the extended conjugate gradient method algorithm for discrete optimal control regulator problems. Comparative results emanating from the use of different penalty constants/algorithms on some problems are given. The obtained numerical results reveal that the proposed optimal penalty constant expression is efficient.

Numerical method for the adhesive normal contact analysis based on a Dugdale approximation

Modeling adhesion between two contacting surfaces plays a vital role in nano-tribology. However, providing analytical models, although desirable, is mostly impossible, in particular for complex geometries. Therefore, much attention has to be paid to numerical modeling of this phenomenon. Based on the adhesive stress description of the Maugis-Dugdale model of adhesion, which is credible over a broad range of engineering applications, an extended Conjugate Gradient Method (CGM) has been developed for adhesive contact problems. To examine the accuracy of the proposed method, the common case of the adhesive contact of a rigid sphere on an elastic half-space is investigated. To further evaluate the accuracy of this method, the adhesive contact of a rigid sphere over a wavy elastic half-space is also studied for different combinations of the amplitude and wavelength. There is good agreement between the analytical solution and the values predicted by the proposed method in the force-approach curves. Moreover, the calculation of pull-off force at a bisinusoidal interface between two surfaces is carried out for various cases to study the effects of different influential parameters including work of adhesion, elastic modulus, radius curvature at a crest, and the wavelength ratio. A curve is fitted on the calculated pull-off force in order to express it as an analytical relation. Similar to the JKR and DMT expressions for the pull-off force of a rigid ball on an elastic half-plane, the fitted curve is not affected by the elastic modulus and is linearly dependent on the radius of curvature and the work of adhesion. In addition, a power law governs the relation between pull-off force and the wavelength ratio. In the end, it is shown that roughness can either increase or decrease the adhesive force at a rough interface depending on the degree of the roughness.

Application of Scaled Extended Conjugate Gradient Method in Solving Continuous-Time Linear Regulator Problems

Application of Scaled Extended Conjugate Gradient Method in Solving Continuous-Time Linear Regulator Problems

Numerical Experiment with the Construction of a Control Operator Applied in ECGM Algorithm

In this paper, we constructed a control operator, G, which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving for the optimal control and trajectories of continuous time linear regulator problems. Similar operators constructed in the past by various authors have limited application. This call for the construction of the control operator that is aimed at taking care of any of the Mayer’s, Lagrange’s and Bolza’s cost form of linear regulator problems. The authors of this paper desire that, with the construction of the operator, one will circumvent the difficulties undergone using the classical methods and its application will further improve the result of the Extended Conjugate Gradient Method in solving this class of optimal control problem. The constructed Linear Control Operator is applied in ECGM algorithm to solve Continuous-Time Linear Regulator Problems with the convergence profile showing the efficiency of the operator.

On Construction of A Control Operator Introduced To ECGM Algorithm for Solving Discrete-Time Linear Quadratic Regulator Control Systems with Delay-I

In this paper, we constructed a control operator sequel to an earlier constructed control operator in one of our papers which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving discrete time linear quadratic regulator problems with delay parameter in the state variable. The construction of the control operator places scalar linear delay problems of the type within the class of problems that can be solved with the ECGM and it is aimed at reducing the rigours faced in using the classical methods in solving this class of problem. More so, the authors of this paper desire that, the application of this control operator will further improve the results of the ECGM as well as increasing the variant approaches used in solving the said class of optimal control problem.

On Construction of a Control Operator Applied In ECGM Algorithm

In this paper, we constructed a control operator, G, which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving for the optimal control and trajectories of continuous time linear regulator problems. Similar operators constructed in the past by various authors have limited application. This call for the construction of the control operator that is aimed at taking care of any of the Mayer's, Lagrange's and Bolza's cost form of linear regulator problems. The authors of this paper desire that, with the construction of the operator, one will circumvent the difficulties undergone using the classical methods and its application will further improve the result of the Extended Conjugate Gradient Method in solving this class of optimal control problem.

On The Extended Conjugate Gradient Method(ECGM) Algorithm For Discrete Optimal Control Problems And Some Of Its Features

The acceptability of an algorithm is a function of its implementability and convergence. In this paper, we examine some features of the extended conjugate gradient method (ECGM) algorithm, one of the optimization techniques for solving continuous/or discrete optimal control problems It is observed while using this algorithm, there is a consistent demand for some of the features of the algorithm. Among these are the step- size, alpha, the gradient(the partial derivatives), the search directions e.t.c. One of these features closely

On Application Of Modified Gradient To Extended Conjugate Gradient Method Algorithm For Solving Optimal Control Problems

This paper investigates and discusses modification on the gradient of continuous function input introduced to Extended Conjugate Gradient Method algorithm (ECGM) employed in solving optimal control problems with the state variable constrained by a differential equation. Numerical results show some improvement over the classical methods.

Discretized Scheme Examining Convergence and Geometric Ratio Profiles for Control Problem Constrained by Evolution Equation

Problem statement: Here, we develop a discretized scheme using only the penalty method without involving the multiplier parameter to examine the convergence and geometric ratio profiles. Approach: This approach reduces computational time arising from less data manipulation. Objectively, we wish to obtain a numerical solution comparing favourably with the analytic solution.. Methodologically, we discretize the given problem, obtain an unconstrained formulation and construct an operator which sets the stage for the application of the discretized extended conjugate gradient method. Results: We analyse the efficiency of the developed scheme by considering an example and examining the generated sequential approximate solutions and the convergence ratio profile computed quadratically per cycle using the discretized conjugate gradient method. Conclusion/Recommendations: Both results, as shown in the table, look comparably and this suggests that the developed scheme may very well approximate an analytic solution of a given problem to an appreciable level of tolerance without its prior knowledge.

Constructing the operator matrix for the optimal control of linear lower order non-dispersive waves

Non-dispersive waves propagate with constant phase velocities therefore they are indispensable in communication and some other areas of application. This paper constructed the Operator, R, for the control of linear lower order non-dispersive wave equation using the extended conjugate gradient Method proposed in [2]. The work of Ibiejugba et al, [3] was on the role of multipliers in the multiplier method which was applied to dynamical system. This work involves system governed by first order partial differential equation, namely linear lower order non-dispersive wave. JONAMP Vol. 11 2007: pp. 191-198

On the convergence of the extended conjugate gradient method for discrete optimal control problems (DOCP)

One striking factor that attracts problem solvers to use particular algorithm is its convergence behavior. In this paper, the convergence of the iterates generated by the ECGM for DOCP as proposed by [14] is examined. Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 493-500

A discretized algorithm for the solution of a constrained, continuous quadratic control problem

Numerical solution techniques such as Function space algorithm (FSA), Extended conjugate gradient method (ECGM) and Imbedding extended conjugate gradient method (MECGM) are common techniques for solving optimal control problems. However, these techniques are computationally expensive and iteratively time consuming. In this paper, a Discretized constrained algorithm (DCA) with an associated operator which replaces the integral features of these techniques by a series of summation is developed. Illustrative examples are presented. The results obtained show that the Discretized constrained algorithm (DCA) is much more accurate and more efficient than some of these techniques, particularly the FSA. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 295-300

On the Discretized Algorithm for Optimal Problems Constrained by Differential Equation with Real Coefficients

A discretized scheme, Discretized Continuous Algorithm (DCA), for solving constrained quadratic optimal control problems was developed to ease the computational cumbersomeness inherent in some existing algorithms, particularly, the Function Space A lgorithm (FSA) by replacing the integral by a series of summation. In order to accomplish this numerical scheme, we resort to a finite approximation of it by discretizing its time interval and using finite difference method for its differential constraint. Using the penalty function method, an unconstrained formulation of the problem was obtained. With the bilinear form expression of the problem, an associated operator was constructed which aided the scheme for the solution of such class of problems. A sample problem was examined to test the effectiveness of the scheme as to convergence with relation to other existing schemes such as Extended Conjugate Gradient Method (ECGM), Multiplier Imbedding Extended Conjugate Gradient Method (MECGM) and Function Space Algorithm (FSA) for solving penalized functional of optimal control problem characterized by non-linear integral quadratic nature.

Optimal control of the wave propagation problem with the extended conjugate gradient method

Thus paper formulates and solves an optimal control problem subject to the wave propagation equation, employing the Extended Conjugate Gradient Method (ECGM). New optimal state and control solutions were obtained with their surface plots.

On the control of the navier- stokes equation with the extended conjugate gradient method

The paper formulates and solves an optimal control problem of the Navier-Stokes equation for an incompressible fluid, employing an Extended Conjugate Gradient algorithm, with computer simulations for the optimal velocity fields.

A control operator and some of its applications

The conventional function space algorithms for solving minimization of penalized cost functional for optimal control problem characterized by linear-system integral quadratic cost due to Di Pillo and others, though falling within the framework pertinent to the conjugate gradient method algorithm, is difficult to apply computationally. The difficulty arises principally because there exists in the algorithm a number of stringent requirements imposed on the minimization procedures to facilitate its convergence. Incidentally, such computations are very cumbersome to carry out numerically. To circumvent this major numerical draw back, we construct here a control operator associated with this class of problems and use our explicit knowledge of the operator to devise an extended conjugate gradient method algorithm for solving this family of problems. Furthermore, the establishment of some functional inequalities which are obtained using the knowledge of the control operator is discussed.