Constraining ill-posed inverse problems often requires regularized optimization. I describe two alternative approaches to regularization. The first approach involves a column operator and an extension of the data space. The second approach constructs a row operator and expands the model space. In large-scale problems, when the optimization is incomplete, the two methods of regularization behave...

This paper considers highly ill-posed surface recovery inverse problems, where the sought surface in 2D or 3D is piecewise constant with several possible level values. These levels may further be potentially unknown. Multiple level set functions are used when there are more than two such levels, and we extend the methods and theory of our previous works to handle such more complex situations. A...

We consider the approximate solution of linear ill-posed inverse problems of high dimension with a simulation-based algorithm that approximates the solution within a low-dimensional subspace. The algorithm uses Tikhonov regularization, regression, and low-dimensional linear algebra calculations and storage. For sampling efficiency, we use variance reduction/importance sampling schemes, speciall...

In this paper, we introduce the L-hypersurface method for use in linear inverse problems. The new method is intended to select multiple regularization parameters simultaneously. It is a multidimensional extension of classical L-curve method and hence does not require any speciic knowledge about the noise level or signal semi-norm. We give examples of the L-hypersurface method applied to the lin...

Using the regularization theory for improperly posed problems, we discuss object restoration beyond the diffraction limit in the presence of noise. Only the case of one-dimensional coherent objects is considered. We focus attention n the estimation of the error on the restored objects, and we show that, in most realistic cases, it is at best proportional to an inverse power of |In epsilon|, whe...

This paper considers highly ill-posed surface recovery inverse problems, where the forward problem involves inverting elliptic PDEs, and where the sought surface in 2D or 3D is piecewise constant with several possible level values. These levels may further be potentially unknown. Multiple level set functions are used when there are more than two such levels, and we extend the methods and theory...

For the approximate solution of ill-posed inverse problems, the formulation of a regularization functional involves two separate decisions: the choice of the residual minimizer and the choice of the regularizor. In this paper, the Kullback–Leibler functional is used for both. The resulting regularization method can solve problems for which the operator and the observational data are positive al...