Abstract
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.
Highlights
In many areas of science and engineering one is faced with the problem of having to compute a meaningful approximate solution, defined in an appropriate way, of linearNumerical Algorithms systems of equations of the form Ax ≈ b, (1)where A ∈ Rm×n is a given matrix, x ∈ Rn is the desired solution, and b ∈ Rm is a data vector
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
This paper develops an automatic strategy for determining the regularization parameter for these minimization problems
Summary
In many areas of science and engineering one is faced with the problem of having to compute a meaningful approximate solution, defined in an appropriate way, of linear. To reduce the propagated error in the computed solution, one typically modifies the problem to be solved This modification is commonly referred to as regularization. The regularization term measures the size of the computed solution by the 0-norm This “norm” counts the number of nonzero entries in the vector x in (2). This paper is organized as follows: Section 2 reviews how the GCV parameter for Tikhonov regularization in general form can be computed fairly inexpensively by projection into a Krylov subspace, and in Section 3 we discuss how to determine the GCV parameter for large-scale minimization problems (2).
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