Iterative methods for approximate solution of inverse problems.

*(English)*Zbl 1070.65038
Mathematics and its Applications (Springer) 577. Dordrecht: Springer (ISBN 1-4020-3121-1/hbk). xv, 291 p. (2004).

The book is devoted to the problem of solving nonlinear (particularly, linear) irregular equations \(F(x)=0\), where \(F: X_1\rightarrow X_2\), \(X_1\) and \(X_2\) are Hilbert or Banach spaces, and the operator \(F\) is a FrĂ©chet differentiable one but its derivative \(F'\) is not continuously invertible. Such equations arise in connection with inverse problems of mathematical modeling in various areas of science and engineering.

Irregular equations with approximate data are ill-posed numerical problems, and their solutions can be sought under usage of some additional information about the desired solution and data errors. The authors propose and investigate some regularization schemes on the base of the classical Newton-Kantorovich method and the Gauss-Newton one. The second method is the first one applied to minimization of the functional \(\| F(x)\| ^2\).

The main assumptions about the data are the next ones: A disturbed operator \(\tilde F(x)\) in some vicinity of the sought unique exact solution \(x^*\) is differentiable and fulfills the error estimates: \(\| F(x)\| _{X_2}\leq\delta, \quad \| \tilde F'(x)-F'(x)\| \leq\delta.\)

The main regularization method is the iterative process \[ x_{n+1}=\xi-\Theta(F'(x_n),\alpha_n)(F(x_n)-F'_{x_n}(x_n-\xi)), \] where \(\xi\) is a given trial point, \(\alpha_n\) is a sequence of regularization parameters, \(\Theta(\lambda,\alpha)\) is some “generating function”. Also processes of iterative regularization of modified minimization methods (unconstrained as well as constrained) and dynamical systems method for minimization of \(\| F(x)\| ^2\) are investigated.

The proposed regularization methods are demonstrated on some examples of inverse problems of gravimetry and acoustics.

Irregular equations with approximate data are ill-posed numerical problems, and their solutions can be sought under usage of some additional information about the desired solution and data errors. The authors propose and investigate some regularization schemes on the base of the classical Newton-Kantorovich method and the Gauss-Newton one. The second method is the first one applied to minimization of the functional \(\| F(x)\| ^2\).

The main assumptions about the data are the next ones: A disturbed operator \(\tilde F(x)\) in some vicinity of the sought unique exact solution \(x^*\) is differentiable and fulfills the error estimates: \(\| F(x)\| _{X_2}\leq\delta, \quad \| \tilde F'(x)-F'(x)\| \leq\delta.\)

The main regularization method is the iterative process \[ x_{n+1}=\xi-\Theta(F'(x_n),\alpha_n)(F(x_n)-F'_{x_n}(x_n-\xi)), \] where \(\xi\) is a given trial point, \(\alpha_n\) is a sequence of regularization parameters, \(\Theta(\lambda,\alpha)\) is some “generating function”. Also processes of iterative regularization of modified minimization methods (unconstrained as well as constrained) and dynamical systems method for minimization of \(\| F(x)\| ^2\) are investigated.

The proposed regularization methods are demonstrated on some examples of inverse problems of gravimetry and acoustics.

Reviewer: Vladimir Gorbunov (Ul’yanovsk)

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |

65J22 | Numerical solution to inverse problems in abstract spaces |

47J06 | Nonlinear ill-posed problems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |