In this work we discuss how to compute KLE with complexity O(k n log n), how to approximate large covariance matrices (in H-matrix format), how to use the Lanczos method.
Data sparse approximation of Karhunen-Loeve Expansion
1. Data sparse approximation of the
Karhunen-Lo`eve expansion
Alexander Litvinenko,
joint with B. Khoromskij (Leipzig) and H. Matthies(Braunschweig)
Institut f¨ur Wissenschaftliches Rechnen, Technische Universit¨at Braunschweig,
0531-391-3008, litvinen@tu-bs.de
March 5, 2008
4. Stochastic PDE
We consider
− div(κ(x, ω)∇u) = f(x, ω) in D,
u = 0 on ∂D,
with stochastic coefficients κ(x, ω), x ∈ D ⊆ Rd
and ω belongs to the
space of random events Ω.
[Babuˇska, Ghanem, Matthies, Schwab, Vandewalle, ...].
Methods and techniques:
1. Response surface
2. Monte-Carlo
3. Perturbation
4. Stochastic Galerkin
5. Examples of covariance functions [Novak,(IWS),04]
The random field requires to specify its spatial correl. structure
covf (x, y) = E[(f(x, ·) − µf (x))(f(y, ·) − µf (y))],
where E is the expectation and µf (x) := E[f(x, ·)].
Let h =
3
i=1 h2
i /ℓ2
i + d2 − d
2
, where hi := xi − yi , i = 1, 2, 3,
ℓi are cov. lengths and d a parameter.
Gaussian cov(h) = σ2
· exp(−h2
),
exponential cov(h) = σ2
· exp(−h),
spherical
cov(h) =
σ2
· 1 − 3
2
h
hr
− 1
2
h3
h3
r
for 0 ≤ h ≤ hr ,
0 for h > hr .
7. KLE
The spectral representation of the cov. function is
Cκ(x, y) = ∞
i=0 λi ki(x)ki (y), where λi and ki(x) are the eigenvalues
and eigenfunctions.
The Karhunen-Lo`eve expansion [Loeve, 1977] is the series
κ(x, ω) = µk (x) +
∞
i=1
λi ki (x)ξi (ω), where
ξi (ω) are uncorrelated random variables and ki are basis functions in
L2
(D).
Eigenpairs λi , ki are the solution of
Tki = λi ki, ki ∈ L2
(D), i ∈ N, where.
T : L2
(D) → L2
(D),
(Tu)(x) := D
covk (x, y)u(y)dy.
9. Computation of eigenpairs by FFT
If the cov. function depends on (x − y) then on a uniform tensor grid
the cov. matrix C is (block) Toeplitz.
Then C can be extended to the circulant one and the decomposition
C =
1
n
F H
ΛF (1)
may be computed like follows. Multiply (1) by F becomes
F C = ΛF ,
F C1 = ΛF1.
Since all entries of F1 are unity, obtain
λ = F C1.
F C1 may be computed very efficiently by FFT [Cooley, 1965] in
O(n log n) FLOPS.
C1 may be represented in a matrix or in a tensor format.
10. Multidimensional FFT
Lemma: The d-dim. FT F (d)
can be represented as following
F (d)
= (F
(1)
1 ⊗ I ⊗ I . . .)(I ⊗ F
(1)
2 ⊗ I . . .) . . . (I ⊗ I . . . ⊗ F
(1)
d ), (2)
and the complexity of F (d)
is O(nd
log n), where n is the number of
dofs in one direction.
11. Discrete eigenvalue problem
Let
Wij :=
k,m D
bi (x)bk (x)dxCkm
D
bj (y)bm(y)dy,
Mij =
D
bi (x)bj (x)dx.
Then we solve
W fh
ℓ = λℓMfh
ℓ , where W := MCM
Approximate C in
◮ low rank format
◮ the H-matrix format
◮ sparse tensor format
and use the Lanczos method to compute m largest eigenvalues.
13. H - Matrices
Comp. complexity is O(kn log n) and storage O(kn log n).
To assemble low-rank blocks use ACA [Bebendorf, Tyrtyshnikov].
Dependence of the computational time and storage requirements of
CH on the rank k, n = 322
.
k time (sec.) memory (MB) C−CH 2
C 2
2 0.04 2e + 6 3.5e − 5
6 0.1 4e + 6 1.4e − 5
9 0.14 5.4e + 6 1.4e − 5
12 0.17 6.8e + 6 3.1e − 7
17 0.23 9.3e + 6 6.3e − 8
The time for dense matrix C is 3.3 sec. and the storage 1.4e + 8 MB.
14. H - Matrices
Let h =
2
i=1 h2
i /ℓ2
i + d2 − d
2
, where hi := xi − yi , i = 1, 2, 3,
ℓi are cov. lengths and d = 1.
exponential cov(h) = σ2
· exp(−h),
The cov. matrix C ∈ Rn×n
, n = 652
.
ℓ1 ℓ2
C−CH 2
C 2
0.01 0.02 3e − 2
0.1 0.2 8e − 3
1 2 2.8e − 6
10 20 3.7e − 9
16. Sparse tensor decompositions of kernels
cov(x, y) = cov(x − y)
We want to approximate C ∈ RN×N
, N = nd
by
Cr =
r
k=1 V 1
k ⊗ ... ⊗ V d
k such that C − Cr ≤ ε.
The storage of C is O(N2
) = O(n2d
) and the storage of Cr is O(rdn2
).
To define V i
k use e.g. SVD.
Approximate all V i
k in the H-matrix format and become HKT format.
See basic arithmetics in [Hackbusch, Khoromskij, Tyrtyshnikov].
Assume f(x, y), x = (x1, x2), y = (y1, y2), then the equivalent approx.
problem is f(x1, x2; y1, y2) ≈
r
k=1 Φk (x1, y1)Ψk (x2, y2).
17. Numerical examples of tensor approximations
Gaussian kernel exp{−|x − y|2
} has the Kroneker rank 1.
The exponen. kernel e{
− |x − y|} can be approximated by a tensor
with low Kroneker rank
r 1 2 3 4 5 6 10
C−Cr ∞
C ∞
11.5 1.7 0.4 0.14 0.035 0.007 2.8e − 8
C−Cr 2
C 2
6.7 0.52 0.1 0.03 0.008 0.001 5.3e − 9
19. Application: covariance of the solution
For SPDE with stochastic RHS the eigenvalue problem and spectral
decom. look like
Cf fℓ = λℓfℓ, Cf = Φf Λf ΦT
f .
If we only want the covariance
Cu = (K ⊗ K)−1
Cf = (K−1
⊗ K−1
)Cf = K−1
Cf K−T
,
one may with the KLE of Cf = Φf Λf ΦT
f reduce this to
Cu = K−1
Cf K−T
= K−1
Φf ΛΦT
f K−T
.
20. Application: higher order moments
Let operator K be deterministic and
Ku(θ) =
α∈J
Ku(α)
Hα(θ) = ˜f(θ) =
α∈J
f(α)
Hα(θ), with
u(α)
= [u
(α)
1 , ..., u
(α)
N ]T
. Projecting onto each Hα obtain
Ku(α)
= f(α)
.
The KLE of f(θ) is
f(θ) = f +
ℓ
λℓφℓ(θ)fl =
ℓ α
λℓφ
(α)
ℓ Hα(θ)fl
=
α
Hα(θ)f(α)
,
where f(α)
= ℓ
√
λℓφ
(α)
ℓ fl .
21. Application: higher order moments
The 3-rd moment of u is
M
(3)
u = E
α,β,γ
u(α)
⊗ u(β)
⊗ u(γ)
HαHβHγ
=
α,β,γ
u(α)
⊗u(β)
⊗u(γ)
cα,β,γ,
cα,β,γ := E (Hα(θ)Hβ(θ)Hγ(θ)) = cα,β · γ!, and cα,β are constants
from the Hermitian algebra.
Using u(α)
= K−1
f(α)
= ℓ
√
λℓφ
(α)
ℓ K−1
fl and uℓ := K−1
fℓ, obtain
M
(3)
u =
p,q,r
tp,q,r up ⊗ uq ⊗ ur , where
tp,q,r := λpλqλr
α,β,γ
φ
(α)
p φ
(β)
q φ
(γ)
r cα,βγ.
23. Conclusion
◮ Covariance matrices allow data sparse low-rank approximations.
◮ With application of H-matrices
◮ we extend the class of covariance functions to work with,
◮ allows non-regular discretisations of the cov. function on large
spatial grids.
◮ Application of sparse tensor product allows computation of k-th
moments.
24. Plans for Feature
1. Convergence of the Lanczos method with H-matrices
2. Implement sparse tensor vector product for the Lanczos method
3. HKT idea for d ≥ 3 dimensions
