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A simple regularization method for the ill-posed evolution equation

Nguyen Huy Tuan, Dang Duc Trong (2011)

Czechoslovak Mathematical Journal

The nonhomogeneous backward Cauchy problem u t + A u ( t ) = f ( t ) , u ( T ) = ϕ , where A is a positive self-adjoint unbounded operator which has continuous spectrum and f is a given function being given is regularized by the well-posed problem. New error estimates of the regularized solution are obtained. This work extends earlier results by N. Boussetila and by M. Denche and S. Djezzar.

Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Monnanda Erappa Shobha, Ioannis K. Argyros, Santhosh George (2014)

Applicationes Mathematicae

We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is y δ with | | y - y δ | | δ , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where F ' ( x ) - 1 exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter...

On the semilocal convergence of a two-step Newton-like projection method for ill-posed equations

Ioannis K. Argyros, Santhosh George (2013)

Applicationes Mathematicae

We present new semilocal convergence conditions for a two-step Newton-like projection method of Lavrentiev regularization for solving ill-posed equations in a Hilbert space setting. The new convergence conditions are weaker than in earlier studies. Examples are presented to show that older convergence conditions are not satisfied but the new conditions are satisfied.

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