PARAMETER ESTIMATION AND INVERSE PROBLEMS
CONTENTS INTRODUCTION CLASSI?CATION OF INVERSE PROBLEMS EXAMPLES OF PARAMETER ESTIMATION PROBLEMS EXAMPLES OF INVERSE PROBLEMS WHY INVERSE PROBLEMS ARE HARD EXERCISES NOTES AND FURTHER READING LINEAR REGRESSION INTRODUCTION TO LINEAR REGRESSION STATISTICAL ASPECTS OF LEAST SQUARES UNKNOWN MEASUREMENT STANDARD DEVIATIONS L1 REGRESSION MONTE CARLO ERROR PROPAGATION DISCRETIZING CONTINUOUS INVERSE PROBLEMS INTEGRAL EQUATIONS QUADRATURE METHODS EXPANSION IN TERMS OF REPRESENTERS EXPANSION IN TERMS OF ORTHONORMAL BASIS FUNCTIONS THE METHOD OF BACKUS AND GILBERT RANK DEFICIENCY AND ILL-CONDITIONING THE SVD AND THE GENERALIZED INVERSE COVARIANCE AND RESOLUTION OF THE GENERALIZED INVERSE SOLUTION INSTABILITY OF THE GENERALIZED INVERSE SOLUTION AN EXAMPLE OF A RANK-DE?CIENT PROBLEM DISCRETE ILL-POSED PROBLEMS TIKHONOV REGULARIZATION SELECTING A GOOD SOLUTION SVD IMPLEMENTATION OF TIKHONOV REGULARIZATION RESOLUTION, BIAS, AND UNCERTAINTY IN THE TIKHONOV SOLUTION HIGHER-ORDER TIKHONOV REGULARIZATION RESOLUTION IN HIGHER-ORDER TIKHONOV REGULARIZATION THE TGSVD METHOD GENERALIZED CROSS VALIDATION ERROR BOUNDS 5 10 NOTES AND FURTHER READING ITERATIVE METHODS INTRODUCTION ITERATIVE METHODS FOR TOMOGRAPHY PROBLEMS THE CONJUGATE GRADIENT METHOD THE CGLS METHOD ADDITIONALREGULARIZATIONTECHNIQUES USING BOUNDS AS CONSTRAINTS MAXIMUM ENTROPY REGULARIZATION TOTAL VARIATION FOURIER TECHNIQUES LINEAR SYSTEMS IN THE TIME AND FREQUENCY DOMAINS DECONVOLUTION FROM A FOURIER PERSPECTIVE LINEAR SYSTEMS IN DISCRETE TIME WATER LEVEL REGULARIZATION NONLINEAR REGRESSION NEWTON’S METHOD THE GAUSS–NEWTON AND LEVENBERG–MARQUARDT METHODS STATISTICAL ASPECTS IMPLEMENTATION ISSUES 10 NONLINEAR INVERSE PROBLEMS 10 1 REGULARIZING NONLINEAR LEAST SQUARES PROBLEMS 10 2 OCCAM’S INVERSION 10 3 EXERCISES 10 4 NOTES AND FURTHER READING 11 BAYESIAN METHODS REVIEW OF THE CLASSICAL APPROACH 11 2 THE BAYESIAN APPROACH 11 3 THE MULTIVARIATE NORMAL CASE MAXIMUM ENTROPY METHODS EPILOGUE