Long range numerical simulation of short waves as nonlinear solitary waves

A new numerical method has been developed to propagate short wave equation pulses over indefinite distances and through regions of varying index of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain 2–3 grid cells in thickness indefinitely with no numerical spreading. With the method, only a simple discretized equation is solved each time step at each grid node. The method can be contrasted to Lagrangian Ray Tracing: it is an Eulerian based method that captures the waves directly on the computational grid, where the basic objects are codimension 1 surfaces (in the fine grid limit), defined on a regular grid, rather than collections of markers. In this way, the complex logic of current ray tracing methods, which involves allocation of markers to each surface and interpolation as the markers separate, is avoided.     

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

In this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.     

Adaptive neural sliding mode compensator for a class of nonlinear systems with unmodeled uncertainties

This paper addresses the problem of adaptive neural sliding mode control for a class of multi-input multi-output nonlinear system. The control strategy is an inverse nonlinear controller combined with an adaptive neural network with sliding mode control using an on-line learning algorithm. The adaptive neural network with sliding mode control acts as a compensator for a conventional inverse controller in order to improve the control performance when the system is affected by variations in its entire structure (kinematics and dynamics). The controllers are obtained by using Lyapunov's stability theory. Experimental results of a case study show that the proposed method is effective in controlling dynamic systems with unexpected large uncertainties.     

Coupling projection domain decomposition method and Kansa's method in electrostatic problems

This paper present a novel approach for solving electrostatic problems associated with an asymmetrical shielded stripline and shielded coupled-striplines. This novel approach is based on combination of radial basis functions-based meshless unsymmetric collocation method (also, Kansa's method) with projection domain decomposition method. Under this new method, we just need to solve a Steklov-Poincaré interface equation and the original problem is solved by computing a series of independent subproblems. Two real problems are solved by the proposed approach to demonstrate the accuracy and efficiency.     

Time series analysis using RBF networks with FIR/IIR synapses

Radial basis functions networks (RBF) with dynamic synapses are introduced. The novelty aspect consists in replacing the standard scalar values of the output weights by discrete-time FIR/IIR filters. LMS-type learning algorithms are derived and simulation results for prediction of chaotic time series are reported.     

A numerical method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation

This paper presents a numerical solution for solving a nonlinear 2-D optimal control problem (2DOP). The performance index of a nonlinear 2DOP is described with a state and a control function. Furthermore, dynamic constraint of the system is given by a classical diffusion equation. It is preferred to use the Ritz method for finding the numerical solution of the problem. The method is based upon the Legendre polynomial basis. By using this method, the given optimisation nonlinear 2DOP reduces to the problem of solving a system of algebraic equations. The benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, compared with the eigenfunction method, the satisfactory results are obtained only in a small number of polynomials order. This numerical approach is applicable and effective for such a kind of nonlinear 2DOP. The convergence of the method is extensively discussed and finally two illustrative examples are included to observe the validity and applicability of the new technique developed in the current work.     

The method of particular solutions for solving nonlinear Poisson problems

Based on the recent development in the method of particular solutions, we re-exam three approaches using different basis functions for solving nonlinear Poisson problems. We further propose to simplify the solution procedure by removing the insolvency condition when the radial basis functions are augmented with high order polynomial basis functions. We also specify the deficiency of some of these methods and provide necessary remedy. The traditional Picard method is introduced to compare with the recent proposed methods using MATLAB optimization toolbox solver for solving nonlinear Poisson equations. Ranking on these three approaches are given based on the results of numerical experiment.     

A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets

Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black–Scholes problem. Nonlinear models appear when transaction costs or illiquid market effects are taken into account. This paper deals with the numerical analysis of nonlinear Black–Scholes equations modeling illiquid markets when price impact in the underlying asset market affects the replication of a European contingent claim. Numerical analysis of a nonlinear model is necessary because disregarded computations may waste a good mathematical model. In this paper we propose a finite-difference numerical scheme that guarantees positivity of the solution as well as stability and consistency.     

SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).     

Adaptive control of a CSTR with a neural network model

An adaptive control algorithm with a neural network model, previously proposed in the literature for the control of mechanical manipulators, is applied to a CSTR (Continuous Stirred Tank Reactor). The neural network model uses either radial Gaussian or “Mexican hat” wavelets as basis functions. This work shows that the addition of linear functions to the networks significantly improves the error convergence when the CSTR is operated for long periods of time in a neighborhood of one operating point, a common scenario in chemical process control. Then, a quantitative comparative study based on output errors and control efforts is conducted where adaptive controllers using wavelets or Gaussian basis functions and PID controllers (IMC tuning with fixed parameters and self tuning PID) are compared. From this comparative study, the practicality and advantages of the adaptive controllers over fixed or adaptive PID control is assessed.     

On the numerical solution of initial value problems for nonlinear trapezoidal formulas with different types

In this paper, the local truncation errors of the trapezoidal formulas such as arithmetic mean (AM), geometric mean (GM), heronian mean (HeM), harmonic mean (HaM), contraharmonic mean (CoM), root mean square (RMS), logarithmic mean (LM) and centroidal mean (CeM) are investigated and the stability analysis of these formulas are found. Finally, it is applied to various initial value problems.     

Radial basis function partition of unity method for modelling water flow in porous media

Water flow in variably-saturated porous media is modelled by using the highly nonlinear parabolic Richards’ equation. The nonlinearity is due to the hydraulic conductivity and moisture content variables. The latter were estimated by using experimental models, including Gardner, Burdine, Mualem and van Genuchten models. The aim of this work is to develop a new technique based on the radial basis function partition of unity method (RBFPUM) and Gardner model in order to solve Richards’ equation in one and two dimensions. We have used Gardner model to handle the nonlinearity issue and the RBFPUM is used to approximate the solution of the linearized Richards’ equation. Our proposed algorithm is based on testing many configurations of the partitions number and selecting the optimal shape parameter for each case. Then we pick up the optimal configuration (partitions number-shape parameter) that yields the best solution in terms of error and conditioning number. By following this procedure, an optimal solution is ensured for our given problem. As numerical tests, we consider the vertical infiltration of water in soils in order to validate our proposed method.     

Using Lyapunov's direct method for wave suppression in reactive systems

This paper proposes a new approach for stabilizing a homogeneous solution in reaction–convection–diffusion system with oscillatory kinetics, in which moving or stationary patterns emerge in the absence of control. Specifically, we aim to suppress patterns by using a spatially weighted finite-dimensional feedback control that assures stability of the solution according to Lyapunov's direct method. A practical design procedure, based on spectral representation of the system and dissipative nature of parabolic PDEs, is presented.     

The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method

In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the radial basis function (RBF) collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.     

Admissible controls and attainable states for a class of nonlinear systems with general constraints

We consider nonlinear control systems where the control and the state variables are submitted to explicit constraints. This paper has two objectives. First, for a class of nonlinear systems with constraints, an existence result of nontrivial admissible controls and some of their interesting properties are proved. Then, we investigate the problem of local controllability in a neighbourhood of an equilibrium point, while observing state and control constraints along the whole trajectory. An iterative procedure is also given, which allows one to compute the steering admissible control function. This procedure is illustrated with a classical example.     

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor-corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.