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Solving large linear inverse problems by projection

SUMMARY 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 X 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.