large-scale inversion of magnetic data using golub-kahan bidiagonalization with truncated generalized cross validation for regularization parameter estimation

In this paper a fast method for largescale sparse inversion of magnetic data is considered. the l1norm stabilizer is used to generate models with sharp and distinct interfaces. to deal with the nonlinearity introduced by the l1norm, a modelspace iteratively reweighted least squares algorithm is used. the original model matrix is factorized using the golubkahan bidiagonalization that projects the problem onto a krylov subspace with a significantly reduced dimension. the model matrix of the projected system inherits the illconditioning of the original matrix, but the spectrum of the projected system accurately captures only a portion of the full spectrum. equipped with the singular value decomposition of the projected system matrix, the solution of the projected problem is expressed using a filtered singular value expansion. this expansion depends on a regularization parameter which is determined using the method of generalized cross validation (gcv), but here it is used for the truncated spectrum. this new technique, truncated gcv (tgcv), is more effective compared with the standard gcv method. numerical results using a synthetic example and real data demonstrate the efficiency of the presented algorithm.

Largescale Inversion of Magnetic Data Using GolubKahan Bidiagonalization with Truncated Generalized Cross Validation for Regularization Parameter Estimation

In this paper a fast method for largescale sparse inversion of magnetic data is considered. The L1norm stabilizer is used to generate models with sharp and distinct interfaces. To deal with the nonlinearity introduced by the L1norm, a modelspace iteratively reweighted least squares algorithm is used. The original model matrix is factorized using the GolubKahan bidiagonalization that projects the problem onto a Krylov subspace with a significantly reduced dimension. The model matrix of the projected system inherits the illconditioning of the original matrix, but the spectrum of the projected system accurately captures only a portion of the full spectrum. Equipped with the singular value decomposition of the projected system matrix, the solution of the projected problem is expressed using a filtered singular value expansion. This expansion depends on a regularization parameter which is determined using the method of Generalized Cross Validation (GCV), but here it is used for the truncated spectrum. This new technique, Truncated GCV (TGCV), is more effective compared with the standard GCV method. Numerical results using a synthetic example and real data demonstrate the efficiency of the presented algorithm.