Beyond Standard Krylov
Subspace Methods
– Desirable Features for Scientific
Software –
Andreas Frommer
Bergische Universität Wuppertal
June 25, 2010
AI
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
The story of Krylov subspaces
History:
I
1952: CG by Hestenes and Stiefel
I
1975: CG rediscoverd as iterative method by Fletcher, BiCG
I
80’s-90’s: General Krylov subspace framework: many authors
I
1984: The Faber-Manteuffel theorem
I
today: established methods
I
Bye bye: Chebyshev, splitting based methods, SOR, . . .
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
2/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Basics about Krylov subspaces
A ∈ Cn×n ,
(A, r 0 )
Definition: Kk
deg(p) ≤ k − 1}.
=
Ax = b,
A non-singular
span{r 0 , Ar 0 , . . . , Ak−1 r 0 }
= {p(A)r 0 :
Known fact: ∃ p, deg(p) ≤ n, p(0) = 1 s.t. p(A)r 0 = 0
(“individualized” Cayley Hamilton)
Idea:
I r 0 = b − Ax 0 initial residual
I p(t) = 1 − tq(t)
I x ∗ = x 0 + q(A)r 0 satisfies b − Ax = p(A)r 0 = 0.
Krylov subspace method:
x k ∈ x 0 + Kk (A, r 0 )
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
3/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
State of the art:
I
Krylov subspace methods as iterative solvers
I
common feature: x k from x 0 + Kk (A, r 0 )
I
differences: variational characterization of x k
available as standard software:
I
I
I
I
I
I
Matlab
NAG
Petsc
ITPACK
...
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
4/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Arnoldi process
Wanted: (Nested) orthogonal basis v 1 , . . . , v k for Kk (A, r 0 ).
Arnoldi: Orthogonalize Av k against v 1 , . . . , v k , normalize → v k+1
Arnoldi summary: Vk = [v 1 | . . . |v k ]





AVk = Vk+1 Hk+1,k , Hk+1,k = 



|
∗
∗
0
0
..
.
∗
∗
∗
0
..
.
∗
∗
∗
∗
..
.
···
···
···
···
..
.
∗
∗
∗
∗
..
.
0
0
···
{z
0
∗
k

 "



=

0


#
Hk
...
∗
}
Note: Hk = Vk∗ AVk represents matrix projected on Kk (A, r 0 )
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
5/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Lanczos process
If A is hermitian it is sufficient to orthogonalize against v k and
v k−1
Consequence: Short recurrences (length 2 instead of length k)
Lanczos summary: Vk = [v 1 | . . . |v k ]





AVk = Vk+1 Tk+1,k , Tk+1,k = 



|
∗
∗
0
0
..
.
∗
∗
∗
0
..
.
0
∗
∗
∗
..
.
···
···
···
···
..
.
0
0
0
0
..
.
0
0
···
{z
0
∗
k

 "



=

0


#
Tk
...
.
∗
}
Note: Tk = Vk∗ AVk is tridiagonal, Hermitian (and real)
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
6/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Nonsymmetric Lanczos process
Idea: Maintain short recurrences for A non-hermitian.
Price: Basis Vk is not orthonormal but bi-orthogonal:
v k+1 ⊥ Kk (A∗ , r̃ ),
w k+1 ⊥ Kk (A, r 0 ).
Nonsymmetric Lanczos summary:





AVk = Vk+1 Tk+1,k , Tk+1,k = 



|
∗
∗
0
0
..
.
∗
∗
∗
0
..
.
0
∗
∗
∗
..
.
···
···
···
···
..
.
0
0
0
0
..
.
0
0
···
{z
0
∗
k

 "



=

0


#
Tk
...
∗
}
Note: Tk = Vk∗ AVk is tridiagonal, non-Hermitian
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
7/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Variational characterizations
Arnoldi or Lanczos based:
CG, FOM: r k = b − Ax k ⊥ Kk (A, r 0 )
MINRES,GMRES: r k ⊥ AKk (A, r 0 )
⇔ x k = argminx∈x 0 +Kk kb − Ax k k2
Nonsymmetric Lanczos based:
BiCG: r k ⊥ Kk (A∗ , r̃ 0 )
QMR: quasi minimal resdiual property
BiCGStab: nonsymmetric Lanczos, but don’t build
Wk
. . . and many more: TFQMR, LSQR, SYMMLQ, CGS, IDR, . . .
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
8/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Contents
In this talk we want to emphasize very useful enhancements for
standard Krylov subspace methods:
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
9/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Basics
Computing the basis
The methods
AI
Contents
In this talk we want to emphasize very useful enhancements for
standard Krylov subspace methods:
I
Shifted linear systems
I
Flexible methods
I
Indefinite inner products
I
Deflation
I
Conclusions
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
10/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Definition
A family of shifted linear systems is given as
(A − σi I )xi = b, i = 1, . . . , p
where σi ∈ C, i = 1, . . . , p.
Crucial fact: Kk (A, b) = Kk (A − σi I , b)
Moreover: In Arnoldi, Lanczos, nonsymmetric Lanczos:
I
vectors v k , w k independent of σi
I
Hk → Hk − σi I
Consequence: There is a potential for doing the iteration on the
shifted systems without any further matrix operations
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
11/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Example: Multishift CG
choose x 0 , compute r 0 = b − Ax 0
initialize other quantities
for j = 1, . . . , until convergence
{begin of j-th Lanczos step}
δ j = hAv j , v j i, δ̂ j = δ j − σ
w j+1 = Av j − δ j v j − β j v j−1
β j+1 = kw j+1 k
v j+1 = w j+1 /β j+1
{end of Lanczos step, compute cg iterates}
if j > 0 then
j
j
j
j
ρj = 1/(1 + τ τj−1 βδj ), ρ̂j = 1/(1 + τ̂ τ̂j−1 βδ̂j )
τ j+1 = −ρj τ j β j+1 /δj
τ̂ j+1 = −ρ̂j τ̂ j β j+1 /δ̂j
x j+1 = ρj (x j + τ j /δ j )v j + (1 − ρj )x j−1
x̂ j+1 = ρ̂j (x̂ j + τ̂ j /δ̂ j )v j + (1 − ρ̂j )x̂ j−1
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
12/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Other methods
I
multishift BiCG
I
multishift QMR (Freund 1994)
I
multishift FOM
I
multishft GMRES
I
multishift restarted FOM (Simoncini 06)
but not for
I
multishift restarted GMRES
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
13/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Restarted Krylov subspace methods
Situation: A non-hermitian, use FOM or GMRES.
When long recurrences become too long . . .
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
14/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Restarted Krylov subspace methods
Situation: A non-hermitian, use FOM or GMRES.
When long recurrences become too long . . .
. . . we stop, and restart with the current iterate x m and residual r m
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
15/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Restarted Krylov subspace methods
Situation: A non-hermitian, use FOM or GMRES.
When long recurrences become too long . . .
. . . we stop, and restart with the current iterate x m and residual r m
Fix cycle length m:
x m ∈ x 0 +Km (A, r 0 ), x 2m ∈ x m +Km (A, r m ), x 3m ∈ x 2m +Km (A, r 2m ), . . .
Restarted methods: FOM(m), GMRES(m)
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
16/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Enforcing collinear residuals
x m ∈ x 0 +Km (A, r 0 ), x 2m ∈ xm +Km (A, r m ), x 3m ∈ x 2m +Km (A, r 2m ), . . .
Problem with shifts in GMRES(m): b − (A − σi I )xim is not
collinear to b − Ax m .
Approach for multishift GMRES(m):
I
Enforce collinearity of residual b − (A − σi I )xim to b − Ax m
I
We do no longer perform GMRES(m) on the shifted system
Theorem [F. 98]: If A is positive real and σi < 0, this approach
works and kb − (A − σi I )xim k2 ≤ kb − Ax m k.
A similar approach yields shifted BiCGStab [F. 03].
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
17/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Issues for Software
Which multshift methods to implement?
I
CG, QMR
I
restarted FOM
I
optional: restarted GMRES, BiCGStab
A warning:
I
There are different algorithmic variants . . .
I
. . . which have different numerical stability
I
Example: van den Eshof and Sleijpen 04 develop more stable
multishift CG
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
18/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Families of shifted linear systems arise for
I
simple parameter dependencies
I
Tichonov-Philips regularization
I
Computation of the action of a matrix function
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
19/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Simple parameter dependencies
Example: Wilson fermion matrix M = κ1 I − D in QCD.
I
κ is a mass parameter
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
20/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Tikhonov-Philips regularization
Ah : discretization of a compact operator A between Hilbert spaces.
A is not invertible.
As h gets smaller, solution xh in Ah x = b is very bad if b is noisy.
Tikhonov-Philips regularization: Constrain kxh k ≤ α.
Resulting system: (A∗h Ah + αI )xh (α) = A∗h b.
Note: Right choice for α found a posteriori.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
21/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
An example
0.3
0.2
0.1
0
0
1
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
22/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Matrix functions
Wanted: f (A)b, where f : C → C, A large and sparse
Examples:
I
Exponential integrators in ODEs, PDEs: exp(tA)b
I
QCD: sign(M)b
Note: f (A)b = p(A)b, where polynomial p interpolates f on
spec(A). Back to Krylov subspaces!
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
23/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Issues for Software
Applications
AI
Short recurrences for f (A)b
Stable short recurrence methods use rational approximations
f (t) ≈
p
X
i=1
p
X
ωi
, ⇒ f (A)b ≈
ωi (A − σi I )−1 b.
t − σi
i=1
20
1
0.8
15
0.6
10
0.4
5
0.2
0
0
−0.2
−5
−0.4
−10
−0.6
−15
−20
−10
−0.8
−8
−6
−4
−2
0
2
4
6
−1
8
−10
6
−10
4
−10
2
−10
0
−10
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
−2
−10
24/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Convergence speeds
“Theorem”:(Most) Krylov subspace methods are direct, i.e.
∗
∃ n∗ = n∗ (A, b) ≤ n s.t. x n = b.
Related polynomial approximation problem:
k := kp (A)bk is
(P) given k, find pk s.t. pk (0) = 1 and cA,b
∗
k
smallest.
k ≤ c k kbk with c k = inf
Note: cA,b
∗
deg p≤k kp(A)k∗ .
A
A
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
25/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Theorem: a) Let A be Hermitian and
positive definite, spec(A) ⊂ [α, β], κ =
β
α . Then for the CG iterates
kx k − A−1 bkA ≤ c k kx 0 − A−1 bkA ,
√
k
κ−1
ck = √
κ+1
α
F(A)
b) Let F (A) = {hx, Axi : kxk2 = 1} and
assume α ≤ F (A) ≤ β. Then for the
GMRES iterates
kb − Ax k k2 ≤ (1 −
β
α
)kb − Ax k−1 |.
β
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
26/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Preconditioning
Idea: Ax = b → M1−1 AM2−1 x̃ = M1−1 b.
I Systems with M1 , M2 easy to solve
I one solve required in each iteration
I spec(M −1 AM −∗ ) ⊆ [1 − , 1 + ] or
1
1
F (M1−1 AM2 −1) ⊆ D(1, ).
Standard:
I ILU and modified ILU preconditioners
I Gauss-Seidel or Jacobi preconditioners
Less standard:
I algebraic multilevel preconditioners
I domain decomposition preconditioners
I recursive preconditioning
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
27/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Variable preconditioners
For simlpicity: Right preconditioing, i.e. M2 = I .
Recursive preconditioning: Use a Krylov subspace method to
(quite inaccurately) solve Ay = u k
Example: GMRESR
Domain decomposition preconditioners: Approximately solve
smaller subsystems Pi∗ APi and combine individual solutions.
Common feature: Preconditioner M depends on iteration number
k or on current vector u k
Problem: We do not build the Krylov subspace Kk (M −1 A, b).
Solution: Flexible methods
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
28/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Examples
Recursive preconditioning for (I + Γ5 sign(Γ5 M))x = b in QCD
0
10
SUMR
relSUMR
relGMRESR(SUMR)
−1
10
−2
norm of residual
10
−3
10
−4
10
−5
10
−6
10
−7
10
0
2
4
6
multiplications with Q
8
10
4
x 10
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
29/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Available methods:
I
Inexact CG: Golub and Ye 99
I
Flexible CG: Notay 00
I
Flexible GMRES: Saad 93
I
recursive GMRES: Van der Vorst and Vuik 94
I
FQMR: Szyld and Vogel, 02
To do:
I
systematic comparison of competing algorithms
I
investigate numerical stability
Example: Recent results by Roloznik, Gutknecht et al., 08, 09, 10
show that GCR is intrinsically less stable than GMRES.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
30/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Flexible methods allow for
I
non-stationary iterative methods as preconditioners
including: multigrid, domain decomposition
I
inexact multiplications with matrix A
Example: A = sign(Q)
I
relaxation strategies for accuracy of multiplication with A
(Simoncini, Szyld 03, Sleijpen, van den Eshof 03)
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
31/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Algorithms
Software issues
Applications
AI
Wilson inverter on QPACE
Features:
I
Nearest neighbour couplings on 4d grid
I
QPACE: no 1 Green TOP 500 parallel
computer absed on the CELL
We use:
I
FGMRES with restarts
I
Multiplicative Schwarz as a domain
decomposition method with red black
ordering of domains
I
On domain O(10) steps of GMRES
Collaborative Research Center “Hadron Physics from Lattice
QCD” (Regensburg, Wuppertal) funded by DFG.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
32/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Non-trivial symmetries
In quite some important applications, A is non-Hermitian, but
AJ = J ∗ A∗
where J is
I
simple
I
hermitian
I
indefinite
Of course AJ is Hermitian, but usually then very indefinite.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
33/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Examples:

I
I
0 ··· 0
 0 ··· 1

Symplectic systems, J =  .
..
..
 ..
.
.
1 0 ···
I 0
Lorentz symmetry: J =
0 −I
1
0
..
.





0
Note: J induces an indefinite inner product hx, y iJ := hx, Jy i in
which A is self-adjoint.
CG with this inner product ≡ BiCG with special choice for r̃ 0
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
34/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
I
Nonsymmetric Lanczos can exhibit premature breakdowns
I
can mostly be cured using look-ahead (Nachtigal, Freund 92)
I
BiCG can have additional sources of breakdown
AI
Suggestion:
I
Implement QMR with look-ahead Lanczos for J-symmetric
systems
I
Incorporate J-symmetric preconditioning
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
35/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Projections
Let A be hermitian and positive definite. Assume we know `
smallest eigenpairs (λi , qi ) of A.
Definitions:
I
Q = [q1 | . . . |q` ] matrix of orthonormal eigenvectors
I
Λ = diag(λ1 , . . . , λ` )
I
P = QQ ∗ orthogonal projector on rangeQ
We have
Ax = b ⇔ x = x1 + x2 where Ax1 = Pb, Ax2 = (I − P)b.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
36/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Ax = b ⇔ x = x1 + x2 where Ax1 = Pb, Ax2 = (I − P)b.
Crucial:
I
x1 = QΛ−1 Pb
I
K (A, (I − P)b) = K ((I − P)A(I − P)∗ , b)
Consequence: Get x2 with CG for A and (I − P)b, effective
n
condition number is reduced from λλn1 to λλ`+1
A non-hermitian: Idea works similarly, but with oblique projector
built form left and right eigenvectors.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
37/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Deflation on the fly?
Facts:
I
The Krylov subspace Kk (A, r 0 ) can be used to obtain
approximation to eigenpairs (Ritz vectors, harmonic Ritz
vectors)
I
Full length methods (like CG, GMRES) can be viewed as
gathering eigenvalue information implicitly and using it to
deflate
⇒ superlinear convergence
I
If we restart (example: GMRES(m)), all this information is
forgotten and lost
⇒ risk of stagnation
Idea for restarted GMRES: Extract eigenpair information from
one cycle and used it in the next.
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
38/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Many approaches suggested in literature.
I
Erhel, Burrage, Pohl 06
I
Wu, Simon 00
I
Morgan 95-09
I
Sorenson 92
I
Eiermann, Ernst, Schneider 00
I
. . . and many more
Today, a standard seems to emerge:
I
deflation via augmentation of the standard Krylov subspace
I
augmentation s.t. subspace is still a Krylov subspace
containing subspaces for each of the approximate eigenvectors
I
harmonic Ritz values for GMRES, Ritz values for FOM
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
39/43
Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Software issues
I
Implement GMRES-DR à la Morgan
I
optional: develop and implement FOM-DR
I
combination with flexible methods (Pinel 10)
I
combination with multishift methods (based on Schäfer 08)
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
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Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Wilson solver on QPACE, again
800
10
’res_12_8_24_5’
’res_nodr’
600
0.01
500
ITERATIONS
0.1
0.001
0.0001
400
300
1e-05
200
1e-06
100
1e-07
0
0.135
1e-08
0
200
400
600
800
1000
1200
1400
sap-fgmres-dr-mx(32,10)
700
1
0.1355
0.136
0.1365
0.137
kappa
0.1375
0.138
1600
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
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Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
Software issues
Deflation
Algorithms
Software issues
Applications
AI
Seismic exploration application
by X.Pinel, CERFACS and TOTAL
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
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Introduction
Shifted linear systems
Flexible Methods
Indefinite inner products
Conclusions
AI
Conclusions
I
I
Parallelization is in the MVM and the preconditioning
4 enhancements:
I
I
I
I
shifted
flexible
J-symmetry
deflation
I
Priority for CG, GMRES(m) . . .
I
. . . then QMR, BiCGStab
I
before you implement professionally, study stability in theory
and practice
I
enhancements would be helpful in many areas of scientific
computing
Beyond Standard Krylov Subspace Methods – Desirable Features for Scientific Software –, Andreas Frommer
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