Null space preconditioners for saddle point problems
Jennifer Pestana
Joint work with Tyrone Rees, STFC Rutherford Appleton Laboratory
February 1, 2017
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Null space preconditioners
February 1, 2017
1 / 34
Where do saddle point systems arise?
How do we minimise the error between measurements and a model?
How do we choose parameters so that our model is as close as
possible to a desired state?
How can we exploit the structure present when models themselves
include a minimisation principle?
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Null space preconditioners
February 1, 2017
3 / 34
Best Linear Unbiased Estimate (BLUE)
Two observations of true state xt :
y1 = xt + 1 ,
y2 = xt + 2 .
Errors 1 , 2
have zero mean,
are uncorrelated,
and have variances σ12 , σ22 .
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Null space preconditioners
February 1, 2017
4 / 34
Best Linear Unbiased Estimate (BLUE)
Two observations: y1 = xt + 1 , y2 = xt + 2 .
BLUE
Linear: xa = v1 y1 + v2 y2
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Null space preconditioners
February 1, 2017
4 / 34
Best Linear Unbiased Estimate (BLUE)
Two observations: y1 = xt + 1 , y2 = xt + 2 .
BLUE
Linear: xa = v1 y1 + v2 y2
Unbiased:
E(xa ) = xt
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Null space preconditioners
February 1, 2017
4 / 34
Best Linear Unbiased Estimate (BLUE)
Two observations: y1 = xt + 1 , y2 = xt + 2 .
BLUE
Linear: xa = v1 y1 + v2 y2
Unbiased:
E(xa ) = xt
= v1 E(y1 ) + v2 E(y2 )
= v1 xt + v2 xt
= (v1 + v2 )xt
⇒ v1 + v2 = 1
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Null space preconditioners
February 1, 2017
4 / 34
Best Linear Unbiased Estimate (BLUE)
Two observations: y1 = xt + 1 , y2 = xt + 2 .
BLUE
Linear: xa = v1 y1 + v2 y2
Unbiased:
E(xa ) = xt
= v1 E(y1 ) + v2 E(y2 )
= v1 xt + v2 xt
= (v1 + v2 )xt
⇒ v1 + v2 = 1
h
i
Best: minimise E (xa − E(xa ))2 = v12 σ12 + v22 σ22
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Null space preconditioners
February 1, 2017
4 / 34
Best Linear Unbiased Estimate (BLUE)
min v 2 σ 2
v1 ,v2 1 1
+ v22 σ22
s.t. v1 + v2 = 1
BLUE
min J(v ) = v T Dv
where
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v
v= 1 ,
v2
s.t. v T 1 = 1,
D=
σ12
Null space preconditioners
σ22
.
February 1, 2017
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Best Linear Unbiased Estimate (BLUE)
BLUE
min J(v ) = v T Dv
or
1
min J(v ) = v T Dv
2
where
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v
v= 1 ,
v2
s.t.
s.t. v T 1 = 1,
b(v , q) = v T 1q = q,
D=
σ12
Null space preconditioners
σ22
.
February 1, 2017
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Numerical weather prediction
Estimate true state
of atmosphere
xiA at time i
Models
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Background
state
Observations
Null space preconditioners
February 1, 2017
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Numerical weather prediction
4D-var
N
1
1X
J(x0 ) = (x0 − x0B )T B −1 (x0 − x0B ) +
(yi − Hi (xi ))T Ri−1 (yi − Hi (xi ))
2
2
i=1
subject to nonlinear model dynamics xi = M(xi−1 ), i = 1, . . . , N.
x0 ∈ Rn : guess x0 = x(t0 ) at the beginning of the assimilation window
x0B : background (prior) at t0
yi ∈ Rm , i = 1, . . . , N: observation vectors at time i
Hi : maps state vector xi from model space to observation space
Mi,i−1 : integration of numerical model fro step i to i + 1
B, Ri : background and observation error covariance matrices
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Null space preconditioners
February 1, 2017
5 / 34
Example: Stokes equations
−∇2 u + ∇p = f
∇·u =0
in Ω with boundary conditions
Z
Z
∇v : ∇v −
J(v ) =
Ω
fv
Ω
Z
v · ∇q = 0 = g (q)
b(v , q) =
Ω
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Null space preconditioners
February 1, 2017
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Mathematical formulation
Variational problem
1
u = arg min J(v ) = a(v , v ) − f (v )
v ∈X
2
such that b(v , q) = g (q) for all q ∈ M
X and M may have finite or infinite dimension,
|a(v , w )| ≤ Ca kv kX kw kX for all v , w ∈ X ,
a(v , w ) = a(w , v ) and a(v , v ) ≥ 0 for all v , w ∈ X ,
|b(v , q)| ≤ Cb kv kX kqkM for all v ∈ X , q ∈ M,
f ∈ X 0 and g ∈ M0 bounded linear functionals
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Null space preconditioners
February 1, 2017
7 / 34
Saddle point problems
Variational problem
1
u = arg min J(v ) = a(v , v ) − f (v )
v ∈X
2
such that b(v , q) = g (q) for all q ∈ M
Lagrange function
L(v , q) = J(v ) + [b(v , q) − g (q)], q ∈ M
coincides with J when the constraints are satisfied.
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Null space preconditioners
February 1, 2017
8 / 34
Saddle point problems
Lagrange function
L(v , q) = J(v ) + [b(v , q) − g (q)], q ∈ M
coincides with J when the constraints are satisfied.
Saddle point problem
Find (u, p) ∈ X × M such that
a(u, v ) + b(v , p) = f (v ) for all v ∈ X ,
b(u, q) = g (q) for all q ∈ M
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Null space preconditioners
February 1, 2017
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Saddle point problems
Lagrange function
L(v , q) = J(v ) + [b(v , q) − g (q)], q ∈ M
coincides with J when the constraints are satisfied.
Saddle point problem
Find (u, p) ∈ X × M such that
a(u, v ) + b(v , p) = f (v ) for all v ∈ X ,
b(u, q) = g (q) for all q ∈ M
(u, q) ≤ (u, p) ≤ (v , p) for all (v , q) ∈ X × M
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Null space preconditioners
February 1, 2017
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Matrix formulation
Find (u, p) ∈ X × M such that
a(u, v ) + b(v , p) = f (v ) for all v ∈ X ,
b(u, q) = g (q) for all q ∈ M
(Potentially after discretization)
f
A BT x
=
y
g
B 0
| {z }
A
where
A ∈ Rn×n , SPSD,
B ∈ Rm×n , m ≤ n , B full rank
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Null space preconditioners
February 1, 2017
9 / 34
BLUE
min
1 T
v Dv
2
s.t. v T 1q = q
leads to saddle point problem
0
D 1T v
=
q
1
1 0
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Null space preconditioners
February 1, 2017
10 / 34
Numerical weather prediction
N
1
1X
J(x0 ) = (x0 − x0B )T B −1 (x0 − x0B ) +
(yi − Hi (xi ))T Ri−1 (yi − Hi (xi ))
2
2
i=1
s.t. xi = M(xi−1 ), i = 1, . . . , N.
After linearisation leads to saddle point problem

   
I
B 0
λ
b
 0 R̂ Ĥ   µ  = d 
δx
0
I Ĥ T 0
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Null space preconditioners
February 1, 2017
11 / 34
Stokes equations
−∇2 ∇ u
f
=
∇·
0 p
0
in Ω with boundary conditions
After discretizing we get
A BT
A=
B 0
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f
u
=
p
g
Null space preconditioners
February 1, 2017
12 / 34
Other sources of saddle point problems
Optimization: quadratic programming, interior point methods,
optimal control
PDEs with conservation laws: biharmonic equations in mixed form,
Maxwell equations, Navier-Stokes equations
Interpolation: hybrid schemes
Constrained or weighted least squares,...
See Benzi, Golub and Liesen (2005)
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Null space preconditioners
February 1, 2017
13 / 34
How do we solve saddle point problems?
Matrix formulation
f
A BT x
=
y
g
B 0
| {z }
A
How do we solve saddle point systems?
Direct methods (Gaussian elimination)
I
I
Perform a fixed number of operations
Obtain a single approximation
Iterative methods (Krylov methods)
I
I
Start with an initial guess that is updated at each iteration
Get successively better approximations
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Null space preconditioners
February 1, 2017
15 / 34
How do we solve large systems?
For large systems direct methods can be infeasible
Most iterative methods rely on matrix-vector products with A
If these matrix-vector products can be applied cheaply then iterative
methods can be used to solve very large problems
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Null space preconditioners
February 1, 2017
16 / 34
How do we solve large systems?
For large systems direct methods can be infeasible
Most iterative methods rely on matrix-vector products with A
If these matrix-vector products can be applied cheaply then iterative
methods can be used to solve very large problems
Sparse matrices
Sparse matrices have few nonzeros, either
because of the problem or
because of the numerical approximation
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Null space preconditioners
February 1, 2017
16 / 34
Krylov subspace methods
Aw = b
Krylov subspace methods
Choose w 0 and compute r 0 = b − Aw 0
For k = 1, 2, . . . choose w k such that
w k − w 0 ∈ Kk (A, r 0 ) = span{r 0 , Ar 0 , . . . , A(k−1) r 0 }
Many methods depending on matrix properties
Conjugate gradient method
MINRES
GMRES, BiCG, BiCGStab, QMR, TFQMR, IDR, ...
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Null space preconditioners
February 1, 2017
17 / 34
MINRES
If A is symmetric we can apply MINRES, which minimizes kr k k2 .
MINRES convergence bound
kr k k2
≤ min
p∈Πk
kr 0 k2
max |p(λ)|
λ∈σ(A)
p(0)=1
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Null space preconditioners
February 1, 2017
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MINRES
MINRES convergence bound
kr k k2
≤ min
p∈Πk
kr 0 k2
max |p(λ)|
λ∈σ(A)
p(0)=1
(0, 1)
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(0, 1)
(0, 1)
Null space preconditioners
(0, 1)
February 1, 2017
18 / 34
MINRES
MINRES convergence bound
kr k k2
≤ min
p∈Πk
kr 0 k2
max |p(λ)|
λ∈σ(A)
p(0)=1
(0, 1)
(0, 1)
(0, 1)
(0, 1)
Clustered eigenvalues, or few distinct eigenvalues, that are bounded
away from the origin are generally good
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Null space preconditioners
February 1, 2017
18 / 34
How do we ensure fast convergence?
Preconditioning
A BT
Aw =
B 0
x
f
=
=b
y
g
For MINRES convergence bounds guarantee fast convergence if
eigenvalues clustered
For GMRES, clustered eigenvalues can also be useful (e.g., Campbell
et al., 1996)
If convergence is poor, solve equivalent system:
1 1
1
1
P − 2 AP − 2 P 2 w = P − 2 b
AP −1 (Pw ) = b
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Null space preconditioners
February 1, 2017
20 / 34
Schur complement (range space) preconditioners
Aw = b where
I
A
A BT
I
A=
=
−1
BA
I
−S
B 0
A−1 B T
I
with S = BA−1 B T
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Null space preconditioners
February 1, 2017
21 / 34
Schur complement (range space) preconditioners
Aw = b where
I
A
A BT
I
A=
=
−1
BA
I
−S
B 0
A−1 B T
I
with S = BA−1 B T
"
Pcs =
Pcons
Ab
#
"
,
Sb
I
=
B Ab−1 I
Pus
"
Ab
#
Ab B T
=
,
−Sb
#
−Sb
I
Ab−1 B T
I
(Kuznetsov, 1995; Murphy, Golub & Wathen, 2000)
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Null space preconditioners
February 1, 2017
21 / 34
Schur complement (range space) preconditioners
Aw = b where
I
A
A BT
I
A=
=
−1
BA
I
−S
B 0
A−1 B T
I
with S = BA−1 B T
If Ab and Sb good approximations of A and S then P −1 A has clustered
eigenvalues
Many successful preconditioners based on this approach
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Null space preconditioners
February 1, 2017
21 / 34
Nonsingularity
A BT
Aw =
B 0
x
f
=
=b
y
g
A ∈ Rn×n and SPSD, B ∈ Rm×n , B full rank, m ≤ n
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Null space preconditioners
February 1, 2017
22 / 34
Nonsingularity
A BT
Aw =
B 0
x
f
=
=b
y
g
A ∈ Rn×n and SPSD, B ∈ Rm×n , B full rank, m ≤ n
Theorem (Theorem 3.2. Benzi, Golub & Liesen (2005))
Assume that A is symmetric positive semidefinite and B has full rank.
Then a necessary and sufficient condition for the saddle point matrix A to
be nonsingular is null(A) ∩ null(B) = {0}.
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Null space preconditioners
February 1, 2017
22 / 34
Nonsingularity
A BT
Aw =
B 0
x
f
=
=b
y
g
A ∈ Rn×n and SPSD, B ∈ Rm×n , B full rank, m ≤ n
Theorem (Theorem 3.2. Benzi, Golub & Liesen (2005))
Assume that A is symmetric positive semidefinite and B has full rank.
Then a necessary and sufficient condition for the saddle point matrix A to
be nonsingular is null(A) ∩ null(B) = {0}.
We need A to be positive definite on null(B)
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Null space preconditioners
February 1, 2017
22 / 34
The nullspace method
A BT
B 0
A
x
y
=
f
g
B
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Null space preconditioners
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x p + x̄
y
=
f
g
x = x p + x̄
Bx p = g
x̄ ∈ null(B)
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Null space preconditioners
B
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x̄
y
=
h
0
h = f − Ax p
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Null space preconditioners
B
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x̄
y
=
h
0
Let Z span the nullspace of B (i.e. BZ = 0)
B
BZ = 0 ⇒ x̄ = Z x n
Z
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Null space preconditioners
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x̄
y
=
h
0
Let Z span the nullspace of B (i.e. BZ = 0)
B
BZ = 0 ⇒ x̄ = Z x n
Z
AZ x n +
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BT y =
h
Null space preconditioners
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x̄
y
=
h
0
Let Z span the nullspace of B (i.e. BZ = 0)
B
Z
Z T AZ x n + Z T B T y = Z T h
Z T AZ
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Null space preconditioners
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x̄
y
=
h
0
Let Z span the nullspace of B (i.e. BZ = 0)
B
Z
Z T AZ x n = Z T h
Z T AZ
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Null space preconditioners
February 1, 2017
23 / 34
The nullspace method
A BT
B 0
A
x̄
y
=
h
0
Let Z span the nullspace of B (i.e. BZ = 0)
B
Z
Z T AZ x n = Z T h
smaller, SPD matrix
This is the nullspace method
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Null space preconditioners
Z T AZ
February 1, 2017
23 / 34
Nullspace factorizations
A BT
B
x
f
=
y
g
Nullspace method
1
2
3
4
Find a particular solution x p ∈ Rn for Bx p = g
Find Z whose columns span the nullspace of B
Then since x = xb + Z x n solve Z T AZ x n = Z T (f − Ax p )
Solve the overdetermined system B T y = f − Ax
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Null space preconditioners
February 1, 2017
24 / 34
Nullspace factorizations
A BT
B
x
f
=
y
g
Nullspace method
1
2
3
4
Find a particular solution x p ∈ Rn for Bx p = g
Find Z whose columns span the nullspace of B
Then since x = xb + Z x n solve Z T AZ x n = Z T (f − Ax p )
Solve the overdetermined system B T y = f − Ax
x = x p + x̄ = Y x r + Z x n , range(Y ) = range(B T ), range(Z ) = null(B)
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Null space preconditioners
February 1, 2017
24 / 34
Nullspace factorization
Want sparse and well conditioned Y and Z


A11 A12 B1T
A = A21 A22 B2T  , B1 ∈ Rm×m nonsingular
B1 B2
0
−B1−1 B2
Zf =
,
I
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−1 B
Yf = 1
0
Null space preconditioners
February 1, 2017
25 / 34
Nullspace factorization


I
0
0
I B1−1 B2
A11 0 B1T
I
A = B2T B1−T I XB1−1   0 N 0  0
B1 0
0
0
0
I
0 B1−T X T
{z
}|
{z
}|
|
{z

L
D
N = ZfT AZf and X = ZfT
LT
A11
A21

0
0,
I
}
Can build preconditioners by approximating and/or dropping blocks.
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Null space preconditioners
February 1, 2017
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Central null preconditioner


I
0
0
I B1−1 B2
A11 0 B1T
I
A = B2T B1−T I XB1−1   0 N 0  0
B1 0
0
0
0
I
0 B1−T X T
{z
}|
{z
}|
|
{z

D
L
Pcn
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LT

0
0,
I
}


A11 0 B1T
=  0 Ne 0 
B1 0
0
Null space preconditioners
February 1, 2017
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Central null preconditioner


I
0
0
I B1−1 B2
A11 0 B1T
I
A = B2T B1−T I XB1−1   0 N 0  0
B1 0
0
0
0
I
0 B1−T X T
{z
}|
{z
}|
|
{z

D
L
Pcn
LT

0
0,
I
}


A11 0 B1T
=  0 Ne 0 
B1 0
0
Eigenvalues depend on N −1 A22
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Null space preconditioners
February 1, 2017
26 / 34
Upper null preconditioner


I
0
0
I B1−1 B2
A11 0 B1T
−T
−1
I
A = B2T B1
I XB1   0 N 0  0
−T T
B1 0
0
0
0
I
0 B1 X
{z
}|
{z
}|
|
{z

D
L
Pun
LT

0
0
I
}


A11 A12 B1T
= 0
Ñ
0 
B1 B2
0
Theorem (P. & Rees, 2016)
Let Ñ be an nonsingular approximation to N. Then
−1 A) = {1} ∪ σ(N
e−1 N).
σ(Pun
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Null space preconditioners
February 1, 2017
27 / 34
Constraint preconditioner


I
0
0
I B1−1 B2
A11 0 B1T
−T
−1
T
I
A = B2 B1
I XB1   0 N 0  0
−T T
B1 0
0
0
0
I
0 B1 X
{z
}|
{z
}|
|
{z

D
L

Pconn
I
T

= B2 B1−T
0
LT


A11 0 B1T
0
0
I B1−1 B2
I
I XB1−1   0 Ne 0  0
0
I
0 B1−T X T
B1 0
0

0
0
I
}

0
0
I
Theorem (P. & Rees, 2016)
The matrix Pconn is an exact constraint preconditioner. Also, if Ñ is
−1 A) = {1} ∪ σ(N
e−1 N).
nonsingular then σ(Pconn
Pconn is equivalent to a preconditioner in GALAHAD
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Null space preconditioners
February 1, 2017
28 / 34
Diagonal preconditioner
Cheap to apply
−1 A depend on N −1 A
Eigenvalues of Pcn
22
Triangular preconditioner
−1 A) = {1} ∪ σ(N
e−1 N)
σ(Pun
Can also apply lower triangular preconditioner using short term
recurrences
Constraint preconditioner
Exact constraint preconditioner
σ(P −1 A) = {1} ∪ σ(Ne−1 N)
un
Can use in projected conjugate gradients (Gould, Hribar & Nocedal,
2001) or projected MINRES (Gould, Orban & Rees, 2014)
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Null space preconditioners
February 1, 2017
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Results: Optimization matrices
min
1 T
x Hx + f T x
2
s.t. Bx = g ,
x ≥ 0.
Primal-dual interior point method:
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H + Xk−1 Zk
B
Null space preconditioners
BT
0
February 1, 2017
30 / 34
Results: Optimization problems
Incomplete Cholesky factorizations:
Matrix
AUG3DC
CONT-200
CVXQP3 S
DPKLO1
DTOC3
GOULDQP3
LISWET1
MOSARQP1
PRIMAL1
QPCSTAIR
STCQP2
YAO
n
3873
40397
100
133
14999
699
10002
2500
325
467
4097
2002
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m
1000
39601
75
77
9998
349
10000
700
85
356
2052
2000
Pus
10
*
7
15
6
7
6
25
4
10
13
5
Pcs
21
*
14
30
12
13
9
51
8
20
26
9
Null space preconditioners
Pcons
9
*
7
15
6
6
4
25
4
10
13
4
Pun
16
24
6
8
5
6
2
8
13
20
21
2
Pcn
33
43
33
28
10
27
4
22
25
40
22
5
Pconn
16
23
5
7
5
6
1
7
12
19
20
1
February 1, 2017
31 / 34
Results: Optimization matrices
Identity matrices:
Matrix
AUG3DC
CONT-200
CVXQP3 S
DPKLO1
DTOC3
GOULDQP3
LISWET1
MOSARQP1
PRIMAL1
QPCSTAIR
STCQP2
YAO
n
3873
40397
100
133
14999
699
10002
2500
325
467
4097
2002
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m
1000
39601
75
77
9998
349
10000
700
85
356
2052
2000
Pus
35
*
82
51
*
20
*
464
77
247
267
*
Pcs
71
*
150
102
*
39
*
927
154
490
528
*
Null space preconditioners
Pcons
35
*
83
51
*
19
*
464
77
247
267
*
Pun
87
28
26
24
6
40
3
15
40
51
85
3
Pcn
166
55
44
50
10
71
5
29
79
93
95
5
February 1, 2017
Pconn
91
28
26
25
6
41
4
15
41
53
93
4
32 / 34
Results: F-matrices
In F-matrices B is a gradient matrix.
Matrix
DORT
DORT2
L3P
M3P
S3P
dan2
n
13360
7515
17280
2160
270
63750
m
9607
5477
12384
1584
207
46661
Pus
121
117
44
24
12
7
Pcs
237
233
87
47
23
13
Pcons
120
116
43
23
11
6
Pun
12
8
16
12
10
9
Pcn
32
30
36
31
28
48
Pconn
12
8
15
11
9
8
Matrices from Tůma (2002); de Niet and Wubs (2008)
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Null space preconditioners
February 1, 2017
33 / 34
Conclusions and outlook
Conclusions
Saddle point problems arise in variety of applications (constrained
minimisation)
For large sparse linear systems, Krylov subspace methods are often a
good choice
BUT only if we have a good preconditioner
Discussed preconditioners based on a nullspace factorization
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Null space preconditioners
February 1, 2017
34 / 34
Conclusions and outlook
Conclusions
Saddle point problems arise in variety of applications (constrained
minimisation)
For large sparse linear systems, Krylov subspace methods are often a
good choice
BUT only if we have a good preconditioner
Discussed preconditioners based on a nullspace factorization
Outlook
What is the best nullspace basis to choose?
What happens when we replace blocks by approximations?
What applications is this approach best suited to?
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Null space preconditioners
February 1, 2017
34 / 34
Thank you!
References I
[BGL05] M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point
problems, Acta Numer. 14 (2005), 1–137.
[FJ97]
R. Fletcher and T. Johnson, On the stability of null-space methods for KKT
systems, SIMAX 18 (1997), 938–958.
[GHN01] N. I. M. Gould, M. E. Hribar, and J. Nocedal, On the solution of equality
constrained quadratic programming problems arising in optimization, SISC 23
(2001), 1376–1395.
[GOR14] N. I.M. Gould, D. Orban, and T. Rees, Projected Krylov methods for
saddle-point systems, SIMAX 35 (2014), 1329–1343.
[PR16]
J. Pestana and T. Rees, Null-space preconditioners for saddle point problems,
SIMAX 37 (2016), 1103–1128.
[PW15]
J. Pestana and A. J. Wathen, Natural preconditioning and iterative methods
for saddle point systems, SIREV 57 (2015), 71–91.
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Null space preconditioners
February 1, 2017
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References II
[RS14]
T. Rees and J. A. Scott, The null-space method and its relationship with
matrix factorizations for sparse saddle point systems, Tech. Report
RAL-TR-2014-016, STFC Rutherford Appleton Laboratory, 2014.
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February 1, 2017
3/3