Open AccessSolving large linear inverse problems by projection
Author(s) -
Nolet Guust,
Snieder Roel
Publication year - 1990
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.1990.tb01792.x
Subject(s) - lanczos resampling , tridiagonal matrix , mathematics , monotonic function , inverse problem , inverse theory , inversion (geology) , inverse , linear system , mathematical analysis , computer science , eigenvalues and eigenvectors , geometry , geology , physics , telecommunications , paleontology , quantum mechanics , structural basin , surface wave
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.