View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767001.png" /> be metric spaces, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767002.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767003.png" /> be a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767004.png" />. A quasi-solution of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767005.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767006.png" /> for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767007.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767008.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q0767009.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670010.png" /> that minimizes the residual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670013.png" />. If equation (1) has a proper solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670016.png" /> is also quasi-solution.

The dependence of the set of quasi-solutions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670017.png" /> is conveniently represented as a superposition of two mappings:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670019.png" /> is the (generally multi-valued) inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670021.png" /> is the metric projection operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670022.png" /> onto the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670023.png" />. This superposition enables one to reduce the study of the properties of quasi-solutions to that of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670025.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670026.png" /> is a [[Chebyshev set|Chebyshev set]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670027.png" /> is single-valued and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670028.png" />, then the problem of finding a quasi-solution is well-posed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670029.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670030.png" /> is many-valued, then stability of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670031.png" /> can be formulated in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670033.png" />-continuity (continuity of set-valued functions).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670035.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670036.png" /> is strictly convex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670037.png" /> is a continuous invertible linear operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670038.png" /> 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670039.png" /> is a closed linear operator but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670040.png" /> is a Hilbert space).

There exists a number of methods for determining sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670041.png" /> that guarantee an effective determination of quasi-solutions. One of the most widespread methods consists in the following: One considers a third space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670042.png" /> (all or some of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670043.png" /> may coincide) and a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670044.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670045.png" /> is unbounded. One takes for the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670046.png" /> the image of a ball

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670047.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670048.png" /></td> </tr></table>

subject to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670049.png" />. For Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670051.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670053.png" /> is given. If the method for determining the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670054.png" /> is as described above, then stability of a quasi-solution is characterized by the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670055.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670056.png" /></td> </tr></table>

The following relation holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670057.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670058.png" /> is the solution of the extremal problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670059.png" />.

For Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670061.png" /> there are expressions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670063.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670064.png" /> is a differential operator. On a suitable space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670065.png" /> is then a compact operator, so that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076700/q07670066.png" /> 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>