Adjustment method

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).

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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].

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