On the Use of Arnoldi and Golub-Kahan Bases to Solve Nonsymmetric Ill-Posed Inverse Problems

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Virginia Tech

Iterative Krylov subspace methods have proven to be efficient tools for solving linear systems of equations. In the context of ill-posed inverse problems, they tend to exhibit semiconvergence behavior making it difficult detect inverted noise" and stop iterations before solutions become contaminated. Regularization methods such as spectral filtering methods use the singular value decomposition (SVD) and are effective at filtering inverted noise from solutions, but are computationally prohibitive on large problems. Hybrid methods apply regularization techniques to the smaller projected problem" that is inherent to iterative Krylov methods at each iteration, thereby overcoming the semiconvergence behavior.

Commonly, the Golub-Kahan bidiagonalization is used to construct a set of orthonormal basis vectors that span the Krylov subspaces from which solutions will be chosen, but seeking a solution in the orthonormal basis generated by the Arnoldi process (which is fundamental to the popular iterative method GMRES) has been of renewed interest recently. We discuss some of the positive and negative aspects of each process and use example problems to examine some qualities of the bases they produce. Computing optimal solutions in a given basis gives some insight into the performance of the corresponding iterative methods and how hybrid methods can contribute.

Ill-posed inverse problems, Krylov subspace, Arnoldi process, Golub-Kahan bidiagonalization