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]

## Contents

* 1 Basic idea
* 2 Implementation
* 3 Modifications and extensions
* 4 References
* 5 Further reading

## Basic idea

We consider a saddle point problem of the form

(ABB*)(x1x2)=(b1b2),

where A is a symmetric positive-definite matrix. Multiplying the first row by B*A−1 and subtracting from the second row yields the upper-triangular system

(AB−S)(x1x2)=(b1b2−B*A−1b1),

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

Sx2=B*A−1b1−b2

in order to compute x2. The vector x1 can be reconstructed by solving

Ax1=b1−Bx2.

It is possible to update x1 alongside x2 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

r2:=B*A−1b1−b2−Sx2=B*A−1(b1−Bx2)−b2=B*x1−b2,

of the Schur complement system, where

x1:=A−1(b1−Bx2)

denotes the upper half of the solution vector matching the initial guess x2 for its lower half. We complete the initialization by choosing the first search direction

p2:=r2.

In each step, we compute

a2:=Sp2=B*A−1Bp2=B*p1

and keep the intermediate result

p1:=A−1Bp2

for later. The scaling factor is given by

α:=p2*a2/p2*r2

and leads to the updates

x2:=x2+αp2,r2:=r2−αa2.

Using the intermediate result p1 saved earlier, we can also update the upper part of the solution vector

x1:=x1−αp1.

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

β:=r2*a2/p2*a2,p2:=r2−βp2.

The iteration terminates if the residual r2 has become sufficiently small or if the norm of p2 is significantly smaller than r2 indicating that the Krylov subspace has been almost exhausted.

## Modifications and extensions

If solving the linear system 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.. Studies in linear and nonlinear programming. Stanford University Press. https://archive.org/details/studiesinlinearn0000arro.
2. ↑ 2.0 2.1 Elman, H. C.; Golub, G. H. (1994). "Inexact and preconditioned Uzawa algorithms for saddle point problems". SIAM J. Numer. Anal. 31 (6): 1645–1661. 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. 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. ↑ 5.0 5.1 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.. 55. pp. 91–102. 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. https://books.google.com/books?id=GvvMfd1chfkC&pg=PA145.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Uzawa iteration. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Uzawa_iteration&oldid=197522"