Standard Picard Iteration Research Articles

On the robust Newton\u2019s method with the Mann iteration and the artistic patterns from its dynamics

There are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications.

There is a huge group of algorithms described in the literature that iteratively find solutions of a given equation. Most of them require tuning. The article presents root-finding algorithms that are based on the Newton–Raphson method which iteratively finds the solutions, and require tuning. The modification of the algorithm implements the best position of particle similarly to the particle swarm optimisation algorithms. The proposed approach allows visualising the impact of the algorithm’s elements on the complex behaviour of the algorithm. Moreover, instead of the standard Picard iteration, various feedback iteration processes are used in this research. Presented examples and the conducted discussion on the algorithm’s operation allow to understand the influence of the proposed modifications on the algorithm’s behaviour. Understanding the impact of the proposed modification on the algorithm’s operation can be helpful in using it in other algorithms. The obtained images also have potential artistic applications.

Newton\u2019s method with fractional derivatives and various iteration processes via visual analysis

The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomial z3 − 1 on a complex plane.

One more look on visualization of operation of a root-finding algorithm

Many algorithms that iteratively find solution of an equation require tuning. Due to the complex dependence of many algorithm’s elements, it is difficult to know their impact on the work of the algorithm. The article presents a simple root-finding algorithm with self-adaptation that requires tuning, similarly to evolutionary algorithms. Moreover, the use of various iteration processes instead of the standard Picard iteration is presented. In the algorithm’s analysis, visualizations of the dynamics were used. The conducted experiments and the discussion regarding their results allow to understand the influence of tuning on the proposed algorithm. The understanding of the tuning mechanisms can be helpful in using other evolutionary algorithms. Moreover, the presented visualizations show intriguing patterns of potential artistic applications.

An assessment of coupling algorithms for nuclear reactor core physics simulations

This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k -eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss–Seidel) to Anderson acceleration and multiple variants of preconditioned Jacobian-free Newton–Krylov (JFNK). The performance of the methods are evaluated over a range of energy group structures and core power levels. A novel physics-based approximation to a Jacobian-vector product has been developed to mitigate the impact of expensive on-line cross section processing steps. Numerical simulations demonstrating the efficiency of JFNK and Anderson acceleration relative to standard Picard iteration are performed on a 3D model of a nuclear fuel assembly. Both criticality ( k -eigenvalue) and critical boron search problems are considered.

S-iteration scheme and polynomiography

The aim of this paper is to present a modification of the visualization process of finding the roots of a given complex polynomial. In this paper S-iteration scheme is used instead of standard Picard iteration. The word “polynomiography coined by Kalantari for that visualization process. The images obtained are called polynomiographs. Polynomiographs have importance for both the art and science aspects. By using S-iteration scheme we obtain quite new nicely looking polynomiographs that are dierent from standard Picard iteration. Presented examples show that we obtain very interesting patterns for complex polynomial equations, permutation and doubly stochastic matrices. We believe that the results of this paper enrich the functionality of the existing polynomiography software. c 2015 All rights reserved.

Solvability of Age-Structured Epidemiological Models with Intracohort Transmission

The standard version of the epidemiological model with continuous age structure (Iannelli, Mathematical Theory of Age-Structured Population Dynamics, 1995) consists of the linear McKendrick model for the evolution of a disease-free population, coupled with one of the classical (SIS, SIR, etc.) models for the spread of the disease. A natural functional space in which the linear McKendrick model is well posed is the space of integrable functions. However, in the so-called intracohort models, the disease term contains pointwise products of the unknown functions; that is, of the age-specific densities of susceptibles, infectives and other classes (if applicable) which render the standard semilinear perturbation technique of proving the well-posedness of the full model not applicable in that space. This is due to the fact that the product of two integrable functions need not be integrable. Therefore, most works on the well-posedness of such problems have been done under additional assumption that the disease-free population is stable and the equilibrium has been reached (Busenberg et al. SIAM J Math Anal 22(4):1065–1080, 1991, Prus, J Math Biol 11:65–84, 1981). This allowed for showing, after some algebraic manipulations, that the order interval [0, 1] in the space of integrable functions was invariant under the action of the linear McKendrick semigroup and thus the usual iteration technique could be applied in this interval, yielding a bounded solution. An additional advantage of adopting the stability assumption was that it eliminated from the model the death rate which, in all realistic cases, is unbounded and creates serious technical difficulties. The aim of this note is to show that a careful modification of the standard Picard iteration procedure allows for proving the same result without the stability assumption. Since in such a case we are not able to suppress the unbounded death rate, we will briefly present the necessary linear results in a way which simplifies and unifies some classical results existing in the literature (Inaba, Math Popul Stud 1:49–77, 1988, Prus, J Math Biol 11:65–84, 1981, Webb, Theory of Nonlinear Age Dependent Population Dynamics, 1985).

Leveraging Anderson Acceleration for improved convergence of iterative solutions to transport systems

In this note we demonstrate that using Anderson Acceleration (AA) in place of a standard Picard iteration can not only increase the convergence rate but also make the iteration more robust for two transport applications. We also compare the convergence acceleration provided by AA to that provided by moment-based acceleration methods. Additionally, we demonstrate that those two acceleration methods can be used together in a nested fashion. We begin by describing the AA algorithm. At this point, we will describe two application problems, one from neutronics and one from plasma physics, on which we will apply AA. We provide computational results which highlight the benefits of using AA, namely that we can compute solutions using fewer function evaluations, larger time-steps, and achieve a more robust iteration.

An efficient nonlinear iteration scheme for nonlinear parabolic–hyperbolic system

A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number $$M > \sqrt{2}$$ and instability when $$M < \sqrt{2}$$ ; and Artola and Majda’s analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when $$M > \sqrt{2}$$ . For the Euler equations in three dimensions, every compressible vortex sheet is violently unstable and this instability is the analogue of the Kelvin–Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss–Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash–Moser–Hörmander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in three-dimensional MHD.

Geometry and Curvature of Diffeomorphism Groups withH1Metric and Mean Hydrodynamics

Holm, Marsden, and Ratiu (Adv. in Math.137(1998), 1–81) derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equationV(t)+∇U(t)V(t)−α2[∇U(t)]t·ΔU(t)=−gradp(t) where divU=0, andV=(1−α2Δ)U. In this model, the momentumVis transported by the velocityU, with the effect that nonlinear interaction between modes corresponding to length scales smaller thanαis negligible. We generalize this equation to the setting of ann-dimensional compact Riemannian manifold. The resulting equation is the Euler–Poincaré equation associated with the geodesic flow of theH1right invariant metric on Dsμ, the group of volume preserving Hilbert diffeomorphisms of classHs. We prove that the geodesic spray is continuously differentiable fromTDsμ(M) intoTTDsμ(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold (Ann. Inst. Grenoble16(1966), 319–361). To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariantH1metric on Dsμis a bounded trilinear map in theHstopology, from which it follows that solutions to Jacobi's equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.