A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft

Mixed Convection Flow of Couple Stress Fluid in a Vertical Channel with Radiation and Soret Effects

The radiation and thermal diffusion effects on mixed convection flow of couple stress fluid through a channel are investigated. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations. The resulting equations are then solved using the Spectral Quasi-linearization Method (QLM). The results, which are discussed with the aid of the dimensionless parameters entering the problem, are seen to depend sensitively on the parameters

On the Comparison between Compact Finite Difference and Pseudospectral Approaches for Solving Similarity Boundary Layer Problems

We introduce two methods based on higher order compact finite differences for solving boundary layer problems. The methods called compact finite difference relaxation method (CFD-RM) and compact finite difference quasilinearization method (CFD-QLM) are an alternative form of the spectral relaxation method (SRM) and spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral-based methods which have been successfully used to solve boundary layer problems. The main objective of this paper is to give a comparison of the compact finite difference approach against the pseudo-spectral approach in solving similarity boundary layer problems. In particular, we seek to identify the most accurate and computationally efficient method for solving systems of boundary layer equations in fluid mechanics. The results of the two approaches are comparable in terms of accuracy for small systems of equations. For larger systems of equations, the proposed compact finite difference approaches are more accurate than the spectral-method-based approaches

Large Scale Constrained Trajectory Optimization Using Indirect Methods

State-of-the-art direct and indirect methods face significant challenges when solving large scale constrained trajectory optimization problems. Two main challenges when using indirect methods to solve such problems are difficulties in handling path inequality constraints, and the exponential increase in computation time as the number of states and constraints in problem increases. The latter challenge affects both direct and indirect methods. A methodology called the Integrated Control Regularization Method (ICRM) is developed for incorporating path constraints into optimal control problems when using indirect methods. ICRM removes the need for multiple constrained and unconstrained arcs and converts constrained optimal control problems into two-point boundary value problems. Furthermore, it also addresses the issue of transcendental control law equations by re-formulating the problem so that it can be solved by existing numerical solvers for two-point boundary value problems (TPBVP). The capabilities of ICRM are demonstrated by using it to solve some representative constrained trajectory optimization problems as well as a five vehicle problem with path constraints. Regularizing path constraints using ICRM represents a first step towards obtaining high quality solutions for highly constrained trajectory optimization problems which would generally be considered practically impossible to solve using indirect or direct methods. The Quasilinear Chebyshev Picard Iteration (QCPI) method builds on prior work and uses Chebyshev Polynomial series and the Picard Iteration combined with the Modified Quasi-linearization Algorithm. The method is developed specifically to utilize parallel computational resources for solving large TPBVPs. The capabilities of the numerical method are validated by solving some representative nonlinear optimal control problems. The performance of QCPI is benchmarked against single shooting and parallel shooting methods using a multi-vehicle optimal control problem. The results demonstrate that QCPI is capable of leveraging parallel computing architectures and can greatly benefit from implementation on highly parallel architectures such as GPUs. The capabilities of ICRM and QCPI are explored further using a five-vehicle constrained optimal control problem. The scenario models a co-operative, simultaneous engagement of two targets by five vehicles. The problem involves 3DOF dynamic models, control constraints for each vehicle and a no-fly zone path constraint. Trade studies are conducted by varying different parameters in the problem to demonstrate smooth transition between constrained and unconstrained arcs. Such transitions would be highly impractical to study using existing indirect methods. The study serves as a demonstration of the capabilities of ICRM and QCPI for solving large-scale trajectory optimization methods. An open source, indirect trajectory optimization framework is developed with the goal of being a viable contender to state-of-the-art direct solvers such as GPOPS and DIDO. The framework, named beluga, leverages ICRM and QCPI along with traditional indirect optimal control theory. In its current form, as illustrated by the various examples in this dissertation, it has made significant advances in automating the use of indirect methods for trajectory optimization. Following on the path of popular and widely used scientific software projects such as SciPy [1] and Numpy [2], beluga is released under the permissive MIT license [3]. Being an open source project allows the community to contribute freely to the framework, further expanding its capabilities and allow faster integration of new advances to the state-of-the-art

Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value and Boundary Value Problems

The solution of initial value problems (IVPs) provides the evolution of dynamic system state history for given initial conditions. Solving boundary value problems (BVPs) requires finding the system behavior where elements of the states are defined at different times. This dissertation presents a unified framework that applies modified Chebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs. Existing methods for solving IVPs and BVPs have not been very successful in exploiting parallel computation architectures. One important reason is that most of the integration methods implemented on parallel machines are only modified versions of forward integration approaches, which are typically poorly suited for parallel computation. The proposed MCPI methods are inherently parallel algorithms. Using Chebyshev polynomials, it is straightforward to distribute the computation of force functions and polynomial coefficients to different processors. Combining Chebyshev polynomials with Picard iteration, MCPI methods iteratively refine estimates of the solutions until the iteration converges. The developed vector-matrix form makes MCPI methods computationally efficient. The power of MCPI methods for solving IVPs is illustrated through a small perturbation from the sinusoid motion problem and satellite motion propagation problems. Compared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, prior to parallel implementation. Modifying the algorithm to do double integration for second order systems, and using orthogonal polynomials to approximate position states lead to additional speedups. Finally, introducing perturbation motions relative to a reference motion results in further speedups. The advantages of using MCPI methods to solve BVPs are demonstrated by addressing the classical Lambert’s problem and an optimal trajectory design problem. MCPI methods generate solutions that satisfy both dynamic equation constraints and boundary conditions with high accuracy. Although the convergence of MCPI methods in solving BVPs is not guaranteed, using the proposed nonlinear transformations, linearization approach, or correction control methods enlarge the convergence domain. Parallel realization of MCPI methods is implemented using a graphics card that provides a parallel computation architecture. The benefit from the parallel implementation is demonstrated using several example problems. Larger speedups are achieved when either force functions become more complicated or higher order polynomials are used to approximate the solutions

Numerical iterative methods for nonlinear problems.

The primary focus of research in this thesis is to address the construction of iterative methods for nonlinear problems coming from different disciplines. The present manuscript sheds light on the development of iterative schemes for scalar nonlinear equations, for computing the generalized inverse of a matrix, for general classes of systems of nonlinear equations and specific systems of nonlinear equations associated with ordinary and partial differential equations. Our treatment of the considered iterative schemes consists of two parts: in the first called the ’construction part’ we define the solution method; in the second part we establish the proof of local convergence and we derive convergence-order, by using symbolic algebra tools. The quantitative measure in terms of floating-point operations and the quality of the computed solution, when real nonlinear problems are considered, provide the efficiency comparison among the proposed and the existing iterative schemes. In the case of systems of nonlinear equations, the multi-step extensions are formed in such a way that very economical iterative methods are provided, from a computational viewpoint. Especially in the multi-step versions of an iterative method for systems of nonlinear equations, the Jacobians inverses are avoided which make the iterative process computationally very fast. When considering special systems of nonlinear equations associated with ordinary and partial differential equations, we can use higher-order Frechet derivatives thanks to the special type of nonlinearity: from a computational viewpoint such an approach has to be avoided in the case of general systems of nonlinear equations due to the high computational cost. Aside from nonlinear equations, an efficient matrix iteration method is developed and implemented for the calculation of weighted Moore-Penrose inverse. Finally, a variety of nonlinear problems have been numerically tested in order to show the correctness and the computational efficiency of our developed iterative algorithms

Existence and approximation of solutions of nonlinear boundary value problems

In chapter two, we establish new results for periodic solutions of some second order non-linear boundary value problems. We develop the upper and lower solutions method to show existence of solutions in the closed set defined by the well ordered lower and upper solutions. We develop the method of quasilinearization to approximate our problem by a sequence of solutions of linear problems that converges to the solution of the original problem quadratically. Finally, to show the applicability of our technique, we apply the theoretical results to a medical problem namely, a biomathematical model of blood flow in an intracranial aneurysm. In chapter three we study some nonlinear boundary value problems with nonlinear nonlocal three-point boundary conditions. We develop the method of upper and lower solutions to establish existence results. We show that our results hold for a wide range of nonlinear problems. We develop the method of quasilinearization and show that there exist monotone sequences of solutions of linear problems that converges to the unique solution of the nonlinear problems. We show that the sequences converge quadratically to the solutions of the problem in the C1 norm. We generalize the technique by introducing an auxiliary function to allow weaker hypotheses on the nonlinearity involved in the differential equations. In chapter four, we extend the results of chapter three to nonlinear problems with linear four point boundary conditions. We generalize previously existence results studied with constant lower and upper solutions. We show by an example that our results are more general. We develop the method of quasilinearization and its generalization for the four point problems which to the best of our knowledge is the first time the method has been applied to such problems. In chapter five, we extend the results to second order problems with nonlinear integral boundary conditions in two separate cases. In the first case we study the upper and lower solutions method and the generalized method of quasilinearization for the Integral boundary value problem with the nonlinearity independent of the derivative. While in the second case we show the nonlinearity to depend also on the first derivative. Finally, in chapter six, we study multiplicity results for three point nonlinear boundary value problems. We use the method of upper and lower solutions and degree arguments to show the existence of at least two solutions for certain range of a parameter r and no solution for other range of the parameter. We show by an example that our results are more general than the results studied previously. We also study existence of at least three solutions in the pressure of two lower and two upper solutions for some three-point boundary value problems. In one problem, we employ a condition weaker than the well known Nagumo condition which allows the nonlinearity f(t, x, x’) to grow faster than quadratically with respect to x’ in some cases

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp