## Computational Methods for Inverse ProblemsInverse problems arise in a number of important practical applications, ranging from biomedical imaging to seismic prospecting. This book provides the reader with a basic understanding of both the underlying mathematics and the computational methods used to solve inverse problems. It also addresses specialized topics like image reconstruction, parameter identification, total variation methods, nonnegativity constraints, and regularization parameter selection methods. Because inverse problems typically involve the estimation of certain quantities based on indirect measurements, the estimation process is often ill-posed. Regularization methods, which have been developed to deal with this ill-posedness, are carefully explained in the early chapters of Computational Methods for Inverse Problems. The book also integrates mathematical and statistical theory with applications and practical computational methods, including topics like maximum likelihood estimation and Bayesian estimation. |

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### Contents

Analytical Tools | 13 |

Numerical Optimization Tools | 29 |

Statistical Estimation Theory | 41 |

Image Deblurring | 59 |

Parameter Identification | 85 |

Regularization Parameter Selection Methods | 97 |

Contents xi | 107 |

Total Variation Regularization | 129 |

Nonnegativity Constraints | 151 |

173 | |

### Common terms and phrases

Algorithm analysis applied assume BTTB CG iterations Chapter components compute constrained minimization convergence rate corresponding defined denotes derivative diagonal discrepancy principle discrete distribution eigenvalues estimation error Euclidean norm Example Exercise Figure filter fv+i given grad gradient projection Hessian Hilbert space implementation initial guess inner product inverse problems iteration count J(fv L-curve least squares functional likelihood function line search linear operator linear systems lower semicontinuous matrix minimization problem Newton's method nonlinear nonnegatively constrained numerical obtain optimization Parameter Selection Methods penalty functional Poisson positive definite preconditioner predictive risk primal-dual Newton probability mass function quadratic random variable random vector reconstruction regularization methods Regularization Parameter Selection regularized solution representation right-hand side singular values solve steepest descent subplot symmetric techniques test problem Theorem Tikhonov regularization Toeplitz total variation total variation regularization TSVD regularization two-dimensional UPRE variation regularization zero