Generalized cross validation for ℓ p-ℓ q minimization

Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p-norm, and a regularization term, that is defined in terms of a q-norm. We allow 0 < p,q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed

On the block Lanczos and block Golub–Kahan reduction methods applied to discrete ill‐posed problems

The reduction of a large-scale symmetric linear discrete ill-posed problem with multiple right-hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill-posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill-posed problems with multiple right-hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution