Uzawa iteration

In numerical mathematics, the Uzawa iteration is an algorithm for solving saddle point problems. It is named after Hirofumi Uzawa and was originally introduced in the context of concave programming. ^[1]

Basic idea

We consider a saddle point problem of the form

${\displaystyle {\begin{pmatrix}A&B\\B^{*}&\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}},}$

where ${\displaystyle A}$ is a symmetric positive-definite matrix. Multiplying the first row by ${\displaystyle B^{*}A^{-1}}$ and subtracting from the second row yields the upper-triangular system

${\displaystyle {\begin{pmatrix}A&B\\&-S\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}-B^{*}A^{-1}b_{1}\end{pmatrix}},}$

where ${\displaystyle S:=B^{*}A^{-1}B}$ denotes the Schur complement. Since ${\displaystyle S}$ is symmetric positive-definite, we can apply standard iterative methods like the gradient descent method or the conjugate gradient method to solve

${\displaystyle Sx_{2}=B^{*}A^{-1}b_{1}-b_{2}}$

in order to compute ${\displaystyle x_{2}}$. The vector ${\displaystyle x_{1}}$ can be reconstructed by solving

${\displaystyle Ax_{1}=b_{1}-Bx_{2}.\,}$

It is possible to update ${\displaystyle x_{1}}$ alongside ${\displaystyle x_{2}}$ 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

${\displaystyle r_{2}:=B^{*}A^{-1}b_{1}-b_{2}-Sx_{2}=B^{*}A^{-1}(b_{1}-Bx_{2})-b_{2}=B^{*}x_{1}-b_{2},}$

of the Schur complement system, where

${\displaystyle x_{1}:=A^{-1}(b_{1}-Bx_{2})}$

denotes the upper half of the solution vector matching the initial guess ${\displaystyle x_{2}}$ for its lower half. We complete the initialization by choosing the first search direction

${\displaystyle p_{2}:=r_{2}.\,}$

In each step, we compute

${\displaystyle a_{2}:=Sp_{2}=B^{*}A^{-1}Bp_{2}=B^{*}p_{1}}$

and keep the intermediate result

${\displaystyle p_{1}:=A^{-1}Bp_{2}}$

for later. The scaling factor is given by

${\displaystyle \alpha$ :=p_{2}^{*}a_{2}/p_{2}^{*}r_{2}}

and leads to the updates

${\displaystyle x_{2}:=x_{2}+\alpha p_{2},\quad r_{2}:=r_{2}-\alpha a_{2}.}$

Using the intermediate result ${\displaystyle p_{1}}$ saved earlier, we can also update the upper part of the solution vector

${\displaystyle x_{1}:=x_{1}-\alpha p_{1}.\,}$

Now we only have to construct the new search direction by the Gram–Schmidt process, i.e.,

${\displaystyle \beta$ :=r_{2}^{*}a_{2}/p_{2}^{*}a_{2},\quad p_{2}:=r_{2}-\beta p_{2}.}

The iteration terminates if the residual ${\displaystyle r_{2}}$ has become sufficiently small or if the norm of ${\displaystyle p_{2}}$ is significantly smaller than ${\displaystyle r_{2}}$ indicating that the Krylov subspace has been almost exhausted.

Modifications and extensions

If solving the linear system ${\displaystyle Ax=b}$ exactly is not feasible, inexact solvers can be applied. ^{[2] [3] [4]}

If the Schur complement system is ill-conditioned, preconditioners can be employed to improve the speed of convergence of the underlying gradient method. ^{[2] [5]}

Inequality constraints can be incorporated, e.g., in order to handle obstacle problems. ^[5]

References

1. Uzawa, H. (1958). "Iterative methods for concave programming". In Arrow, K. J.; Hurwicz, L.; Uzawa, H. (eds.). Studies in linear and nonlinear programming . Stanford University Press.
2. Elman, H. C.; Golub, G. H. (1994). "Inexact and preconditioned Uzawa algorithms for saddle point problems". SIAM J. Numer. Anal. 31 (6): 1645–1661. CiteSeerX   10.1.1.307.8178 . doi:10.1137/0731085.
3. Bramble, J. H.; Pasciak, J. E.; Vassilev, A. T. (1997). "Analysis of the inexact Uzawa algorithm for saddle point problems". SIAM J. Numer. Anal. 34 (3): 1072–1982. CiteSeerX   10.1.1.52.9559 . doi:10.1137/S0036142994273343.
4. Zulehner, W. (1998). "Analysis of iterative methods for saddle point problems. A unified approach". Math. Comp. 71 (238): 479–505. doi: 10.1090/S0025-5718-01-01324-2 .
5. Gräser, C.; Kornhuber, R. (2007). "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. Vol. 55. pp. 91–102. CiteSeerX   10.1.1.72.9238 . doi:10.1007/978-3-540-34469-8_8. ISBN   978-3-540-34468-1.

Further reading

• Chen, Zhangxin (2006). "Linear System Solution Techniques". Finite Element Methods and Their Applications. Berlin: Springer. pp. 145–154. ISBN   978-3-540-28078-1.