Numerical methods for the design of large-scale nonlinear discrete ill-posed inverse problems

Abstract

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 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.

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

“…In [15,23,43], for instance, the authors consider bilevel optimisation for finite-dimensional Markov random field models. In inverse problems, the optimal inversion and experimental acquisition setup is discussed in the context of optimal model design in works by Haber, Horesh and Tenorio [25,26], as well as Ghattas et al [3,9]. Recently, parameter learning in the context of functional variational regularisation models (1.1) also entered the image processing community with works by the authors [10,22], Kunisch, Pock and co-workers [14,33], Chung et al [16] and Hintermüller et al [30].…”

Nonlinear ill-posed problem analysis in model-based parameter estimation and experimental design

“…To further improve the precision of parameter estimates, model-based experimental design techniques have been developed with the aim to maximize the information content in experimental data (Franceschini and Macchietto, 2008;Pukelsheim, 1993). Theory and applications for the model-based optimal experimental design (OED) for well-posed problems have been addressed repeatedly in the last decades Macchietto, 2000, 2002;Franceschini and Macchietto, 2008;Körkel et al, 2004;Pukelsheim, 1993), applications to ill-posed problems can also be found (Bardow, 2008;Bitterlich and Knabner, 2003;Haber et al, 2008Haber et al, , 2010Lahmer, 2011;O'Sullivane, 1986). However, ill-posed problems have to be handled carefully as they yield a biased estimator.…”

“…It is important to point out that this analysis in the context of OED is not yet available in literature. Moreover, different regularization techniques for handling nonlinear ill-posed PE problems from singular value analysis point of view are discussed, namely, orthogonal decomposition based techniques (i.e., SsS and TSVD) (Burth et al, 1999;Golub and Van Loan,1996;Hansen, 1998;López et al, 2013;Marquardt, 1970;Xu, 1998) and the Tikhonov strategy (Bardow, 2008;Haber et al, 2008Haber et al, , 2010Hansen, 1998Hansen, , 2007Johansen, 1997;Tikhonov and Arsenin, 1977). These techniques are then applied to ill-posed OED problems.…”

“…These techniques are then applied to ill-posed OED problems. To avoid the computational demanding bi-level optimization approach addressed by Haber et al, (2010), Horesh et al, (2010), we consider the variance contribution of the biased estimator only (Bard, 1974;Fessler, 1996). The application of these techniques is presented for three literature case studies with increasing complexity i.e., a semi continuous (fed-batch) fermentation reactor (Asprey and Macchietto, 2002), a reconstruction of biochemical networks (Kremling et al, 2004) and a biological system to water treatment (Cruz et al,2012; Kaelin et al, 2009).…”

Optimal design of simultaneous source encoding

“…Other than recent numerous exceptions (see [8,5,12,15,9,14] and references therein), optimal experimental design of ill-posed problems has been somewhat an under-researched topic. In the case of ill-posed problems, the selection of optimal weights for (8) is more difficult because this estimate is biased and its bias depends on the unknown m: bias( m) = −λ 2 C(w) −1 M m, where the inverse Fisher matrix is given by…”