Maximum entropy regularization for Fredholm integral equations of the first kind. (English) Zbl 0791.65099
In theory and practice of linear ill-posed problems the approximate solution of linear Fredholm integral equations plays an important role. The paper considers the equation \(Kf(x)=g(x)\), \(x \in \Sigma\), with \(g \in L^ 2 (\Sigma)\) given, possibly subject to error. Variations of the Phillips-Tikhonov regularization
\[
\text{minimize } \| Kf-g \|^ 2_{L^ 2(\Sigma)}+\alpha^ 2D(f,\varphi) \text{ subject to } f \geq 0,
\]
are studied where \(D(f,\varphi)\) is the cross-entropy functional
\[
D(f,\varphi)=\int_ \Omega \left\{ f(y) \log {f(y) \over \varphi (y)}+\varphi (y)-f(y) \right\} d \mu(y).
\]
A series of analytic properties of this approach, which is motivated also from a stochastic point of view, are presented in this paper. A very helpful overview of literature concerning different versions of this so-called maximum entropy regularization method is given. Further paragraphs of the paper deal with finite-dimensional approximations of the method and some results associated with moment problems.
The paper is a valuable contribution to the theory and numerical analysis of linear ill-posed problems.
The paper is a valuable contribution to the theory and numerical analysis of linear ill-posed problems.
Reviewer: B.Hofmann (Chemnitz)
MSC:
65R20 | Numerical methods for integral equations |
65R30 | Numerical methods for ill-posed problems for integral equations |
45B05 | Fredholm integral equations |