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X2 Tests For The Choice Of The Regularization Parameter In Nonlinear Inverse Problems, J. L. Mead, C. C. Hammerquist
X2 Tests For The Choice Of The Regularization Parameter In Nonlinear Inverse Problems, J. L. Mead, C. C. Hammerquist
Jodi Mead
We address discrete nonlinear inverse problems with weighted least squares and Tikhonov regularization. Regularization is a way to add more information to the problem when it is ill-posed or ill-conditioned. However, it is still an open question as to how to weight this information. The discrepancy principle considers the residual norm to determine the regularization weight or parameter, while the χ2 method [J. Mead, J. Inverse Ill-Posed Probl., 16 (2008), pp. 175–194; J. Mead and R. A. Renaut, Inverse Problems, 25 (2009), 025002; J. Mead, Appl. Math. Comput., 219 (2013), pp. 5210–5223; R. A. Renaut, I. Hnetynkova, and J. L. …
Discontinuous Parameter Estimates With Least Squares Estimators, J. L. Mead
Discontinuous Parameter Estimates With Least Squares Estimators, J. L. Mead
Jodi Mead
We discuss weighted least squares estimates of ill-conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices. Least squares estimators are the maximum likelihood estimate for normally distributed data and parameters, but here we do not assume particular probability distributions. Weights for the estimator are found by ensuring its minimum follows a χ2 distribution. Previous work with this approach has shown that it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [20]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization parameter. …
A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut
A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut
Jodi Mead
We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of …
Characterization And Properties Of $(R,S)$-Symmetric, $(R,S)$-Skew Symmetric, And $(R,S)$-Conjugate Matrices, William F. Trench
Characterization And Properties Of $(R,S)$-Symmetric, $(R,S)$-Skew Symmetric, And $(R,S)$-Conjugate Matrices, William F. Trench
William F. Trench
No abstract provided.