Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales

AbstractIn this paper, we study the simplified generalized Gauss–Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form {F(x)=y} where {F:D(F)\subseteq X\to Y} is a nonlinear operator between Hilbert spaces X and Y. Under suitable nonlinearly conditions on F, we obtain an order optimal error estimate under the Morozov type stopping rule.

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ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

The ANZIAM Journal ◽

10.1017/s1446181109000418 ◽

2009 ◽

Vol 51 (2) ◽

pp. 191-217 ◽

Cited By ~ 8

Author(s):

P. MAHALE ◽

M. T. NAIR

Keyword(s):

Nonlinear Operator ◽

Optimal Error Estimate ◽

Positive Real ◽

Iterative Regularization ◽

Optimal Error ◽

Lavrentiev Regularization ◽

Inexact Data ◽

Ill Posed ◽

Funct Anal ◽

General Source

AbstractWe consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with $\|y-y^\d \|\leq \d $ resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

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Newton-Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems

Journal of Mathematics ◽

10.1155/2013/439316 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Santhosh George

Keyword(s):

Newton Method ◽

Hilbert Spaces ◽

Nonlinear Integral Equation ◽

Nonlinear Operator ◽

Regularization Parameter ◽

Adaptive Method ◽

Majorizing Sequence ◽

Source Condition ◽

Ill Posed ◽

General Source

Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equationF(x)=y, whereF:D(F)⊆X→Yis a nonlinear operator between the Hilbert spacesXandY. The analysis in George, 2010 was carried out using a majorizing sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure.

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A Variant of Projection-Regularization Method for Ill-Posed Linear Operator Equations

International Journal of Computational Methods ◽

10.1142/s0219876221500080 ◽

2020 ◽

pp. 2150008

Author(s):

Bechouat Tahar ◽

Boussetila Nadjib ◽

Rebbani Faouzia

Keyword(s):

Approximate Solution ◽

Linear Operator ◽

Tikhonov Regularization ◽

Hilbert Spaces ◽

Regularization Method ◽

Tikhonov Regularization Method ◽

Operator Equations ◽

Ill Posed ◽

Rank Approximation ◽

Linear Operator Equations

In this paper, we report on a strategy for computing the numerical approximate solution for a class of ill-posed operator equations in Hilbert spaces: [Formula: see text]. This approach is a combination of Tikhonov regularization method and the finite rank approximation of [Formula: see text]. Finally, numerical results are given to show the effectiveness of this method.

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Optimal error estimate of a compact scheme for nonlinear Schrödinger equation

Applied Numerical Mathematics ◽

10.1016/j.apnum.2017.05.004 ◽

2017 ◽

Vol 120 ◽

pp. 68-81 ◽

Cited By ~ 6

Author(s):

Jialin Hong ◽

Lihai Ji ◽

Linghua Kong ◽

Tingchun Wang

Keyword(s):

Schrödinger Equation ◽

Error Estimate ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Compact Scheme ◽

Optimal Error Estimate ◽

Optimal Error ◽

Nonlinear Schrödinger

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An optimal error estimate for the two-dimensional nonlinear time fractional advection–diffusion equation with smooth and non-smooth solutions

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2019.12.013 ◽

2020 ◽

Vol 79 (10) ◽

pp. 2819-2831

Author(s):

Hui Zhang ◽

Junqing Jia ◽

Xiaoyun Jiang

Keyword(s):

Diffusion Equation ◽

Error Estimate ◽

Optimal Error Estimate ◽

Two Dimensional ◽

Smooth Solutions ◽

Optimal Error ◽

Advection Diffusion Equation ◽

Advection Diffusion

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UNIFORM INF–SUP CONDITIONS FOR THE SPECTRAL DISCRETIZATION OF THE STOKES PROBLEM

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202599000208 ◽

1999 ◽

Vol 09 (03) ◽

pp. 395-414 ◽

Cited By ~ 47

Author(s):

C. BERNARDI ◽

Y. MADAY

Keyword(s):

Error Estimate ◽

Error Estimates ◽

Best Approximation ◽

Stokes Problem ◽

Optimal Error Estimate ◽

Optimal Error ◽

Discretization Parameter ◽

The Best Approximation ◽

Discrete Spaces ◽

Spectral Discretization

In standard spectral discretizations of the Stokes problem, error estimates on the pressure are slightly less accurate than the best approximation estimates, since the constant of the Babuška–Brezzi inf–sup condition is not bounded independently of the discretization parameter. In this paper, we propose two possible discrete spaces for the pressure: for each of them, we prove a uniform inf–sup condition, which leads in particular to an optimal error estimate on the pressure.

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