Χ Tests for the Choice of the Regularization Parameter in Nonlinear Inverse Problems∗

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. 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 χ2 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 χ2 method has been for linear problems, and here we extend it to nonlinear problems. In particular, we determine the appropriate χ2 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].