scholarly journals Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

2016 ◽  
Vol 54 (1) ◽  
pp. 17-33
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Computational Cost ◽  
Nonlinear Least Squares ◽  
Least Square ◽  
Error Tolerance ◽  
Local Analysis ◽  
Large Radius ◽  
Special Cases ◽  
Local Convergence Analysis ◽  
Least Square Problems

Abstract We present a local convergence analysis of inexact Gauss-Newton-like method (IGNLM) for solving nonlinear least-squares problems in a Euclidean space setting. The convergence analysis is based on our new idea of restricted convergence domains. Using this idea, we obtain a more precise information on the location of the iterates than in earlier studies leading to smaller majorizing functions. This way, our approach has the following advantages and under the same computational cost as in earlier studies: A large radius of convergence and more precise estimates on the distances involved to obtain a desired error tolerance. That is, we have a larger choice of initial points and fewer iterations are also needed to achieve the error tolerance. Special cases and numerical examples are also presented to show these advantages.

scholarly journals On Local Convergence Analysis of Inexact Newton Method for Singular Systems of Equations under Majorant Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Fangqin Zhou
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Local Convergence ◽  
Singular Systems ◽  
Inexact Newton Method ◽  
Systems Of Equations ◽  
Convergence Ball ◽  
Special Cases ◽  
Majorant Condition ◽  
Local Convergence Analysis

We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases.

scholarly journals On an improved convergence analysis of a two-step Gauss-Newton type method under generalized Lipschitz conditions

2020 ◽  
Vol 36 (3) ◽  
pp. 365-372
Author(s):  
I. K. ARGYROS ◽  
R. P. IAKYMCHUK ◽  
S. M. SHAKHNO ◽  
H. P. YARMOLA
Keyword(s):  
Convergence Analysis ◽  
Computational Effort ◽  
Convergence Criteria ◽  
Type Method ◽  
Precise Information ◽  
Special Cases ◽  
Original Domain ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Theoretical Results

We present a local convergence analysis of a two-step Gauss-Newton method under the generalized and classical Lipschitz conditions for the first- and second-order derivatives. In contrast to earlier works, we use our new idea using a center average Lipschitz conditions through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain: weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates and more precise information on the location of the solution. These advantages are obtained under the same computational effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works.

Extending the convergence domain of Newton’s method for twice Fréchet differentiable operators

2016 ◽  
Vol 14 (02) ◽  
pp. 303-319
Author(s):  
Ioannis K. Argyros ◽  
Á. Alberto Magreñán
Keyword(s):  
Convergence Analysis ◽  
Newton’S Method ◽  
Newton's Method ◽  
Computational Cost ◽  
Semilocal Convergence ◽  
Convergence Criteria ◽  
Fréchet Differentiable ◽  
Local Convergence Analysis ◽  
Appl Math ◽  
Semilocal Convergence Theorem

We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79 (1997) 131–145]. Numerical examples where the old convergence criteria cannot apply to solve nonlinear equations but the new convergence criteria are satisfied are also presented at the concluding section of this paper.

scholarly journals Determining Robust Reaction Kinetics from Limited Data

2021 ◽  
Author(s):  
Gizem Ozbuyukkaya ◽  
Robert Parker ◽  
Goetz Veser
Keyword(s):  
Parameter Estimation ◽  
Kinetic Parameter ◽  
Cross Validation ◽  
Computational Cost ◽  
Synthetic Data ◽  
Nonlinear Least Squares ◽  
Least Square ◽  
Data Sets ◽  
Limited Data ◽  
Kinetic Parameter Estimation

Accurate chemical kinetics are essential for reactor design and operation. However, despite recent advances in “big data” approaches, availability of kinetic data is often limited in industrial practice. Herein, we present a comparative proof-of-concept study for kinetic parameter estimation from limited data. Cross-validation (CV) is implemented to nonlinear least-squares (LS) fitting and evaluated against Markov chain Monte Carlo (MCMC) and genetic algorithm (GA) routines using synthetic data generated from a simple model reaction. As expected, conventional LS is fastest but least accurate in predicting true kinetics. MCMC and GA are effective for larger data sets but tend to overfit to noise for limited data. Cross-validation least-square (LS-CV) strongly outperforms these methods at much reduced computational cost, especially for significant noise. Our findings suggest that implementation of cross-validation with conventional regression provides an efficient approach to kinetic parameter estimation with high accuracy, robustness against noise, and only minimal increase in complexity.

scholarly journals TMF: A GNSS Tropospheric Mapping Function for the Asymmetrical Neutral Atmosphere

2021 ◽  
Vol 13 (13) ◽  
pp. 2568
Author(s):  
Di Zhang ◽  
Jiming Guo ◽  
Tianye Fang ◽  
Na Wei ◽  
Wensheng Mei ◽  
...  
Keyword(s):  
Large Scale ◽  
Least Squares Method ◽  
Computational Cost ◽  
Elevation Angle ◽  
Mapping Function ◽  
Nonlinear Least Squares ◽  
Least Square ◽  
Neutral Atmosphere ◽  
Medium Range Weather Forecast ◽  
Mapping Functions

Tropospheric mapping function plays a vital role in the high precision Global Navigation Satellites Systems (GNSS) data processing for positioning. However, most mapping functions are derived under the assumption that atmospheric refractivity is spherically symmetric. In this paper, the pressure, temperature, and humidity fields of ERA5 data with the highest spatio-temporal resolution available from the European Centre for Medium-range Weather Forecast (ECMWF) were utilized to compute ray-traced delays by the software WHURT. Results reveal the universal asymmetry of the hydrostatic and wet tropospheric delays. To accurately represent these highly variable delays, a new mapping function that depends on elevation and azimuth angles—Tilting Mapping Function (TMF)—was applied. The basic idea is to assume an angle between the tropospheric zenith direction and the geometric zenith direction. Ray-traced delays served as the reference values. TMF coefficients were fitted by Levenberg–Marquardt nonlinear least-squares method. Comparisons demonstrate that the TMF can improve the MF-derived slant delay’s accuracy by 73%, 54% and 29% at the 5° elevation angle, against mapping functions based on the VMF3 concept, without, with a total and separate estimation of gradients, respectively. If all coefficients of a symmetric mapping function are determined together with gradients by a least-square fit at sufficient elevation angles, the accuracy is only 6% lower than TMF. By adopting the b and c coefficients of VMF3, TMF can keep its high accuracy with less computational cost, which could be meaningful for large-scale computing.

Local convergence of deformed Jarratt-type methods in Banach space without inverses

2016 ◽  
Vol 09 (01) ◽  
pp. 1650015
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Computational Cost ◽  
Fréchet Derivative ◽  
Convergence Conditions ◽  
Type Method ◽  
Frechet Derivative ◽  
The Third ◽  
Local Convergence Analysis

We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study.

Fast Bilinear Algorithms for Symmetric Tensor Contractions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Symmetric Matrix ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Symmetric Tensor ◽  
Partial Sums ◽  
Symmetric Tensors ◽  
Outer Product ◽  
Bilinear Complexity ◽  
Special Cases ◽  
Tensor Contractions

AbstractIn matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

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