03/29/2006, City Univ 1

Iterative Methods with Inexact Preconditioners

and Applications to Saddle-point Systems &

Electromagnetic Maxwell Systems

Jun Zou Department of Mathematics

The Chinese University of Hong Kong

http://www.math.cuhk.edu.hk/~zou

Joint work withQiya Hu (CAS, Beijing)

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Outline of the Talk

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Inexact Uzawa Methods for SPPs

• Linear saddle-point problem:

where A, C : SPD matrices ; B : n x m ( n > m )

• Applications:

Navier-Stokes eqns, Maxwell eqns, optimizations, purely algebraic systems , … …

• Well-posedness : see Ciarlet-Huang-Zou, SIMAX 2003

• Much more difficult to solve than SPD systems

• Ill-conditioned: need preconditionings, parallel type

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Why need preconditionings ?

• When solving a linear system

• A is often ill-conditioned if it arises from discretization of PDEs

• If one finds a preconditioner B s.t. cond(BA) is small, then we solve

• If B is optimal, i.e. cond (BA) is independent of h, then the number of iterations for solving a system of h=1/100 will be the same as for solving a system of h=1/10 • Possibly with a time difference of hours & days, or days & months, especially for time-dependent problems

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Schur Complement Approach

A simple approach: first solve for p ,

Then solve for u ,

We need other more effective methods !

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Preconditioned Uzawa Algorithm

Given two preconditioners:

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Preconditioned inexact Uzawa algorithm

• Algorithm

• Randy Bank, James Bramble, Gene Golub, ... ...

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Preconditioned inexact Uzawa algorithm• Algorithm

• Question :

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Uzawa Alg. with Relaxation Parameters(Hu-Zou, SIAM J Maxtrix Anal, 2001)

• Algorithm I

• How to choose

Uzawa Alg with Relaxation Parameters

• Algorithm with relaxation parameters:

• Implementation

• Unfortunately, convergence guaranteed under

But ensured for any preconditioner for C ; scaling invariant

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(Hu-Zou, Numer Math, 2001)

• Algorithm with relaxation parameter

• This works well only when both

• This may not work well in the cases

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(Hu-Zou, Numer Math, 2001)

• Algorithm with relaxation parameter

• For the case :

more efficient algorithm:

• Convergence guaranteed if

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(Hu-Zou, SIAM J Optimization, 2005)

Inexact Preconditioned Methods for NL SPPs

• Nonlinear saddle-point problem:

• Arise from NS eqns, or nonlinear optimiz :

Time-dependent Maxwell System

● The curl-curl system: Find u such that

● Eliminating H to get the E - equation:

● Eliminating E to get the H - equation:

● Edge element methods (Nedelec’s elements) : see Ciarlet-Zou : Numer Math 1999; RAIRO Math Model & Numer Anal 1997

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Time-dependent Maxwell System

● The curl-curl system: Find u such that

● At each time step, we have to solve

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Non-overlapping DD Preconditioner I (Hu-Zou, SIAM J Numer Anal, 2003)

● The curl-curl system: Find u such that

● Weak formulation: Find

● Edge element of lowest order :

● Nodal finite element :

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Edge Element Method

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Additive Preconditioner Theory

● Additive Preconditioner Theory

● Given an SPD S, define an additive Preconditioner M :

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DDMs for Maxwell Equations

• 2D, 3D overlapping DDMs: Toselli (00), Pasciak-Zhao (02), Gopalakrishnan-Pasciak (03)

• 2D Nonoverlapping DDMs : Toselli-Klawonn (01), Toselli-Widlund-Wohlmuth (01)

• 3D Nonoverlapping DDMs : Hu-Zou (2003), Hu-Zou (2004)

• 3D FETI-DP: Toselli (2005)

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Nonoverlapping DD Preconditioner I (Hu-Zou, SIAM J Numer Anal, 2003)

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Interface Equation on

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Global Coarse Subspace

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Two Global Coarse Spaces

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Nonoverlapping DD Preconditioner I

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Nonoverlapping DD Preconditioner II(Hu-Zou, Math Comput, 2003)

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Variational Formulation

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Equivalent Saddle-point System

can not apply Uzawa iteration

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Equivalent Saddle-point System

Write the system

into equivalent saddle-point system :

Convergence rate depends on

Important : needed only once in Uzawa iter.

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DD PreconditionersLet

Theorem

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DD Preconditioner II

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Local & Global Coarse Solvers

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Stable Decomposition of VH

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Condition Number Estimate

The additive preconditioner

Condition number estimate:

Independent of jumps in coefficients

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Mortar Edge Element Methods

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Mortar Edge Element Methods

See Ciarlet-Zou, Numer Math 99:

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Mortar Edge M with Optim Convergence(nested grids on interfaces)

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Local Multiplier Spaces: crucial !

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Near Optimal Convergence

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Auxiliary Subspace Preconditioner(Hiptmair-Zou, Numer Math, 2006)

Solve the Maxwell system :

by edge elements on unstructured meshes

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Optimal DD and MG Preconditioners

• Edge element of 1st family for discretization

• Edge element of 2nd family for preconditioning

• Mesh-independent condition number

• Extension to elliptic and parabolic equations

Thank You !