#### Document Type

Article

#### Publication Date

2013

#### Abstract

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 *χ*2 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. Mead, *Comput.Statist.Data Anal*., 54 (2010), pp. 3430–3445] uses the regularized residual. Using the regularized residual has the benefit of giving a clear *χ*2 test with a fixed noise level when the number of parameters is equal to or greater than the number of data. Previous work with the *χ*2 method has been for linear problems, and here we extend it to nonlinear problems. In particular, we determine the appropriate *χ*2 tests for Gauss–Newton and Levenberg–Marquardt algorithms, and these tests are used to find a regularization parameter or weights on initial parameter estimate errors. This algorithm is applied to a two-dimensional cross-well tomography problem and a one-dimensional electromagnetic problem from [R. C. Aster, B. Borchers, and C. Thurber, *Parameter Estimation and Inverse Problems*, Academic Press, New York, 2005].

#### Copyright Statement

First published in *SIAM Journal on Matrix Analysis and Applications *in Vol. 34(3) 2013, published by the Society of Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

#### Publication Information

Mead, J. L. and Hammerquist, C. C.. (2013). "*x*^{2} Tests for the Choice of the Regularization Parameter in Nonlinear Inverse Problems". *SIAM Journal on Matrix Analysis and Applications,** 34*(3), 1213-1230. https://doi.org/10.1137/12088447X