On convergence of regularized modified Newton's method for nonlinear ill-posed problems

George, Santhosh

doi:10.1515/jiip.2010.004

Cited by 29 publications

(19 citation statements)

References 15 publications

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“…Any of such functions is called admissible for̂and it can be used as a measure for the convergence of →̂(see [19]). The main result of this section is the following theorem, proof of which is analogous to the proof of Theorem 4.4 in [13]. …”

Section: Balancing Principlementioning

confidence: 87%

“…In the last few years many authors considered iterative methods, for example, Landweber method [7,8], LevenbergMarquardt method [9], Gauss-Newton [10,11], Conjugate Gradient [12], Newton-like methods [13,14], and TIGRA (Tikhonov gradient method) [6]. For Landweber's and TIGRA method, one has to evaluate the operator and the adjoint of the Fréchet derivative of .…”

Section: Introductionmentioning

confidence: 99%

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Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

Argyros

George

Jidesh

2014

International Journal of Mathematics and Mathematical Sciences

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We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equation ( ) = . The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.

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Section: Balancing Principlementioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

Argyros

George

Jidesh

2014

International Journal of Mathematics and Mathematical Sciences

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“…This method is a simplified version of iteratively regularized Gauss-Newton method (IRGNM), which falls within the class of regularized Gauss-Newton method, and studied recently in details by Mahale & Nair (Mahale and Nair, 2009), Jin (Jin, 2010), and George (George, 2010).…”

Section: Numerical Implementation (Discrete Approximation)mentioning

confidence: 99%

Application of Core Identification to Star-Shape Constant Conductivity Inclusion

Garnadi¹

2014

Int. J. of Pure and Appl. Math.

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“…is the Fréchet derivative of F at q 0 , and α k is regularization parameter, see Jin [3] and George [1]. Introduce the space T M of trigonometric polynomials of degree ≤ M, and solve the following minimization problem Numerical Test Case.…”

Section: Numerical Implementationmentioning

confidence: 99%

Identification of Starshape Constant Conductivity Inclusion from Boundary Measurement

Garnadi

2011

Proc Appl Math and Mech

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The problem of determining the interface separating a starshape support of constant conductivity inclusion from boundary measurement data of a solution of the corresponding PDEs is considered. An equivalent statement as a nonlinear integral equation is obtained. The problem is implemented using Fourier method. Numerical experiment based on simplified iteratively regularized Gauss-Newton method (sIRGNM) is presented. MSC classes: 31A25, 35R30, 65N21. Consider the following boundary value problemwhere Ω 1 is a unit circular domain with boundary ∂Ω 1 , and Ω S ⊂ Ω 1 is the support of S, and χ Ω S denoted the characteristic function of Ω S . The inverse problem consists in identifying the shape of Ω S given the Neumann data ∂u ∂n on ∂Ω 1 . Therefore, we define F as the operator mapping q to Assuming that the conductivity perturbation a is small, we linearize the equations w.r.t. the conductivity perturbation. The linearized equation for small a can be obtained by solvingwhere u = w − u 0 . Assume that supp(a) =Ω,Ω ⊂⊂ Ω, and suppose a ∈ L 2 (Ω). Then from (1.1) equivalently we haveHere g is applied voltage, u 0 is a harmonic reference potential, and u is fluctuation of the potential due to the inclusion a. From (1.2), we reduce our problem to finding the support of the source in the Poisson equation: ∆u = S(x) in Ω, u = 0 on ∂Ω, where the source is expressed formally as S(x) = − ∇ · a ∇ u 0 , and u 0 is the corresponding harmonic reference potential. The inverse linearized problem is to find a conductivity distribution a which fits the measured currents on the surface, ∂u ∂ν | ∂Ω 1 , of potential difference u satisfying (1.2)for the different applied voltages g. Starshape object inclusion problem as core identification.It is well known that the unique solution to the problem (0.1) is given bywhere G(x, ξ) is the Green's function of Ω 1 . Assume that a is a constant, Ω S to be star-shaped with respect to the origin. Then ∂Ω S = {q(t)(cos t, sin t) : t ∈ [0, 2π]} for q(t) > 0 and 2π−periodic. Consider the case a = χ Ω S , hence a| ∂Ω 1 = 0. Let Z 0 = Z\{0}, and let the Poisson Kernel: P (r, s) = 1 2π n∈Z (r) |n| e i ns , the Green's function of the unit disk Ω 1 in polar coordinates. We choose the harmonic reference potential u 0 of the form u 0 := U k (r, s) =k |k| e i ks , k ∈ Z 0 , which is the solution of Laplace equations with a scaled trigonometric function g = e i ks |k| , k ∈ Z 0 , as the Dirichlet

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On convergence of regularized modified Newton's method for nonlinear ill-posed problems

Cited by 29 publications

References 15 publications

Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

Application of Core Identification to Star-Shape Constant Conductivity Inclusion

Identification of Starshape Constant Conductivity Inclusion from Boundary Measurement

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