Solving large linear inverse problems by projection

As originally formulated by Backus & Gilbert (1970), ill-posed linear inverse problems possess a unique minimum norm solution, and a locally averaged property of the model may be estimated with a resolution that is a monotonic function of its variance. Application of Backus–Gilbert theory requires the inversion of an N × N matrix, where N is the number of data, and therefore becomes cumbersome for large N. In this paper we show how Lanczos iteration may be used to project the original linear problem on a problem of much smaller size in order to obtain an approximation to the Backus–Gilbert solution without the need of matrix inversion. To calculate the resolution in the projected system one only needs to invert a symmetric tridiagonal matrix.