Jodi Mead

A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

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 χ2 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 the system matrix, is found to be very efficient for parameter estimation, requiring on average about 10 Newton steps. Additionally, the theory and algorithm apply for generalized Tikhonov regularization using the generalized singular value decomposition. The performance of the Newton algorithm is contrasted to standard techniques, including the L-curve, generalized cross validation and unbiased predictive risk estimation. This χ2-curve Newton method of parameter estimation is seen to be robust and cost effective in comparison to other methods, when white or colored noise information on the measured data is incorporated.

Accuracy, Resolution, and Stability Properties of a Modified Chebyshev Method

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order

Discontinuous parameter estimates with least squares estimators

M

Stability of a Pivoting Strategy for Parallel Gaussian Elimination

requires time steps of approximately

Geophysical imaging of subsurface structures with least squares estimates

O(N^{-2M})

Pseudospectral Iterated Method for Differential Equations with Delay Terms

for stable explicit solvers. Theoretically, time steps may be increased to

Regularization parameter estimation for large-scale Tikhonov regularization using a priori information

O(N^{-M})

Least squares problems with inequality constraints as quadratic constraints

with the use of a parameter,

Parameter estimation: A new approach to weighting a priori information

\alpha

Estimating Unsaturated Hydraulic Functions for Coarse Sediment from a Field-Scale Infiltration Experiment

-dependent mapped method introduced by Kosloff and Tal-Ezer [{\em J.\ Comput.\ Phys}., 104 (1993), pp. 457--469]. Our analysis focuses on the utilization of this method for reasonable practical choices for