On the Equivalence of Singular and Ill-Posed Problems: The p-Factor Regularization Method

The local equivalence of singular and ill-posed problems in a class of sufficiently smooth mappings is shown, which justifies the use of the p-factor regularization method to solve them. The main constructions in p-regularity theory that are necessary for stable solution of approximate problems are described, and estimation theorems for regularizing algorithms are proved.

We consider the problem (equation) (1) where , X and Y are Banach spaces, F is a sufficiently smooth mapping, and = 0} is the solution set of Eq. (1). Along with (1), we consider the approximate equation (2) where as , that is, is an approximation of the mapping F. Let be a neighborhood of a point x*, and let . Definition 1. Problem (1) is said to be ill-posed at a point x* if there are mappings such that, for sufficiently small (3) (the so-called approximation condition) and, for the sets X(ε) = of solutions of (2), there is a sequence , as such that either or Problem (1) is called ideal, while problem (2) is called ε-perturbed (or ε-approximate). Similarly, the mapping F is called ideal, while the mapping is called -perturbed (or ε-approximate). Definition 2. Problem (1) is said to be singular at a point x* if is singular, that is, . Along with Definition 1, we introduce the concept of weakly ill-posed problem (1). Definition 3. Problem (1) is said to be weakly illposed at a point x* if there is a mapping such that (4) and the problem is ill-posed. Evidently, Definition 3 covers a much wider class of mappings than Definition 1. Note that Definition 1 actually coincides with the classical definition of illposedness (see, for example, [7]), except that the smoothness condition for mappings and the approximation condition for the derivatives and are added in the former case. The class of problems considered in this paper is also much wider that the classical class of conditionally well-posed problems [7].
In real situations, we often do not know F, but know its approximation ; thus, problems of this type are called inverse. However, the ideal mapping F is suggested to exist (otherwise, we cannot speak about the solvability of the problem). The relationship between approximate and singular problems in a class of sufficiently smooth mappings is established by the following theorem.

Theorem 1. Suppose that and . Then problem (1) is weakly ill-posed at x* if and only if problem (1) is singular at x*.
Proof. Assume that problem (1) is ill-posed. We show that the mapping is singular at x*. In fact, when is nonsingular, the implicit function theorem [1] applied to the mappings yields the existence of such that and → 0 as uniformly with respect to all satisfying approximation condition (3). In turn, this means the well-posedness of (1).
Assume now that is singular. Then there exists such that , . We show that problem (1) is weakly ill-posed. We consider the mappings and + εξ. Obviously, ; however, Eq. (2) ( = 0) has no solution for , since and . That is, the problem is ill-posed, which means the weak ill-posedness of the original problem (1).

Corollary 1. Let be a linear mapping, . Then problem (1) is ill-posed at if and only if problem (1) is singular at x*.
If F is ill-posed at x*, then is singular, and the linearity is needed only for the reverse implication.
So, when approximate problems are solved, it is reasonable to use methods adapted to singular problems. Methods of this type and their basic constructions are described in p-regularity theory (see, for example, [2][3][4]).

ELEMENTS OF
where and . Let Z 2 denote the closed complement of Y 1 in Y (provided that it exists) and let be the projection operator onto Z 2 parallel to Y 1 . Then Y 2 is the closure of the span for the image of the quadratic form . Furthermore, by induction, where Z i is the closed complement of in Y and is the projection operator onto Z i parallel to , . Finally, . The order p is chosen to be the minimum number (provided that it exists) such that representation (5) holds.
We introduce the mappings (6) where is the projection operator onto parallel to . Then the mapping F is representable as or in the vector form  (7) is called the p-factor operator specified by the element h, or merely the p-factor operator when this is clear from the context.
We introduce a nonlinear operator by the formula Note that .

p-FACTOR METHOD FOR SOLVING SINGULAR NONLINEAR EQUATIONS
We consider Eq. (1) with and in the case when is singular at a solution x*. Then the principal scheme of the p-factor method is as follows: where , , and h, , is chosen in such a way that the p-factor operator + is nonsingular, which means that the mapping F is p-regular at x* for h.
For the p-factor method (8), the following theorem is true.

Theorem 2. Suppose that and there exist an element h,
, such that the p-factor operator is nonsingular. Then for (ε > 0 is sufficiently small), the sequence defined by (8) converges to the solution x* and the rate of convergence is estimated as follows: where is an independent constant.
3. p-FACTOR REGULARIZATION METHOD (p-FACTOR REGULARIZATION) Let x* be a solution of the ideal problem (equation) (1). The regularity (nonsingularity) of F at x* is not assumed. Simultaneously, we consider the approximate equation (10) , where ε > 0 is sufficiently small and . We assume that . The main idea of the p-factor regularization lies in the following. We need to replace (1) by another equation which is guaranteed to have x* as its solution and is nonsingular at x*. We are going to implement this idea based on the p-regularity construction. The following theorem is the main result.
Theorem 3. Let and be Banach spaces, F, , , and be p-regular for , . Assume that , where is sufficiently small and is a neighborhood of x*.
Then the mapping is p-factor regularizing for approximate equation (10), that is, and (12) Here, denotes the preimage of the point y = 0 for the mapping , that is, = 0}.

Remark 1.
If the p-factor operator F'(x*) + is bijective, then there exists a locally unique solution of (10): in a neighborhood , which we denote by . Then the main result is as follows: ( 13) and (14) For example, the bijection condition is certainly satisfied if X and Y are finite-dimensional Euclidean spaces of the same dimension.
Proof. We introduce the notation . Applying the classical implicit function theorem [1] to at the point and , we deduce that the operator is nonsingular at this point . Therefore, there is a function that is a local solution of the equation = 0 and estimates (12) and (14) hold, since for an independent constant C > 0.
The exact operators P k , , are used in constructing the mapping intentionally, since this aspect is not the main one in the ideology of the proposed approach, but rather a technical one in the practical implementation of the method. It is a fortiori so, since there already are methods for constructing the operators P k , , based on approximate information (see, for example, [5,6]). In the general case, constructing the operators P k or their approximations with needed properties requires using the specifics of the problem under consideration (see [2]). .