Regularization matrices determined by matrix nearness problems
For practitioners solving ill-posed inverse problems, this provides a principled way to design regularization matrices that enforce known solution features, though the improvement over existing methods is not quantified.
The paper addresses the problem of choosing regularization matrices in Tikhonov regularization for large-scale ill-posed problems. It proposes a novel method to determine such matrices by solving a matrix nearness problem, yielding matrices with desired properties like smoothness and monotonicity, and demonstrates effectiveness through numerical examples.
This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained.