Solution of linear ill-posed operator equations by modified truncated singular value expansion

In much of the literature on the solution of linear ill-posed operator equations in a Hilbert space, the operator equation first is discretized, then the discretized operator is regularized, and finally, the computed solution of the regularized discrete problem is projected into a Hilbert space. In order for this solution approach to give an accurate approximate solution, the regularization method has to correspond to a meaningful analogue in Hilbert space. Moreover, the regularization method chosen may only be applicable to certain linear ill-posed operator equations. However, these issues typically are not discussed in the literature on solution methods based on discretization. One approach to circumvent this difficulty is to avoid discretization. This paper describes how regularization by a modified truncated singular value decomposition introduced in [S. Noschese and L. Reichel, A modified TSVD method for discrete ill‐posed problems, Numerical Linear Algebra with Applications, 21 (2014), pp. 813-822] for finite-dimensional problems can be extended to operator equations. In finite dimensions, this regularization method yields approximate solutions of higher quality than standard truncated singular value decomposition. Our analysis in a Hilbert space setting is of practical interest, because the solution method presented avoids the introduction of discretization errors during the solution process, since we compute regularized solutions without discretization by using the program package Chebfun. While this paper focuses on a particular regularization method, the analysis presented and Chebfun also can be applied to other regularization techniques.