Nonlinearly Preconditioned FETI Solver for Substructured Formulations of Nonlinear Problems

Negrello, Camille; Gosselet, Pierre; Rey, Christian

doi:10.3390/math9243165

Cited by 6 publications

(4 citation statements)

References 46 publications

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“…Algorithmic methods do not involve the inclusion of any additional functional block or element in the structure of the measuring instrument and are based only on the processing of information from the measuring instrument by certain algorithms by a microprocessor computing device within the system [11][12][13][14][15][16][17][18][19][20].…”

Section: Analysis Of Recent Research and Publicationsmentioning

confidence: 99%

Multiplicative Approximation Method of Functional Dependencies by Line Segments

Khudaverdiyeva¹

2022

CNCS

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The article is devoted to the approximation problems of functional dependencies during conversions performed in intelligent measurement devices. Non-linearities are essential parts of most control processes and systems. When using a nonlinear transmitter with a conversion function in measurement information systems used in various fields, it is necessary to perform nonlinear functional conversion operations on numbers in microprocessors/microcontrollers during direct and indirect measurements. For this purpose, various approximation methods are used. The purpose of the approximation is to describe nonlinear functions in a simpler, more convenient way for utilization and calculations, with an insignificantly small loss of accuracy. Existent methods for linearization, although some of them are effective, can be burdensome for implementation in microprocessor-based systems. Here, one of the proposed methods for the approximation of nonlinear functional dependencies by line segments is proposed. In this method, the range of the argument changes in the function is divided into line segments, and the parts of the coordinate system bisector, remaining within the line segments of the function, is swapped to perform approximation. Having involved few simple mathematical operations, the proposed method can be implemented efficiently in microprocessors/microcontrollers to perform approximations in measurement systems.

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Section: Analysis Of Recent Research and Publicationsmentioning

confidence: 99%

Multiplicative Approximation Method of Functional Dependencies by Line Segments

Khudaverdiyeva¹

2022

CNCS

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“…In other words, linearization is a linear approximation to a given equation. Although mathematical functions have different distributions specific to their structures, linearization is required to analyze and generalize these formulas [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

“…2 ranges from 0.228 to 0.106 indicating deviation around 12.2%. Cubic and quadratic are the best fitted models, whereas compound, growth and exponential are the less fitted models.…”

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confidence: 99%

A Multi-disciplinary Investigation of Linearization Deviations in Different Regression Models

Yılmaz

Turanli

2023

AJPAS

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Aim: This study's objective is to examine linearization deviations in various regression models using a multidisciplinary approach. Methods: Curve estimate models' accuracy for each type of data was tested using social, financial, and medical data sets. Results: Although the power of a given model reduces with non-normal distributions, linearization in the social sciences is more efficient and has less variation from regression points in parametric equations. However, single-center distributions in the social sciences typically lead to nonparametric distributions. When compared to social sciences and health sciences, the effectiveness of linearization in financial sciences is higher. The original essence of financial techniques and models is frequently present, and they test their presumptions. Financial models and assumptions are more linear than those seen in real life since they correspond to artificial systems that individuals have developed, making them better suited for predetermined formulas. The rise in linearization issues in the field of finance, which has been increasingly common in recent years, is a symptom of this together with behavioral finance. Studies in the realm of health and well-known models were discovered to have the highest linearization deviations. Exponential or growth functions exhibit the highest linearization deviations in processes like growth, proliferation, and the spread of disease or pandemics. The data display considerable departures from normality and linearization, particularly in animal trials with very small statistical units or research conducted on a particular population. Conclusion: Despite research on R2's explanatory capacity in regression, there aren't enough studies in various fields that concentrate on R2's departures from linearization. Additionally, no study was located in which the subject's mathematical foundation was examined and cross-compared across various data sets. As a result of this feature, the research represents a field first. The research's ability to pragmatically assess the distinctions between disciplines, made possible by its multi-disciplinary nature, is another unique aspect of the work.

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“…Originally introduced in [5,6], they are nonlinear generalizations of standard FETI-DP domain decomposition methods [22]. Related methods are, e.g., nonlinear BDDC (Balancing Domain Decomposition by Constraints) methods [10,11], nonlinear FETI-1 methods [18,19,21], ASPIN (Additive Schwarz Preconditioned Inexact Newton) methods [3], or ASPEN (Additive Schwarz Preconditioned Exact Newton) methods [4,9]. Nonlinear FETI-DP domain decomposition methods are robust and scalable [12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

Köhler

Rheinbach

2022

Vietnam J. Math.

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The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

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Nonlinearly Preconditioned FETI Solver for Substructured Formulations of Nonlinear Problems

Cited by 6 publications

References 46 publications

Multiplicative Approximation Method of Functional Dependencies by Line Segments

Multiplicative Approximation Method of Functional Dependencies by Line Segments

A Multi-disciplinary Investigation of Linearization Deviations in Different Regression Models

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

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