The active-set method for nonnegative regularization of linear ill-posed problems

Abstract

In this work, we analyze the behavior of the active-set method for the nonnegative regularization of discrete ill-posed problems. In many applications, the solution of a linear ill-posed problem is known to be nonnegative. Standard Tikhonov regularization often provides an approximated solution with negative entries. We apply the active-set method to find a nonnegative approximate solution of the linear system starting from the Tikhonov regularized one. Our numerical experiments show that the active-set method is effective in reducing the oscillations in the Tikhonov regularized solution and in providing a nonnegative regularized solution of the original linear system.

Introduction

Many applications give raise to ill-posed problems whenever the underlying physical or technical problem is modelled by an integral equation of the first kind with a smooth kernel. The discretization of such continuous inverse problems leads to linear systems of the form

$$\mathit{Ax}=b\text{,}$$

where $b=\hat{b}+e$ is obtained by adding a white noise error vector e to the noise free right-hand side vector $\hat{b}$. The vector x represents the true solution and the matrix A models the data acquisition process and is defined by the particular application. Since the continuous inverse problem is ill-posed, then the matrix A is ill-conditioned and regularization methods are required. In this paper we apply the Tikhonov regularization method [1] solving the following minimization problem:

$$\underset{x}{\min }\frac{1}{2}||b-\mathit{Ax}|{|}^2+\frac{\gamma }{2}||x|{|}^2\text{,}\gamma >0$$

for a suitable value of the regularization parameter γ.

In many problems the true solution is known to have only nonnegative entries; then, in order to take into account the prior knowledge on the solution, the following constrained optimization problem is solved:

$$\begin{array}{cc}\hfill & \mathrm{minimize}\frac{1}{2}||\mathit{Ax}-b|{|}^2+\frac{\gamma }{2}||x|{|}^2\text{,}\hfill \\ \hfill & \mathrm{subject}\mathrm{to}x⩾0\text{.}\hfill \end{array}$$

To solve quadratic bound constrained problems, such as (3), a variety of minimization methods can be found in the literature (see [6] and references therein for a survey). The application of such methods to ill-posed problems can be found in the area of image restoration and deblurring. In [7] the quasi-Newton method is applied to a parametrization of the problem that incorporates the positivity constraints. Projected Newton methods [5], [9], including gradient projection method [4], are applied in [3] for image restoration. An active-set method, based on a quasi-Newton conjugate gradient preconditioner, is used in astronomical image reconstruction [2].

In this paper we present an efficient implementation of the active-set method coupled with the truncated conjugate gradient iterations. The unconstrained Tikhonov regularized solution is used as a starting iterate for a two levels iterative scheme where in the outer iterations the active-set method is used to impose the nonnegativity constraint, while in the inner iterations the search direction is determined by truncated conjugate gradient method.

The paper is organized as follows. In Section 2 we introduce our implementation of the active-set method; in Section 3 the numerical results are presented for both one dimensional and two dimensional test problems. Finally, in Section 4 conclusions are reported.