View source for Minimal discrepancy method

''minimal residual method''

An iteration method for the solution of an operator 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/m/m063/m063810/m0638101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

with a self-adjoint positive-definite bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m0638102.png" /> acting in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m0638103.png" />, and with a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m0638104.png" />. The formula for the minimal discrepancy method takes the form

<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/m/m063/m063810/m0638105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the parameter

<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/m/m063/m063810/m0638106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

is chosen in each step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m0638107.png" /> from the condition for maximal minimization of the norm of the discrepancy (or residual vector) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m0638108.png" />; that is, it is required that

<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/m/m063/m063810/m0638109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

If the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381010.png" /> belongs to an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381011.png" /> on the real line, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381012.png" /> are positive numbers, then the sequence of approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381013.png" /> in the method (2)–(3) converges to a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381014.png" /> of (1) with the speed of a geometric progression with multiplier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381015.png" />.

Different ways of defining the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381016.png" /> lead to different iteration methods. In particular, for special scalar products the formulas for the minimal discrepancy method coincide with the formulas for the method of steepest descent (cf. [[Steepest descent, method of|Steepest descent, method of]]) and the method of minimal errors (see [[#References|[2]]]).

The condition for the convergence of the minimal discrepancy method may be weakened in comparison with that given above if it is considered on certain subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381017.png" />. For example, if the minimal discrepancy method is considered only in real spaces, then it is possible to drop the requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063810/m06381018.png" /> be self-adjoint (see [[#References|[3]]]–[[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Krasnosel'skii,   S.G. Krein,   "An iteration process with minimal residuals"  ''Mat. Sb.'' , '''31'''  (1952)  pp. 315–334  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Samarskii,   "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.I. Marchuk,   Yu.A. Kuznetsov,   "Iterative methods and quadratic functionals" , Novosibirsk  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.I. Marchuk,   Yu.A. Kuznetsov,   "Sur les méthodes numériques en sciences physique et economique" , Dunod  (1974)  (Translated from Russian)</TD></TR></table>