Inexact Newton-like Methods Research Articles

The effect of rounding errors on Newton methods

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newtonlike methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitztype hypotheses on the second Frechet-derivative instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.

Newton methods on Banach spaces with a convergence structure and applications

In this study, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semilocal results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately we can find sharper error bounds on the distances involved than before. Finally, our methods are used to solve integral equations that cannot be solved with existing methods.

The effect of rounding errors on a certain class of iterative methods

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitztype hypotheses on the mth Frechet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.

Convergence rates for inexact Newton-like methods at singular points and applications

Newton's method converges linearly rather than quadratically if the initial guess is close to the solution and the Fréchet derivative is singular at the solution. Several authors have already tried to improve the rate of convergence for this method in this case. The limitations of their approach is twofold. First, the residuals at each step are assumed to be zero, which is not the case in general. Second, the previous works are limited only to the ordinary Newton method. Here we extend and improve on these results by providing convergence rates for inexact Newton-like methods on a Banach space setting. Moreover our results reduce to the ones already in the literature for special choices of the operators involved. We complete our study with some numerical examples appearing in connection with some “predator–prey” problems appearing in population modeling that show how our results improve on the earlier ones.

Affine invariant local convergence theorems for inexact Newton-like methods

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Frechet-derivative whereas the second theorem employs hypotheses on the second. Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the second Frechet-derivative our radius of convergence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].

Relations between forcing sequences and inexact newton-like iterates in banach space

We use inexact Newton-like iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton-like iterates at each stage is very expensive in general. That is why we consider inexact Newton-like methods, where the Newton- like equations are solved only approximately and in some unspecified manner. In the elegant paper [6] natural assumptions under which the forcing sequence is uniformly less than one were given based on the first-Fréchet derivative of the operator involved in the special case of inexact Newton iterates. Here, we use assumptions on the first and second Fréchet-derivative. This way, we essentially reproduce all results found earlier but for inexact Newton-like iterates. However, our upper error bounds on the distances involved are smaller

Convergence Domains for Some Iterative Processes in Banach Spaces Using Outer or Generalized Inverses

We provide a semilocal Ptak–Kantorovich-type analysis for inexact Newton-like methods using outer and generalized inverses to approximate a locally unique solution of an equation in a Banach space containing a nondifferentiable term. We use Banach-type lemmas and perturbation bounds for outer as well as generalized inverses to achieve our goal. In particular we determine a domain Ω such that starting from any point of Ω our method converges to a solution of the equation. Our results can be used to solve undetermined systems, nonlinear least-squares problems, and ill-posed nonlinear operator equations in Banach spaces. Finally, we provide two examples to show that our results compare favorably with earlier ones.

Sufficient conditions for constructing methods faster than Newton's

In this study we use inexact Newton-like methods to approximate a locally unique solution of a nonlinear operator equation in a Banach space. Using the majorant method and Newton Kantorovich-type hypotheses we provide sufficient convergence conditions and an error analysis for our method. Our method is also shown under very natural assumptions to be faster than Newton's under the same computational cost. Finally we apply our results to solve nonlinear integral equations appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field.

On the convergence of a certain class of iterative procedures under relaxed conditions with applications

We provide sufficient convergence conditions for a certain class of inexact Newton-like methods to a locally unique solution of a nonlinear equation in a Banach space. The equation contains a nondifferentiable term and at each step we use the inverse of the same linear operator. We use Ptak-like conditions that are weaker than earlier ones. Our results apply whenever earlier ones do but not vice versa. A semilocal convergence result is also given based on the contraction mapping principle. Finally, our results apply to solve a nonlinear integral equation of Uryson type appearing in elasticity theory that cannot be solved with existing methods.

Generalized conditions for the convergence of inexact Newton-like methods on banach spaces with a convergence structure and applications

In this study, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semi-local results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately we can find sharper error bounds on the distances involved than before. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, our results are used to solve integral equations that cannot be solved with existing methods.

On a new Newton-Mysovskii-type theorem with applications to inexact Newton-like methods and their discretizations

In this manuscript we first introduce a new Newton-Mysovskii-type theorem to study the convergence of inexact Newton-like methods to a locally unique solution of a nonlinear operator equation with a nondifferentiable term. We then study their discretized versions in connection with the mesh independence principle. This principle asserts that the behaviour of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform discretization. In some of our earlier papers we extended these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use perturbed Newton-like methods and even more general conditions. Our results, on the one hand, extend and on the other hand, make more practical and applicable all previous results. In particular, we solve a nonlinear integral equation appearing in radioactive transfer.

A mesh independence principle for inexact Newton-like methods and their discretizations under generalized Lipschitz conditions

In this manuscript, we study inexact Newton-like methods for the solution of nonlinear operator equations in a Banach space involving a nondifferentiable term, and their discretized versions in connection with the mesh independence principle. This principle asserts that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform (or not) discretization. In some of our earlier papers we extended these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use inexact Newton-like methods and even more general conditions. Our results, on the one hand, extend, and on the other hand, make more practical and applicable previous results. Moreover, we can solve a wider range of problems and find sharper error bounds. Several applications of our results to two point boundary value problems are illustrated.

Improving the rate of convergence of Newton methods on Banach spaces with a convergence structure and applications

In this note, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semilocal results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately, we can find sharper error bounds on the distances involved than before. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, several examples are being provided where our results compare favorably with earlier ones.

Convergence of inexact Newton-like iterations in incremental finite element analysis of elasto-plastic problems

Convergence of two inexact Newton-like methods suitable for application in incremental finite element analysis of problems of elasto-plasticity is investigated by a new technique based on a certain approximation condition. It is shown that the convergence can be controlled by the size of load increments. A numerical example illustrates the developed theoretical results.

Solution of large nonlinear systems in elastoplastic incremental analysis

The paper concerns the solution of (three-dimensional) problems of elastoplasticity by the incremental finite element method. Attention is devoted to the inexact Newton-like methods for solving large nonlinear systems arising in load steps and to the preconditioned conjugate gradient method for solving large linear systems within Newton-like iterations. Numerical experiments illustrate the performance of the described numerical technique.

On the convergence of inexact Newton-like methods

On the convergence of inexact Newton-like methods

Efficient inexact Newton-like methods with application to problems of the deformation theory of plasticity

Efficient inexact Newton-like methods with application to problems of the deformation theory of plasticity

On Newton’s Method for Singular Problems

Sufficient conditions are given for Newton’s method to converge when the derivative is singular at the root.