Generalized Newton's Method based on Graphical Derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and $B$-differentiable versions of Newton's method for nonsmooth Lipschitzian equations

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

Convergence analysis of a proximal Gauss-Newton method

An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate of the radius of the convergence ball. Some applications for solving constrained nonlinear equations are discussed and the numerical performance of the method is assessed on some significant test problems

Rigorous Multicomponent Reactive Separations Modelling : Complete Consideration of Reaction-Diffusion Phenomena

This paper gives the first step of the development of a rigorous multicomponent reactive separation model. Such a model is highly essential to further the optimization of acid gases removal plants (CO2 capture, gas treating, etc.) in terms of size and energy consumption, since chemical solvents are conventionally used.Firstly, two main modelling approaches are presented: the equilibrium-based and the rate-based approaches. Secondly, an extended rate-based model with rigorous modelling methodology for diffusion-reaction phenomena is proposed. The film theory and the generalized Maxwell-Stefan equations are used in order to characterize multicomponent interactions. The complete chain of chemical reactions is taken into account. The reactions can be kinetically controlled or at chemical equilibrium, and they are considered for both liquid film and liquid bulk. Thirdly, the method of numerical resolution is described. Coupling the generalized Maxwell-Stefan equations with chemical equilibrium equations leads to a highly non-linear Differential-Algebraic Equations system known as DAE index 3. The set of equations is discretized with finite-differences as its integration by Gear method is complex. The resulting algebraic system is resolved by the Newton- Raphson method. Finally, the present model and the associated methods of numerical resolution are validated for the example of esterification of methanol. This archetype non-electrolytic system permits an interesting analysis of reaction impact on mass transfer, especially near the phase interface. The numerical resolution of the model by Newton-Raphson method gives good results in terms of calculation time and convergence. The simulations show that the impact of reactions at chemical equilibrium and that of kinetically controlled reactions with high kinetics on mass transfer is relatively similar. Moreover, the Fick’s law is less adapted for multicomponent mixtures where some abnormalities such as counter-diffusion take place

A Newton Root-Finding Algorithm For Estimating the Regularization Parameter For Solving Ill-Conditioned Least Squares Problems

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of the system matrix is found to be very efficient for parameter estimation, requiring on average about 10 Newton steps. Additionally, the theory and algorithm apply for Generalized Tikhonov regularization using the generalized singular value decomposition. The performance of the Newton algorithm is contrasted with standard techniques, including the L-curve, generalized cross validation and unbiased predictive risk estimation. This X2-curve Newton method of parameter estimation is seen to be robust and cost effective in comparison to other methods, when white or colored noise information on the measured data is incorporated

On implementation of the semi-smooth Newton method

Import 03/11/2016Cílem mé bakalářské práce je implementace semihladké Newtonovy metody. Newtonova metoda je numerickou metodou, která se využívá při řešení soustavy rovnic. Tato metoda si ovšem neporadí s řešením rovnic, které obsahují nediferencovatelné funkce. Právě z tohoto důvodu si zavedeme semihladkou Newtonovu metodu, která je založena na využití Clarkeova kalkulu. V této práci postupně rozebírám Clarkeův kalkul, pod který spadá zobecněná derivace se zobecněným gradientem a zobecněný Jacobián. Dále popisuju "klasickou" Newtonovu metodu a hlavně semihladkou Newtonovu metodu. Poté popisuju samotnou implementaci semihladké Newtonovy metody v programu Matlab, dále srovnávám řešení Newtonovou metodou pomocí analyticky spočteného zobecněného Jacobiánu a numericky spočteného zobecněného Jacobiánu. Na závěr řeším praktickou úlohu, v které zkoumám deformaci struny při kontaktu s překážkou.The goal of this work is an implementation of semismooth Newton method. Newton method is a numerical method, which is proposed to solve the system of equations. This method is not able to solve the system of equations with nodifferentable functions. Because of it we will define semismooth Newton metod which is based on Clarke calculus. I need some basic definitions from Clarke calculus like generalized derivative, generalized gradient and generalized Jacobian. Then I describe "classical" Newton method and mainly semismooth Newton method. The next part of my work is devoted to implementation of semismooth Newthon method in program Matlab and comparison of solutions based on generalized Jacobian and numerical approximation of generalized Jacobian. In the end of my work I show you some numerical experiments with deformation of string in contact with rigid obstacle.470 - Katedra aplikované matematikyvelmi dobř