Cauchy-like Preconditioners for 2-D Ill-posed Problems

Mrs. Misha E. Kilmer

Applied Mathematics Program, University of Maryland, College Park

Ill-conditioned matrices with block Toeplitz, Toeplitz block (BTTB) structure arise from the discretization of certain ill-posed problems in signal and image processing. We use a preconditioned conjugate gradient algorithm to compute a regularized solution to this linear system given noisy data. Our preconditioner is a row rank, Cauchy-like block diagonal approximation to the unitarily transformed BTTB matrix, augmented by an identity matrix. We show our preconditioner has desirable properties and demonstrate its effectiveness on some examples.