# 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)

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.