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Some matrix-splitting iterative methods for solving systems of linear equations contain parameters that need to be specified in advance, and the choice of these parameters directly affects the efficiency of the corresponding iterative methods. This paper uses a Bayesian inference-based Gaussian process regression (GPR) method to predict the relatively optimal parameters of some HSS-type iteration methods and provide extensive numerical experiments to compare the prediction performance of the GPR method with other existing methods. Numerical results show that using GPR to predict the parameters of the matrix-splitting iterative methods has the advantage of smaller computational effort, predicting more optimal parameters and universality compared to the currently available methods for finding the parameters of the HSS-type iteration methods.

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs

This paper provides theoretical justification that Anderson acceleration (AA) improves the convergence rate of contractive fixed-point iterations in the vicinity of a fixed-point. A A has been used for decades to speed up nonlinear solvers in many applications. However, a rigorous mathematical justification of the improved convergence rate has remained lacking. The key ideas of the analysis presented here are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed-point iteration. The main result shown here is that AA improves the convergence rate of a fixed-point iteration to first order by a factor of the gain at each step. While the acceleration reduces the contribution from the first-order terms in the residual expansion, additional superlinear terms arise. In agreement with the theory, numerical tests are given which illustrate improved linear convergence rates, but for quadratically converging fixed-point iterations the rate is slowed. Our tests further illustrate how AA can perform similarly to damping in enlarging the domain of convergence.

Changes in the way customers shop, accompanied by an explosion of customer touchpoints and fast-changing competitive and technological dynamics, have led to an increased emphasis on agile marketing. The objective of this article is to conceptualize and investigate the emerging concept of marketing agility. The authors synthesize the literature from marketing and allied disciplines and insights from in-depth interviews with 22 senior managers. Marketing agility is defined as the extent to which an entity rapidly iterates between making sense of the market and executing marketing decisions to adapt to the market. It is conceptualized as occurring across different organizational levels and shown to be distinct from related concepts in marketing and allied fields. The authors highlight the firm challenges in executing marketing agility, including ensuring brand consistency, scaling agility across the marketing ecosystem, managing data privacy concerns, pursuing marketing agility as a fad, and hiring marketing leaders. The authors identify the antecedents of marketing agility at the organizational, team, marketing leadership, and employee levels and provide a roadmap for future research. The authors caution that marketing agility may not be well-suited for all firms and all marketing activities.

In this article, we establish a concept of fixed point result in Uniformly convex Banach space. Our main finding uses the Ishikawa iteration technique in uniformly convex Banach space to demonstrate strong convergence. Additionally, we use our primary result to demonstrate some corollaries.

The l₂ normalized inverse, shifted inverse, and Rayleigh quotient iterations are classic algorithms for approximating an eigenvector of a symmetric matrix. This work establishes rigorously that each iterate produced by one of these three algorithms can be viewed as a Newton's method iterate followed by a normalization. The equivalences given here are not meant to suggest changes to the implementations of the classic eigenvalue algorithms. However, they add further understanding to the formal structure of these iterations, and they provide an explanation for their good behavior despite the possible need to solve systems with nearly singular coefficient matrices. A historical development of these eigenvalue algorithms is presented. Using our equivalences and traditional Newton's method theory helps to gain understanding as to why normalized Newton's method, inverse iteration, and shifted inverse iteration are only linearly convergent and not quadratically convergent, as would be expected, and why a new linear system need not be solved at each iteration. We also give an explanation as to why our normalized Newton's method equivalent of Rayleigh quotient iteration is cubically convergent and not just quadratically convergent, as would be expected.

The aim of this paper is two fold: the first is to define two new classes of mappings and show the existence and iterative approximation of their fixed points; the second is to show that the Ishikawa, Mann, and Krasnoselskij iteration methods defined for such classes of mappings are equivalent. An application of the main results to solve split feasibility and variational inequality problems are also given.

When simulating groundwater flow in unconfined and convertible aquifers using a groundwater model with the block‐centered finite‐difference approach, such as MODFLOW, it frequently encounters drying and rewetting of cells. Although many drying and rewetting simulation methods have been proposed in the past, balancing simulation accuracy and convergence capability all at once is difficult. MODFLOW‐2005, which has second‐order accuracy, employs a trial‐and‐error method, but it suffers from computational instability when large quantities of grid cells are dried. MODFLOW‐NWT adopts the upstream‐weighting approach and Newton iteration method to ensure the stability of the drying and rewetting simulations. However, the upstream‐weighting approach has only first‐order accuracy, and the Newton iteration method is complex to implement because it necessitates the establishment of an additional Jacobian matrix. The methods employed by MODFLOW‐NWT are also available in MODFLOW 6, therefore it inherits both the strengths and weaknesses of MODFLOW‐NWT. In this study, a new method, Picard iteration‐based always active cell (PAAC), is proposed. Similar to MODFLOW‐NWT, the PAAC method also uses dry cells as active cells. The PAAC method, however, does not use the upstream‐weighting approach and has second‐order accuracy. Moreover, it ensures good convergence stability even under the Picard iteration method. In addition to discussing the algorithm, five cases were used to comprehensively compare the simulation effects of the PAAC method with MODFLOW‐2005 and MODFLOW‐NWT, including an analytical solution, repeated drying‐rewetting of multi‐layer grids, pumping well problem, perched aquifer problem and a nearly dry single‐layer grid, which verified the practicability of the PACC method. Key Points A new physically‐based method to simulating the drying‐rewetting problems of groundwater model, Picard iteration‐based always active cell The new method performed robust convergence even with the Picard iteration method and a general PCG solver The new method achieved second‐order accuracy

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.