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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Krylov subspace methods for linear systemswith tensor product structure
Christine Tobler
Seminar for Applied Mathematics,ETH Zürich
19. August 2009
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Outline
1 Introduction
2 Basic Algorithm
3 Convergence bounds
4 Solving the compressed equation
5 Numerical experiments
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Outline
1 Introduction
2 Basic Algorithm
3 Convergence bounds
4 Solving the compressed equation
5 Numerical experiments
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Linear system with tensor product structure
We consider the linear system
Ax = b
with
A =d∑
s=1
In1 ⊗ · · · ⊗ Ins−1 ⊗ As ⊗ Ins+1 ⊗ · · · ⊗ Ind ,
b = b1 ⊗ · · · ⊗ bd ,
As ∈ Rns×ns positive definite, bs ∈ Rns .
Example for 3 dimensions:
(A1 ⊗ I ⊗ I + I ⊗ A2 ⊗ I + I ⊗ I ⊗ A3)x = b1 ⊗ b2 ⊗ b3
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Linear system with tensor product structure
We consider the linear system
Ax = b
with
A =d∑
s=1
In1 ⊗ · · · ⊗ Ins−1 ⊗ As ⊗ Ins+1 ⊗ · · · ⊗ Ind ,
b = b1 ⊗ · · · ⊗ bd ,
As ∈ Rns×ns positive definite, bs ∈ Rns .
Example for 3 dimensions:
(A1 ⊗ I ⊗ I + I ⊗ A2 ⊗ I + I ⊗ I ⊗ A3)x = b1 ⊗ b2 ⊗ b3
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensor decompositions
CP decomposition:
vec(X ) =k∑
r=1
ar ⊗ br ⊗ cr
ar ∈ Rm,br ∈ Rn, cr ∈ Rp
c1
a1
b1
ck
ak
bk
X
Tucker decomposition:
vec(X ) =m∑
i=1
n∑s=1
p∑l=1
Gijl ai ⊗ bj ⊗ cl
=(A⊗ B ⊗ C) vec(G)
G ∈ Rm×n×p,
ai ∈ Rm,bj ∈ Rn, cl ∈ Rp
X GA
B
C
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
About this system
The eigenvalues of the matrix A are given by all possible sums
λ(1)i1 + λ
(2)i2 + ...+ λ
(d)id
where λ(s)is denotes an eigenvalue of As. For As positive definite, the
system has a unique solution.
Note that x and b are vector representations of tensors in Rn1×···×nd ,and b represents a rank-one tensor.
A tensor arising from the discretization of a sufficiently smoothfunction f can be approximated by a short sum of rank-one tensors.By superposition, we can insert such a tensor as right-hand side intoour system.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
About this system
The eigenvalues of the matrix A are given by all possible sums
λ(1)i1 + λ
(2)i2 + ...+ λ
(d)id
where λ(s)is denotes an eigenvalue of As. For As positive definite, the
system has a unique solution.
Note that x and b are vector representations of tensors in Rn1×···×nd ,and b represents a rank-one tensor.
A tensor arising from the discretization of a sufficiently smoothfunction f can be approximated by a short sum of rank-one tensors.By superposition, we can insert such a tensor as right-hand side intoour system.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
About this system
The eigenvalues of the matrix A are given by all possible sums
λ(1)i1 + λ
(2)i2 + ...+ λ
(d)id
where λ(s)is denotes an eigenvalue of As. For As positive definite, the
system has a unique solution.
Note that x and b are vector representations of tensors in Rn1×···×nd ,and b represents a rank-one tensor.
A tensor arising from the discretization of a sufficiently smoothfunction f can be approximated by a short sum of rank-one tensors.By superposition, we can insert such a tensor as right-hand side intoour system.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensorized Krylov subspaces
A Krylov subspace is defined as
Kk (A,b) = span
b, Ab, . . . , Ak−1b.
We define a tensorized Krylov subspace as
K⊗K (A,b) := span(Kk1 (A1,b1)⊗ · · · ⊗ Kkd (Ad ,bd )
).
Note thatKk0 (A,b) ⊂ K⊗K (A,b)
for K = (k0, . . . , k0).⇒ Tensorized Krylov subspaces are richer than standard Krylovsubspaces.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensorized Krylov subspaces
A Krylov subspace is defined as
Kk (A,b) = span
b, Ab, . . . , Ak−1b.
We define a tensorized Krylov subspace as
K⊗K (A,b) := span(Kk1 (A1,b1)⊗ · · · ⊗ Kkd (Ad ,bd )
).
Note thatKk0 (A,b) ⊂ K⊗K (A,b)
for K = (k0, . . . , k0).⇒ Tensorized Krylov subspaces are richer than standard Krylovsubspaces.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensorized Krylov subspaces
A Krylov subspace is defined as
Kk (A,b) = span
b, Ab, . . . , Ak−1b.
We define a tensorized Krylov subspace as
K⊗K (A,b) := span(Kk1 (A1,b1)⊗ · · · ⊗ Kkd (Ad ,bd )
).
Note thatKk0 (A,b) ⊂ K⊗K (A,b)
for K = (k0, . . . , k0).⇒ Tensorized Krylov subspaces are richer than standard Krylovsubspaces.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Outline
1 Introduction
2 Basic Algorithm
3 Convergence bounds
4 Solving the compressed equation
5 Numerical experiments
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Reminder: CG method (Ax = b)
The best approximation of x in Kk (A,b) is defined by:
‖xk − x‖A = minx∈Kk (A,b)
‖x − x‖A.
Find Uk with (orthonormal) columns that span the Krylov subspaceKk (A,b). Set xk = Uk y , then y is the solution of the compressedsystem
Hk y = b, Hk = U>k AUk , b = U>k b.
Convergence bound (κ = κ2(A))
‖xk − x‖A ≤ C(A,b, κ)(√κ− 1√
κ+ 1
)k.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Reminder: CG method (Ax = b)
The best approximation of x in Kk (A,b) is defined by:
‖xk − x‖A = minx∈Kk (A,b)
‖x − x‖A.
Find Uk with (orthonormal) columns that span the Krylov subspaceKk (A,b). Set xk = Uk y , then y is the solution of the compressedsystem
Hk y = b, Hk = U>k AUk , b = U>k b.
Convergence bound (κ = κ2(A))
‖xk − x‖A ≤ C(A,b, κ)(√κ− 1√
κ+ 1
)k.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Reminder: CG method (Ax = b)
The best approximation of x in Kk (A,b) is defined by:
‖xk − x‖A = minx∈Kk (A,b)
‖x − x‖A.
Find Uk with (orthonormal) columns that span the Krylov subspaceKk (A,b). Set xk = Uk y , then y is the solution of the compressedsystem
Hk y = b, Hk = U>k AUk , b = U>k b.
Convergence bound (κ = κ2(A))
‖xk − x‖A ≤ C(A,b, κ)(√κ− 1√
κ+ 1
)k.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensorized Krylov: A vec(X ) = b
Applying the Arnoldi method to As,bs results in Us,Hs with
U>s AsUs = Hs,
Us column-orthogonal, Hs upper Hessenberg matrix.
The columns of Us span Kks (As,bs), and similarly the columns ofU := U1 ⊗ · · · ⊗ Ud span K⊗K (A,b).
Solve the compressed system
Hy = b
with xK = Uy , b = U>b and H = U>AU .
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensorized Krylov: A vec(X ) = b
Applying the Arnoldi method to As,bs results in Us,Hs with
U>s AsUs = Hs,
Us column-orthogonal, Hs upper Hessenberg matrix.
The columns of Us span Kks (As,bs), and similarly the columns ofU := U1 ⊗ · · · ⊗ Ud span K⊗K (A,b).
Solve the compressed system
Hy = b
with xK = Uy , b = U>b and H = U>AU .
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Tensorized Krylov: A vec(X ) = b
Applying the Arnoldi method to As,bs results in Us,Hs with
U>s AsUs = Hs,
Us column-orthogonal, Hs upper Hessenberg matrix.
The columns of Us span Kks (As,bs), and similarly the columns ofU := U1 ⊗ · · · ⊗ Ud span K⊗K (A,b).
Solve the compressed system
Hy = b
with xK = Uy , b = U>b and H = U>AU .
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Compressed system
The structure of Hy = b corresponds to that of Ax = b:
H =d∑
s=1
Ik1 ⊗ · · · ⊗ Iks−1 ⊗ Hs ⊗ Iks+1 ⊗ · · · ⊗ Ikd
b = b1 ⊗ · · · ⊗ bd = ‖b‖2 (e1 ⊗ · · · ⊗ e1)
For dense core tensor y , xK is a tensor in Tucker decomposition:
xK = (U1 ⊗ · · · ⊗ Ud )y
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Outline
1 Introduction
2 Basic Algorithm
3 Convergence bounds
4 Solving the compressed equation
5 Numerical experiments
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Convergence, s.p.d. case (1)
Similarly to CG, we have:
‖xK − x‖A = minx∈K⊗K (A,b)
‖x − x‖A
Every vector in a Krylov subspace Kks (As,bs) can be seen asp(As)bs, with p a polynomial of order at most ks − 1. A similar thoughtreduces the bound calculation to the min-max problem
EΩ(K) := minp∈Π⊗K
‖p(λ1, . . . , λd )− 1λ1 + · · ·+ λd
‖Ω,
where Π⊗K is a space of multivariate polynomials, and Ω contains theeigenvalues (λ1, . . . , λd ), with λi ∈ [λmin(Ai ), λmax(Ai )].
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Convergence, s.p.d. case (2)
Inserting an upper bound for EΩ(K), we find
‖xK − x‖A ≤d∑
s=1
C(A,b, κs)(√κs − 1√κs + 1
)ks
,
with κs = 1 + λmax(As)−λmin(As)λmin(A) .
For the case As, ks constant:
‖xK − x‖A ≤ C(A,b,d)(√κ− 1√
κ+ 1
)k,
with κ = d−1d + κ2(A)
d .
Note that the convergence rate improves with increasing dimension.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Convergence, non-symmetric positive definite case
General convergence bound:
‖xK − x‖2 ≤d∑
s=1
∫ ∞0
e−αs t ‖Use−tHs e1 − e−tAs bs‖2 dt
with αs :=∑
j 6=s αj and αj = λmin(Aj + A>j )/2.
The bound on ‖Use−tHs e1 − e−tAs bs‖2 will depend on additionalknowledge on As.
For example, when the field of values of each matrix As is containedin a known ellipse, an explicit convergence bound can be found.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Outline
1 Introduction
2 Basic Algorithm
3 Convergence bounds
4 Solving the compressed equation
5 Numerical experiments
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Grasedyck’s method, system Hy = b
For H positive definite:
H−1 =
∫ ∞0
exp(−tH)dt
The exponential of H has a Kronecker product structure, too:
exp(−tH) = exp(−td∑
s=0
Hs) =d∏
s=1
exp(−tHs) =d⊗
s=1
exp(−tHs),
Approximation ym:
ym =m∑
j=1
ωj
d⊗s=1
exp(− αjHs
)bs,
with certain coefficients αj , ωj .
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Coefficients of the exponential sum (1)
The coefficients αj , ωj should minimize
supz∈Λ(H)
∣∣1z−
m∑j=1
ωje−αj z∣∣
Case 1: H symmetric and condition number knownThe eigenvalues of H are real: Λ(H) ⊂ [λmin, λmax]. There arecoefficients αj , ωj s.t.
‖y − ym‖2 ≤ C(H, b) exp( −mπ2
log(8κ2(H))
).
These coefficients may be found using a variant of the Remezalgorithm. Tabellated values have been made available byHackbusch.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Coefficients of the exponential sum (2)
Case 2: Nonsymmetric H and/or unknown condition numberThere is an explicit formula for αj , ωj , where the eigenvalues of H onlyneed to have positive real part.
‖y − y2m+1‖2 ≤ C(H, b) exp(µ/π) exp(−√
m),
where µ = max|=m(Λ(H))|.
The convergence is significantly slower than for Case 1.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Theoretical convergence bounds for the coefficients
0 50 100 150 20010
−8
10−6
10−4
10−2
100
102
k
||y −
yk|| 2
Case 1 for cond. 1e5Case 1 for cond. 1e15Case 2
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Outline
1 Introduction
2 Basic Algorithm
3 Convergence bounds
4 Solving the compressed equation
5 Numerical experiments
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Symmetric example: Poisson Equation
Finite difference discretization of the Poisson equation in ddimensions:
∆u = f in Ω = [0,1]d
u = 0 on Γ := ∂Ω,
where the right-hand side f is a separable function.
As =1h2
2 −1
−1 2. . .
. . . . . . −1−1 2
,bs : random numbers.
The approximation error is measured by relative residual, ‖AxK−b‖2‖b‖2
.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Convergence for system size 200d
Numerical convergence
50 100 150 20010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
Theoretical convergence rate
50 100 150 20010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Convergence for system size 1000d
Numerical convergence
200 400 600 800 100010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
Theoretical convergence rate
200 400 600 800 100010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Extended Krylov subspaces
Using extended Krylov subspaces
Kks (As,bs) := span(Kks (As,bs) ∪ Kks+1(A−1s ,bs)),
the algorithm works analogously.Convergence bound (As, ks constant):
‖xK − x‖2 ≤ C(A,b,d)(√κ− 1√
κ+ 1
)k.
The convergence rate depends on κ:
κ ≈ d − 1d
+1d2 (d − 1)
d−1d κ2(A)
d−1d
d = 2 : κ =12
+
√κ2(A)
4, d 0 : κ ≈ d − 1
d+κ2(A)
d.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Extended Krylov, system size 200d
Numerical convergence
5 10 15 20 25 30 35 4010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
Theoretical convergence rate
50 100 150 20010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Non-symmetric example
Finite difference discretization in d dimensions of theconvection-diffusion equation
∆u − (c, . . . , c)∇u = f in Ω = [0,1]d
u = 0 on Γ := ∂Ω,
where f is again a separable function.
As =1h2
2 −1−1 2 −1
. . . . . . . . .−1 2 −1
−1 2
+c
4h
3 −5 1
1 3 −5. . .
. . . . . . . . . 11 3 −5
1 3
.
For c = 10, all eigenvalues of A are real. For c = 100, this is not thecase, and the convergence is significantly worsened.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Non-symmetric case, system size 200d
Convergence for c = 10
50 100 150 20010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 5d = 10d = 50d = 100
Convergence for c = 100
50 100 150 20010
−8
10−6
10−4
10−2
100
102
k
rela
tive
resi
dual
d = 2
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Conclusions
An efficient algorithm to calculate a low-rank approximation ofthe solution tensorThe computational complexity is linear in the number ofdimensionsOnly matrix-vector operations with the full system matrices arerequiredA theoretical convergence bound was found.
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Conclusions
An efficient algorithm to calculate a low-rank approximation ofthe solution tensorThe computational complexity is linear in the number ofdimensionsOnly matrix-vector operations with the full system matrices arerequiredA theoretical convergence bound was found.
Thank you for your attention!
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Introduction Basic Algorithm Convergence bounds Solving the compressed equation Numerical experiments
Literature
D. Kressner, C. Tobler, “Krylov subspace methods for linear systemswith tensor product structure”, Research report No. 2009-16, SAM,ETH Zurich , 2009.
L. Grasedyck, “Existence and computation of low Kronecker-rankapproximations for large linear systems of tensor product structure”,Computing, 72(3-4):247-265, (2004).
D. Braess, W. Hackbusch, “Approximation of 1/x by exponential sumsin [1,∞)”, IMA J. Numer. Anal., 25(4):685-697, (2005).