Variable-smoothing Regularization Methods for Inverse Problems

Many inverse problems of practical interest are ill-posed in the sense that solutions do not depend continuously on data. To eeectively solve such problems, regularization methods are typically used. One problem associated with classical regularization methods is that the solution may be oversmoothed in the process. We present an alternative \local regularization" approach in which a decomposition of the problem into \local" and \global" parts permits varying amounts of local smoothing to be applied over the domain of the solution. This allows for more regularization in regions where the solution is likely to be more smooth, and less regularization in regions where sharp features are likely to be present. We illustrate this point with several numerical examples. We consider here the inverse problem of nding u 2 L 2 (0; 1) solving Au(t) = f(t); a:a: t 2 (0; 1); (1) where A is a Volterra operator given by Au(t) = Z t 0 k(t ?)u() dd; t 2 (0; 1); (2) and where the kernel k is assumed to be uniformly HH older continuous on the interval 0; 1], k(t) > 0 for t 2 (0; 1]. We also assume that f is HH older continuous on 0; 1] and is such that u uniquely solves problem (1). It is well-known that (1) is an ill-posed problem due to lack of continuous dependence of the solution u on data f 2 L 2 (0; 1). Thus, in the usual case where only a measured or computed approximation f to f is available , with kf ? fk < , some kind of regularization or stabilization method is required in order to obtain a reasonable approximation u to u. In the above, kk denotes the usual L 2 (0; 1) norm. Classical Tikhonov regularization is based on nding u solving the minimization problem min u2dom L kAu ? f k 2 + kLuk 2 (3) where L is a densely deened closed operator on L 2 (0; 1) and > 0 is known as the Tikhonov regularization paper. Standard regularization theory guarantees that a choice of = () may be made such that () ! 0 and u () ! u as ! 0. One problem associated with classical regularization methods such as the Tikhonov method above is that the regularized solution u is usually oversmoothed in the process. This is due to the fact that smoothing occurs globally (via …