Transcript Inexact Preconditioned Uzawa Methods & Non

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 with
Qiya Hu (CAS, Beijing)
03/29/2006, City Univ
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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: nxm (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
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Uzawa Alg with Relaxation Parameters
• Algorithm with relaxation parameters:
• Implementation
• Unfortunately, convergence guaranteed under
But ensured for any preconditioner for C ; scaling invariant
(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)
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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
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
● Given an SPD S, define an additive Preconditioner M :
● Additive Preconditioner Theory
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DDMs for Maxwell Equations
• 2D, 3D overlapping DDMs:
Toselli (00), Pasciak-Zhao (02),
Gopalakrishnan-Pasciak (03)
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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 :
Important :
needed only once in Uzawa iter.
Convergence rate depends on
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DD Preconditioners
Let
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 !
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