Quasi-solution

A generalized solution of a certain ill-posed problem that (under sufficiently general conditions) satisfies, in contrast to a proper solution, the condition of being well-posed in the sense of Hadamard.

Let $ X , Y $ be metric spaces, let $ A : X \rightarrow Y $ and let $ M $ be a subset of $ X $. A quasi-solution of the equation

$$ \tag{1 } A x = y $$

on the set $ M $ for given $ y $ in $ Y $ is an element $ \overline{x}\; $ in $ M $ that minimizes the residual $ \rho ( A x , y ) $ for $ x $ in $ M $. If equation (1) has a proper solution $ x _ {0} $ on $ M $, then $ x _ {0} $ is also quasi-solution.

The dependence of the set of quasi-solutions on $ y $ is conveniently represented as a superposition of two mappings:

$$ \overline{x}\; = A ^ {- 1} P y , $$

where $ A ^ {- 1} $ is the (generally multi-valued) inverse of $ A $ and $ P $ is the metric projection operator in $ Y $ onto the set $ N = A M $. This superposition enables one to reduce the study of the properties of quasi-solutions to that of the mappings $ A ^ {- 1} $ and $ P $. For example, if $ N $ is a Chebyshev set and $ A ^ {- 1} $ is single-valued and continuous on $ N $, then the problem of finding a quasi-solution is well-posed. If $ P $ or $ A ^ {- 1} $ is many-valued, then stability of the set $ K $ can be formulated in terms of $ \beta $-continuity (continuity of set-valued functions).

$ X $ and $ Y $ are generally taken to be normed linear spaces, which enables one to obtain more complete and definitive results. Then the problem of finding quasi-solutions is well-posed if $ Y $ is strictly convex, $ A $ is a continuous invertible linear operator and $ M $ is a convex compactum. There are a number of other combinations of conditions that ensure the well-posedness of the problem of finding quasi-solutions, in which some conditions are strengthened while others are weakened (for example, $ A $ is a closed linear operator but $ Y $ is a Hilbert space).

There exists a number of methods for determining sets $ M $ that guarantee an effective determination of quasi-solutions. One of the most widespread methods consists in the following: One considers a third space $ Z $ (all or some of the spaces $ X , Y , Z $ may coincide) and a linear operator $ B : Z \rightarrow X $ for which $ B ^ {- 1} $ is unbounded. One takes for the set $ M = M _ {r} $ the image of a ball

$$ M _ {r} = B S _ {r} ; \ \ S _ {r} = \{ {z \in Z } : {\| z \| \leq r } \} . $$

In this form the problem of finding a quasi-solution is a mathematical programming problem: To minimize the functional

$$ f ( z) = \| A B z - y \| $$

subject to $ \| z \| \leq r $. For Hilbert spaces $ Y $ and $ Z $ one obtains a quadratic programming problem.

In the case of well-posedness of quasi-solutions, of great importance in applications are stability estimates, in which the dependence of $ \| \overline{x}\; _ {1} - \overline{x}\; _ {2} \| $ on $ \| y _ {1} - y _ {2} \| $ is given. If the method for determining the set $ M $ is as described above, then stability of a quasi-solution is characterized by the function

$$ \Omega ( \tau , r ) = $$

$$ = \ \sup \{ {\| x _ {1} - x _ {2} \| } : {x _ {i} = B z _ {i} , \| z _ {i} \| \leq r , \| A x _ {1} - A x _ {2} \| \leq \tau , i = 1 , 2 } \} . $$

The following relation holds:

$$ \Omega ( \tau , r ) = \omega ( \tau , 2 r ) , $$

where $ \omega ( \tau , r ) $ is the solution of the extremal problem $ \omega ( \tau , r ) = \sup \{ {\| B z \| } : {\| z \| \leq r, \| A B z \| \leq \tau } \} $.

For Hilbert spaces $ Z $ and $ Y $ there are expressions for $ \omega ( \tau , r ) $ in closed form.

References

[1]	V.K. Ivanov, "On linear problems which are not well-posed" Soviet Math. Dokl. , 4 : 3 (1962) pp. 981–983 Dokl. Akad. Nauk SSSR , 145 : 2 (1962) pp. 270–272
[2]	V.K. Ivanov, "On ill-posed problems" Mat. Sb. , 61 : 2 (1962) pp. 211–223 (In Russian)
[3]	O.A. Liskovets, "Stability of quasi-solutions of equations with a closed operator" Diff. Eq. , 7 : 9 (1971) pp. 1300–1303 Differentsial. Uravn. , 7 : 9 (1971) pp. 1707–1709
[4]	V.A. Morozov, "Linear and nonlinear ill-posed problems" J. Soviet Math. , 4 : 6 (1975) pp. 706–755 Itogi Nauk. i Tekhn. Mat. Anal. , 11 (1973) pp. 129–178
[5]	A.N. Tikhonov, V.I. [V.I. Arsenin] Arsenine, "Solution of ill-posed problems" , Wiley (1977) (Translated from Russian)
[6]	V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Computing methods of higher mathematics" , 1–2 , Minsk (1972–1975) (In Russian)

Comments

Frequently the operator $ B $ is such that $ B ^ {- 1} $ is a differential operator. On a suitable space, $ B $ is then a compact operator, so that the set $ M $ is a compactum.

References

[a1]	B. Hofmann, "Regularization for applied inverse and ill-posed problems" , Teubner (1986)
[a2]	C.W. Groetsch, "The theory of Tikhonov regularization for Fredholm equations of the first kind" , Pitman (1984)
[a3]	J. Baumeister, "Stable solution of inverse problems" , Vieweg (1987)
[a4]	M.Z. Nashed (ed.) , Genealized inverses and applications , Acad. Press (1976)
[a5]	V.A. Morozov, "Methods for solving incorrectly posed problems" , Springer (1984) (Translated from Russian)