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Difference between revisions of "Discrepancy of an approximation"

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One of the characteristics of the quality of an approximate solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033040/d0330401.png" /> of an operator equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033040/d0330402.png" /> (e.g. a linear algebraic system, a differential equation). The discrepancy is defined as the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033040/d0330403.png" /> or a norm of this quantity, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033040/d0330404.png" />. If the estimate
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One of the characteristics of the quality of an approximate solution $\overline u$ of an operator equation $P(u)=0$ (e.g. a linear algebraic system, a differential equation). The discrepancy is defined as the quantity $P(\overline u)$ or a norm of this quantity, e.g., $\|P(\overline u)\|_2$. If the estimate
  
<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/d/d033/d033040/d0330405.png" /></td> </tr></table>
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$$\|u_1-u_2\|_1\leq C\|P(u_1)-P(u_2)\|_2$$
  
 
holds, then the error of the solution may be estimated in terms of the discrepancy:
 
holds, then the error of the solution may be estimated in terms of the discrepancy:
  
<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/d/d033/d033040/d0330406.png" /></td> </tr></table>
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$$\|\overline u-u\|_1\leq C\|P(\overline u)\|_2.$$
  
 
If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution.
 
If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution.

Latest revision as of 08:54, 25 November 2018

One of the characteristics of the quality of an approximate solution $\overline u$ of an operator equation $P(u)=0$ (e.g. a linear algebraic system, a differential equation). The discrepancy is defined as the quantity $P(\overline u)$ or a norm of this quantity, e.g., $\|P(\overline u)\|_2$. If the estimate

$$\|u_1-u_2\|_1\leq C\|P(u_1)-P(u_2)\|_2$$

holds, then the error of the solution may be estimated in terms of the discrepancy:

$$\|\overline u-u\|_1\leq C\|P(\overline u)\|_2.$$

If no such estimate is available, the discrepancy provides an indirect indication of the quality of the approximate solution.

References

[1] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[2] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
How to Cite This Entry:
Discrepancy of an approximation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrepancy_of_an_approximation&oldid=14211
This article was adapted from an original article by N.S. Bakhvalov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article