Abstract
A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f TV * . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f TV * as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach.
Highlights
In many physical problems, the relation between the quantity observed and the quantity to be measured can be formulated as a Fredholm integral equation of the first kind:Z 1 k( x, t) f (t)dt = g( x ), 0 ≤ x ≤ 1, (1)where the kernel function k and the right-hand side g are known, while f is the unknown to be determined
The relation between the quantity observed and the quantity to be measured can be formulated as a Fredholm integral equation of the first kind: Z 1 k( x, t) f (t)dt = g( x ), 0 ≤ x ≤ 1, (1)
We show the numerical results carried out by the damped Newton (DN) method for minimization of the function corresponding to the total variation (TV) functional and those carried out by the modified Gauss–Newton (MGN) method for minimization of the function corresponding to the weighted total variation (WTV) functional
Summary
The relation between the quantity observed and the quantity to be measured can be formulated as a Fredholm integral equation of the first kind:. The Fredholm integral equation of the first kind is ill-posed; see for instance [1], Chapter 2. Numerical methods for obtaining a reasonable approximate solution to the Fredholm integral equation of the first kind have attracted many researchers, and many research results have been achieved; see, for instance, ([2], Chapter 12), and [3,4,5,6,7]. (MTSVD) method [11], the Chebyshev interpolation method [12], the collocation method [13,14], the projected Tikhonov regularization method [15], and so on, are applied to obtain approximate continuous solutions of Equation (2). We focus on the case where the solution of Equation (1) is piecewise-constant and the possible function values are known, that is: m.
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