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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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.K. Ivanov,   "On ill-posed problems"  ''Mat. Sb.'' , '''61''' :  2  (1962)  pp. 211–223  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. Tikhonov,   V.I. [V.I. Arsenin] Arsenine,   "Solution of ill-posed problems" , Wiley  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Krylov,   V.V. Bobkov,   P.I. Monastyrnyi,   "Computing methods of higher mathematics" , '''1–2''' , Minsk  (1972–1975)  (In Russian)</TD></TR></table>

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