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In [[numerical mathematics]], the '''Uzawa iteration''' is an algorithm for solving saddle point problems. It is named after [[Hirofumi Uzawa]] <!--http://www.amazon.com/Studies-Linear-Non-Linear-Programming-Mathematical/dp/1258444062/ref=sr_1_1?s=books&ie=UTF8&qid=1386043803&sr=1-1--> and was originally introduced in the context of concave programming.<ref name="UZ58">H. Uzawa, ''Iterative methods for concave programming'', in K. J. Arrow, L. Hurwicz, H. Uzawa, ''Studies in linear and nonlinear programming'', Stanford University Press 1958</ref>
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== Basic idea ==
We consider a saddle point problem of the form
 
: <math> \begin{pmatrix} A & B\\ B^* & \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}
        = \begin{pmatrix} b_1\\ b_2 \end{pmatrix},</math>
 
where <math>A</math> is a symmetric [[positive-definite matrix]].
Multiplying the first row by <math>B^* A^{-1}</math> and subtracting from the second row yields the upper-triangular system
 
: <math> \begin{pmatrix} A & B\\ & -S \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}
        = \begin{pmatrix} b_1\\ b_2 - B^* A^{-1} b_1 \end{pmatrix},</math>
 
where <math>S := B^* A^{-1} B</math> denotes the [[Schur complement]].
Since <math>S</math> is symmetric positive-definite, we can apply standard iterative methods like the [[gradient descent]]
method or the [[conjugate gradient method]] to
 
: <math> S x_2 = B^* A^{-1} b_1 - b_2</math>
 
in order to compute <math>x_2</math>.
The vector <math>x_1</math> can be reconstructed by solving
 
: <math> A x_1 = b_1 - B x_2. \, </math>
 
It is possible to update <math>x_1</math> alongside <math>x_2</math> during the iteration for the Schur complement system and thus obtain an efficient algorithm.
 
== Implementation ==
 
We start the conjugate gradient iteration by computing the residual
 
: <math> r_2 := b_2 - B^* A^{-1} b_1 - S x_2 = b_2 - B^* A^{-1} (b_1 - B x_2) = b_2 - B^* x_1,</math>
 
of the Schur complement system, where
 
: <math> x_1 := A^{-1} (b_1 - B x_2) </math>
 
denotes the upper half of the solution vector matching the initial guess <math>x_2</math> for its lower half. We complete the initialization by choosing the first search direction
 
: <math> p_2 := r_2.\, </math>
 
In each step, we compute
 
: <math> a_2 := S p_2 = B^* A^{-1} B p_2 = B^* p_1 </math>
 
and keep the intermediate result
 
: <math> p_1 := A^{-1} B p_2 </math>
 
for later.
The scaling factor is given by
 
: <math> \alpha := p_2^* r_2 / p_2^* a_2 </math>
 
and leads to the updates
 
: <math> x_2 := x_2 + \alpha p_2, \quad r_2 := r_2 - \alpha a_2. </math>
 
Using the intermediate result <math>p_1</math> saved earlier, we can also update the upper part of the solution vector
 
: <math> x_1 := x_1 - \alpha p_1.\, </math>
 
Now we only have to construct the new search direction by the [[Gram–Schmidt process]], i.e.,
 
: <math> \beta := r_2^* a_2 / p_2^* a_2,\quad p_2 := r_2 - \beta p_2. </math>
 
The iteration terminates if the residual <math>r_2</math> has become sufficiently small or if the norm of <math>p_2</math> is significantly smaller than <math>r_2</math> indicating that the [[Krylov subspace]] has been almost exhausted.
 
== Modifications and extensions ==
If solving the linear system <math>A x=b</math> exactly is not feasible, inexact solvers can be applied.
<ref name="ELGO94">H. C. Elman and G. H. Golub, ''Inexact and preconditioned Uzawa algorithms for saddle point problems'', SIAM J. Num. Anal. 31(6):1645–1661 (1994)</ref>
<ref>J. H. Bramble, J. E. Pasciak and A. T. Vassilev, ''Analysis of the inexact Uzawa algorithm for saddle point problems'', SIAM J. Num. Anal. 34:1072–1982 (1997)</ref>
<ref>W. Zulehner, ''Analysis of iterative methods for saddle point problems. A unified approach'', Math. Comp. 71:479–505 (1998)</ref>
 
If the Schur complement system is ill-conditioned, preconditioners can be employed to improve the speed of convergence of the underlying gradient method.<ref name="ELGO94"/>
<ref name="GRKO07">C. Gräser and R. Kornhuber, ''On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints'', Domain Decomposition Methods in Science and Engineering XVI, Lec. Not. Comp. Sci. Eng. 55:91–102 (2007)</ref>
 
Inequality constraints can be incorporated, e.g., in order to handle obstacle problems.<ref name="GRKO07"/>
 
== Literature ==
<references/>
 
[[Category:Numerical analysis]]

Latest revision as of 19:20, 6 November 2014

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