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A two step modified Newton method is considered for obtaining
an approximate solution for the nonlinear ill-posed equation

Recently George and Elmahdy [

Throughout this paper, the inner product and the corresponding norm on the Hilbert space

In application, usually only noisy data

For monotone operators, one usually uses the Lavrentiev regularization method (see [

The optimality of the Lavrentiev method was proved in [

Thus in the last few years, more emphasis was put on the investigation of iterative regularization methods (see [

The organization of this paper is as follows. Convergence analysis of TSMNLM is given in Section

We need the following assumptions for the convergence analysis of TSMNLM.

There exists a constant

Let

Note that if

Let

Let q and r be as in (

Observe that if

The main result of this section is the following theorem.

Let

Using the relation (b) and (c) of Theorem

The objective of this section is to obtain an error estimate for

There exists a continuous, strictly monotonically increasing function

It can be seen that functions

We will be using the error estimates in the following proposition, which can be found in [

Let

The following theorem can be found in [

Let

Combining the estimates in Proposition

Let

Let

Let

Note that the error estimate

Now using the function

Let

In this subsection, we will present a parameter choice rule based on the adaptive method studied in [

Let

Assume that there exists

Following steps are involved in implementing the adaptive choice rule.

Choose

Choose

Finally the adaptive algorithm associated with the choice of the parameter specified in Theorem

Set

Choose

Solve

If

Else set

In this section, we consider the example considered in [

Let

Thus the operator

Note that for

Observe that

In our computation, we take

Observe that while performing numerical computation on finite dimensional subspace

Let

We choose

8 | 2 | 3 | 0.1016 | 0.3428 | 0.2634 | 0.8266 |

16 | 2 | 3 | 0.1004 | 0.3388 | 0.1962 | 0.6191 |

32 | 2 | 3 | 0.1001 | 0.3378 | 0.1429 | 0.4518 |

64 | 2 | 3 | 0.1000 | 0.3376 | 0.1036 | 0.3275 |

128 | 2 | 3 | 0.1000 | 0.3375 | 0.0755 | 0.2387 |

256 | 2 | 3 | 0.1000 | 0.3375 | 0.0560 | 0.1772 |

512 | 2 | 3 | 0.1000 | 0.3375 | 0.0430 | 0.1360 |

1024 | 2 | 3 | 0.1000 | 0.3375 | 0.0347 | 0.1096 |

Curves of the exact and approximate solutions.

Curves of the exact and approximate solutions.

The last column of Table

We considered a two step method for obtaining an approximate solution for a nonlinear ill-posed operator equation

S. Pareth thanks the National Institute of Technology Karnataka, Surathkal for the financial support.