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A method in which the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a0109101.png" /> of a stationary problem
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<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/a/a010/a010910/a0109102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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is regarded as the steady-state limit solution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a0109103.png" /> of a Cauchy initial value problem for a non-stationary evolution equation involving the same operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a0109104.png" /> (cf. [[Cauchy problem|Cauchy problem]]). This evolution equation may e.g. be of the form
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A method in which the solution $  u $
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of a stationary problem
  
<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/a/a010/a010910/a0109105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
 +
A u  = f
 +
$$
  
<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/a/a010/a010910/a0109106.png" /></td> </tr></table>
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is regarded as the steady-state limit solution for  $  t \rightarrow \infty $
 +
of a Cauchy initial value problem for a non-stationary evolution equation involving the same operator  $  A $(
 +
cf. [[Cauchy problem|Cauchy problem]]). This evolution equation may e.g. be of the form
  
Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a0109107.png" /> are suitable operators which guarantee the existence of the "adjustment limit" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a0109108.png" />.
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$$ \tag{2 }
 +
\sum _ { i=1 } ^ { m }
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C _ {i}
 +
\frac{d ^ {i} u (t) }{d t ^ {i} }
 +
  = \
 +
f - A u (t) ,
 +
$$
  
A result of using adjustment is that it permits one to use approximate solution methods of (2) in order to construct iteration algorithms for solving equation (1) (cf. [[Iteration algorithm|Iteration algorithm]]). Thus, for the non-stationary equation (2) one could employ a discretization (differencing) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a0109109.png" /> solution method which is convergent and stable to obtain approximate solutions. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091010.png" />, an explicit method of the form
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$$
 +
\left .  
 +
\frac{d  ^ {k} u }{d t  ^ {k} }
 +
\right | _ {t=0= u _ {0k} ,\  k = 0 \dots m - 1 .
 +
$$
  
<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/a/a010/a010910/a01091011.png" /></td> </tr></table>
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Here the  $  C _ {i} $
 +
are suitable operators which guarantee the existence of the  "adjustment limit"   $  \lim\limits _ {t \rightarrow \infty }  u (t) = u $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091012.png" />. And then this method can be interpreted as an iteration algorithm
+
A result of using adjustment is that it permits one to use approximate solution methods of (2) in order to construct iteration algorithms for solving equation (1) (cf. [[Iteration algorithm|Iteration algorithm]]). Thus, for the non-stationary equation (2) one could employ a discretization (differencing) with respect to  $  t $
 +
solution method which is convergent and stable to obtain approximate solutions. For example, for  $  m = 1 $,
 +
an explicit method of 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/a/a010/a010910/a01091013.png" /></td> </tr></table>
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$$
 +
C _ {1}
 +
\frac{u ( t _ {n+1} ) - u ( t _ {n} ) }{\tau _ {n} }
 +
  = \
 +
f - A u ( t _ {n} )
 +
$$
  
for solving equation (1), in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091015.png" /> are now seen as characterizing this (iteration) method.
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where  $  \tau _ {n} = t _ {n+3} - t _ {n} > 0 $.  
 +
And then this method can be interpreted as an iteration algorithm
  
Varying the form of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091016.png" /> and considering different discretizations with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091017.png" /> in equation (2) (explicit schemes, implicit schemes, splitting schemes, etc.) gives the possibility of obtaining a wide variety of iteration methods for solving equation (1). For these methods equation (2) will be the closure of the computational algorithm (cf. [[Closure of a computational algorithm|Closure of a computational algorithm]]). A generalization of the adjustment method is the [[Continuation method (to a parametrized family)|continuation method (to a parametrized family)]].
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$$
 +
C _ {1} ( u  ^ {n+1} - u  ^ {n} )  = \
 +
\tau _ {n} ( f - A u  ^ {n} ) ,\ \
 +
n = 0 , 1 \dots \ \
 +
u  ^ {0}  =  u _ {00} ,
 +
$$
 +
 
 +
for solving equation (1), in which  $  C _ {1} $
 +
and  $  \tau _ {n} $
 +
are now seen as characterizing this (iteration) method.
 +
 
 +
Varying the form of the operators $  C _ {i} $
 +
and considering different discretizations with respect to $  t $
 +
in equation (2) (explicit schemes, implicit schemes, splitting schemes, etc.) gives the possibility of obtaining a wide variety of iteration methods for solving equation (1). For these methods equation (2) will be the closure of the computational algorithm (cf. [[Closure of a computational algorithm|Closure of a computational algorithm]]). A generalization of the adjustment method is the [[Continuation method (to a parametrized family)|continuation method (to a parametrized family)]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Godunov,  V.S. Ryaben'kii,  "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.I. Marchuk,  V.I. Lebedev,  "Numerical methods in the theory of neutron transport" , Harwood  (1986)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Godunov,  V.S. Ryaben'kii,  "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.I. Marchuk,  V.I. Lebedev,  "Numerical methods in the theory of neutron transport" , Harwood  (1986)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The adjustment method is also called the time-stepping method. Such a time-stepping method with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010910/a01091018.png" /> arises quite naturally when an elliptic boundary value problems is viewed as a steady state of a (dissipative) parabolic problem (cf. [[#References|[a2]]]). Because of the inherent (numerical) stiffness an implicit discretization method, such as BDF, should be advocated.
+
The adjustment method is also called the time-stepping method. Such a time-stepping method with $  m = 1 $
 +
arises quite naturally when an elliptic boundary value problems is viewed as a steady state of a (dissipative) parabolic problem (cf. [[#References|[a2]]]). Because of the inherent (numerical) stiffness an implicit discretization method, such as BDF, should be advocated.
  
 
Similar ideas can be used in optimization theory by e.g. constructing a mechanical system whose stable equilibrium state is the desired optimum. Again this leads to all kinds of iteration algorithms, as in [[#References|[a3]]].
 
Similar ideas can be used in optimization theory by e.g. constructing a mechanical system whose stable equilibrium state is the desired optimum. Again this leads to all kinds of iteration algorithms, as in [[#References|[a3]]].

Latest revision as of 16:09, 1 April 2020


A method in which the solution $ u $ of a stationary problem

$$ \tag{1 } A u = f $$

is regarded as the steady-state limit solution for $ t \rightarrow \infty $ of a Cauchy initial value problem for a non-stationary evolution equation involving the same operator $ A $( cf. Cauchy problem). This evolution equation may e.g. be of the form

$$ \tag{2 } \sum _ { i=1 } ^ { m } C _ {i} \frac{d ^ {i} u (t) }{d t ^ {i} } = \ f - A u (t) , $$

$$ \left . \frac{d ^ {k} u }{d t ^ {k} } \right | _ {t=0} = u _ {0k} ,\ k = 0 \dots m - 1 . $$

Here the $ C _ {i} $ are suitable operators which guarantee the existence of the "adjustment limit" $ \lim\limits _ {t \rightarrow \infty } u (t) = u $.

A result of using adjustment is that it permits one to use approximate solution methods of (2) in order to construct iteration algorithms for solving equation (1) (cf. Iteration algorithm). Thus, for the non-stationary equation (2) one could employ a discretization (differencing) with respect to $ t $ solution method which is convergent and stable to obtain approximate solutions. For example, for $ m = 1 $, an explicit method of the form

$$ C _ {1} \frac{u ( t _ {n+1} ) - u ( t _ {n} ) }{\tau _ {n} } = \ f - A u ( t _ {n} ) $$

where $ \tau _ {n} = t _ {n+3} - t _ {n} > 0 $. And then this method can be interpreted as an iteration algorithm

$$ C _ {1} ( u ^ {n+1} - u ^ {n} ) = \ \tau _ {n} ( f - A u ^ {n} ) ,\ \ n = 0 , 1 \dots \ \ u ^ {0} = u _ {00} , $$

for solving equation (1), in which $ C _ {1} $ and $ \tau _ {n} $ are now seen as characterizing this (iteration) method.

Varying the form of the operators $ C _ {i} $ and considering different discretizations with respect to $ t $ in equation (2) (explicit schemes, implicit schemes, splitting schemes, etc.) gives the possibility of obtaining a wide variety of iteration methods for solving equation (1). For these methods equation (2) will be the closure of the computational algorithm (cf. Closure of a computational algorithm). A generalization of the adjustment method is the continuation method (to a parametrized family).

References

[1] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[2] S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)
[3] G.I. Marchuk, V.I. Lebedev, "Numerical methods in the theory of neutron transport" , Harwood (1986) (Translated from Russian)

Comments

The adjustment method is also called the time-stepping method. Such a time-stepping method with $ m = 1 $ arises quite naturally when an elliptic boundary value problems is viewed as a steady state of a (dissipative) parabolic problem (cf. [a2]). Because of the inherent (numerical) stiffness an implicit discretization method, such as BDF, should be advocated.

Similar ideas can be used in optimization theory by e.g. constructing a mechanical system whose stable equilibrium state is the desired optimum. Again this leads to all kinds of iteration algorithms, as in [a3].

References

[a1] I. Babushka, S.L. Sobolev, "The optimization of numerical processes" Appl. Mat. , 10 (1965) pp. 96–130
[a2] M. Kubicek, V. Hlavacek, "Numerical solution of nonlinear boundary value problems with applications" , Prentice-Hall (1983)
[a3] B.S. Razumikhin, "Physical models and equilibrium methods in programming and economics" , Reidel (1984) (Translated from Russian)
[a4] W.C. Rheinboldt, "Numerical analysis of parametrized nonlinear equations" , Wiley (1986)
How to Cite This Entry:
Adjustment method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjustment_method&oldid=13996
This article was adapted from an original article by V.I. Lebedev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article