Haber, E.; Horesh, L.; Tenorio, L. Numerical methods for the design of large-scale nonlinear discrete ill-posed inverse problems. (English) Zbl 1189.65073 Inverse Probl. 26, No. 2, Article ID 025002, 14 p. (2010). Summary: Design of experiments for discrete ill-posed problems is a relatively new area of research. While there has been some limited work concerning the linear case, little has been done to study design criteria and numerical methods for ill-posed nonlinear problems. We present an algorithmic framework for nonlinear experimental design with an efficient numerical implementation. The data are modeled as indirect, noisy observations of the model collected via a set of plausible experiments. An inversion estimate based on these data is obtained by a weighted Tikhonov regularization whose weights control the contribution of the different experiments to the data misfit term. These weights are selected by minimization of an empirical estimate of the Bayes risk that is penalized to promote sparsity. This formulation entails a bilevel optimization problem that is solved using a simple descent method. We demonstrate the viability of our design with a problem in electromagnetic imaging based on direct current resistivity and magnetotelluric data. Cited in 32 Documents MSC: 65F22 Ill-posedness and regularization problems in numerical linear algebra 65H10 Numerical computation of solutions to systems of equations 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 65C60 Computational problems in statistics (MSC2010) 35R30 Inverse problems for PDEs 62K05 Optimal statistical designs Keywords:inverse problems; undetermined coefficients; ill-posedness; numerical examples; nonlinear experimental design; Tikhonov regularization; electromagnetic imaging Software:TRON PDFBibTeX XMLCite \textit{E. Haber} et al., Inverse Probl. 26, No. 2, Article ID 025002, 14 p. (2010; Zbl 1189.65073) Full Text: DOI Link