![4*
Two variants
Either we assume that matrix of snapshots is given
[q(x, θ1), ..., q(x, θnq )]
Or we assume that the covariance function is of a certain type:
The Mat´ern class of covariance functions is defined as
C(r) := Cν, (r) =
2σ2
Γ(ν)
r
2
ν
Kν
r
, (1)
Center for Uncertainty
Quantification
tion Logo Lock-up
2 / 12](https://image.slidesharecdn.com/talklitvinenkopriorcov-161029105144/85/Talk-litvinenko-prior_cov-2-320.jpg)
![4*
Comparison
[q(x, θ1), ..., q(x, θnq )] ≈ ABT
.
n rank k size, MB t, sec. ε max
i=1..10
|λi − ˜λi |, i ε2
for ˜C C ˜C C ˜C
4.0 · 103
10 48 3 0.8 0.08 7 · 10−3
7.0 · 10−2
, 9 2.0 · 10−4
1.05 · 104
18 439 19 7.0 0.4 7 · 10−4
5.5 · 10−2
, 2 1.0 · 10−4
2.1 · 104
25 2054 64 45.0 1.4 1 · 10−5
5.0 · 10−2
, 9 4.4 · 10−6
Table : Accuracy of H-matrix approximation, l1 = l3 = 0.1, l2 = 0.5.
Center for Uncertainty
Quantification
tion Logo Lock-up
5 / 12](https://image.slidesharecdn.com/talklitvinenkopriorcov-161029105144/85/Talk-litvinenko-prior_cov-5-320.jpg)
![4*
Convergence of KLD with increasing the rank k
k KLD C − CH
2 C(CH
)−1
− I 2
L = 0.25 L = 0.75 L = 0.25 L = 0.75 L = 0.25 L = 0.75
5 0.51 2.3 4.0e-2 0.1 4.8 63
6 0.34 1.6 9.4e-3 0.02 3.4 22
8 5.3e-2 0.4 1.9e-3 0.003 1.2 8
10 2.6e-3 0.2 7.7e-4 7.0e-4 6.0e-2 3.1
12 5.0e-4 2e-2 9.7e-5 5.6e-5 1.6e-2 0.5
15 1.0e-5 9e-4 2.0e-5 1.1e-5 8.0e-4 0.02
20 4.5e-7 4.8e-5 6.5e-7 2.8e-7 2.1e-5 1.2e-3
50 3.4e-13 5e-12 2.0e-13 2.4e-13 4e-11 2.7e-9
Table : Dependence of KLD on the approximation H-matrix rank k,
Matern covariance with parameters L = {0.25, 0.75} and ν = 0.5,
domain G = [0, 1]2
, C(L=0.25,0.75) 2 = {212, 568}.
Center for Uncertainty
Quantification
tion Logo Lock-up
9 / 12](https://image.slidesharecdn.com/talklitvinenkopriorcov-161029105144/85/Talk-litvinenko-prior_cov-9-320.jpg)
![4*
Convergence of KLD with increasing the rank k
k KLD C − CH
2 C(CH
)−1
− I 2
L = 0.25 L = 0.75 L = 0.25 L = 0.75 L = 0.25 L = 0.75
5 nan nan 0.05 6e-2 2.1e+13 1e+28
10 10 10e+17 4e-4 5.5e-4 276 1e+19
15 3.7 1.8 1.1e-5 3e-6 112 4e+3
18 1.2 2.7 1.2e-6 7.4e-7 31 5e+2
20 0.12 2.7 5.3e-7 2e-7 4.5 72
30 3.2e-5 0.4 1.3e-9 5e-10 4.8e-3 20
40 6.5e-8 1e-2 1.5e-11 8e-12 7.4e-6 0.5
50 8.3e-10 3e-3 2.0e-13 1.5e-13 1.5e-7 0.1
Table : Dependence of KLD on the approximation H-matrix rank k,
Matern covariance with parameters L = {0.25, 0.75} and ν = 1.5,
domain G = [0, 1]2
, C(L=0.25,0.75) 2 = {720, 1068}.
Center for Uncertainty
Quantification
tion Logo Lock-up
10 / 12](https://image.slidesharecdn.com/talklitvinenkopriorcov-161029105144/85/Talk-litvinenko-prior_cov-10-320.jpg)
Talk litvinenko prior_cov
- 1. Initial covariance matrix for the Kalman Filter Alexander Litvinenko Group of Raul Tempone, SRI UQ, and Group of David Keyes, Extreme Computing Research Center KAUST Center for Uncertainty Quantification ntification Logo Lock-up http://sri-uq.kaust.edu.sa/
- 2. 4* Two variants Either we assume that matrix of snapshots is given [q(x, θ1), ..., q(x, θnq )] Or we assume that the covariance function is of a certain type: The Mat´ern class of covariance functions is defined as C(r) := Cν, (r) = 2σ2 Γ(ν) r 2 ν Kν r , (1) Center for Uncertainty Quantification tion Logo Lock-up 2 / 12
- 3. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 Matern covariance (nu=1) σ=0.5, l=0.5 σ=0.5, l=0.3 σ=0.5, l=0.2 σ=0.5, l=0.1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 nu=0.15 nu=0.3 nu=0.5 nu=1 nu=2 nu=30 Figure : Matern function for different parameters (computed in sglib). Center for Uncertainty Quantification tion Logo Lock-up 3 / 12
- 4. 4* Types of Matern covariance Cν=3/2(r) = − √ 2νr Γ(p + 1) Γ(2p + 1) p i=0 (p + i)! i!(p − i)! ( √ 8νr )p−i . (2) The most interesting cases are ν = 3/2: Cν=3/2(r) = 1 + √ 3r exp − √ 3r (3) and ν = 5/2, for which Cν=5/2(r) = 1 + √ 5r + 5r2 3 2 exp − √ 5r (4) Center for Uncertainty Quantification tion Logo Lock-up 4 / 12
- 5. 4* Comparison [q(x, θ1), ..., q(x, θnq )] ≈ ABT . n rank k size, MB t, sec. ε max i=1..10 |λi − ˜λi |, i ε2 for ˜C C ˜C C ˜C 4.0 · 103 10 48 3 0.8 0.08 7 · 10−3 7.0 · 10−2 , 9 2.0 · 10−4 1.05 · 104 18 439 19 7.0 0.4 7 · 10−4 5.5 · 10−2 , 2 1.0 · 10−4 2.1 · 104 25 2054 64 45.0 1.4 1 · 10−5 5.0 · 10−2 , 9 4.4 · 10−6 Table : Accuracy of H-matrix approximation, l1 = l3 = 0.1, l2 = 0.5. Center for Uncertainty Quantification tion Logo Lock-up 5 / 12
- 6. 4* Storage cost and computing time k size, MB t, sec. 1 1548 33 2 1865 42 3 2181 50 4 2497 59 6 nem - k size, MB t, sec. 4 463 11 8 850 22 12 1236 32 16 1623 43 20 nem - Table : Dependence of the computing time and storage requirement on the H-matrix rank k, l1 = 0.1, l2 = 0.5, n = 2.3 · 105 . (right) l1 = 0.1, l2 = 0.5, l3 = 0.1, n = 4.61 · 105 . Center for Uncertainty Quantification tion Logo Lock-up 6 / 12
- 7. 4* Examples of H-matrix approximation 25 20 20 20 20 16 20 16 20 20 16 16 20 16 16 16 19 20 20 19 32 19 19 16 16 32 19 20 20 19 19 16 19 16 32 32 20 20 20 20 32 32 32 20 19 19 19 32 20 19 16 16 32 32 20 32 32 20 32 32 32 32 20 32 32 20 19 19 19 20 16 19 16 32 32 20 32 32 32 32 32 20 32 32 32 20 32 20 20 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 20 20 19 20 20 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 20 20 19 20 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 20 20 20 20 32 32 32 20 19 20 19 32 32 32 20 20 19 19 32 32 32 20 20 20 20 32 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 20 20 20 2019 20 20 20 32 32 32 32 20 32 32 20 32 32 32 32 20 32 32 20 20 20 20 20 20 20 20 20 20 20 32 20 32 32 20 20 20 20 20 20 20 20 20 20 20 32 20 32 32 20 20 20 20 20 20 20 20 20 20 20 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 32 20 20 20 20 20 20 20 20 19 20 20 20 32 32 32 20 32 32 32 32 32 20 32 32 32 20 32 20 20 20 20 20 20 20 20 20 20 20 32 32 20 32 20 20 20 20 20 20 20 20 20 20 20 32 32 20 32 20 20 20 20 20 20 20 20 20 20 20 32 32 20 32 32 32 20 32 32 32 20 32 32 32 20 32 20 20 20 20 20 20 20 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 19 19 20 20 32 32 32 32 20 32 32 20 32 32 32 32 20 32 32 19 20 19 20 32 32 32 20 32 32 32 32 32 20 32 32 32 20 32 20 20 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 20 32 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 32 32 32 32 20 20 20 32 20 32 32 20 20 20 32 32 20 32 20 20 20 32 32 32 32 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 20 32 32 32 20 32 20 32 20 32 32 32 20 32 20 32 32 20 32 32 32 20 32 32 32 20 20 32 32 20 20 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 25 9 9 20 9 9 20 7 7 16 9 9 20 9 9 20 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 20 9 9 20 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 20 9 9 20 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 9 9 32 Center for Uncertainty Quantification tion Logo Lock-up 7 / 12
- 8. 4* Kullback-Leibler divergence (KLD) DKL(P Q) is measure of the information lost when distribution Q is used to approximate P: DKL(P Q) = i P(i) ln P(i) Q(i) , DKL(P Q) = ∞ −∞ p(x) ln p(x) q(x) dx, where p, q densities of P and Q. For miltivariate normal distributions (µ0, Σ0) and (µ1, Σ1) 2DKL(N0 N1) = tr(Σ−1 1 Σ0)+(µ1 −µ0)T Σ−1 1 (µ1 −µ0)−k −ln det Σ0 det Σ1 Center for Uncertainty Quantification tion Logo Lock-up 8 / 12
- 9. 4* Convergence of KLD with increasing the rank k k KLD C − CH 2 C(CH )−1 − I 2 L = 0.25 L = 0.75 L = 0.25 L = 0.75 L = 0.25 L = 0.75 5 0.51 2.3 4.0e-2 0.1 4.8 63 6 0.34 1.6 9.4e-3 0.02 3.4 22 8 5.3e-2 0.4 1.9e-3 0.003 1.2 8 10 2.6e-3 0.2 7.7e-4 7.0e-4 6.0e-2 3.1 12 5.0e-4 2e-2 9.7e-5 5.6e-5 1.6e-2 0.5 15 1.0e-5 9e-4 2.0e-5 1.1e-5 8.0e-4 0.02 20 4.5e-7 4.8e-5 6.5e-7 2.8e-7 2.1e-5 1.2e-3 50 3.4e-13 5e-12 2.0e-13 2.4e-13 4e-11 2.7e-9 Table : Dependence of KLD on the approximation H-matrix rank k, Matern covariance with parameters L = {0.25, 0.75} and ν = 0.5, domain G = [0, 1]2 , C(L=0.25,0.75) 2 = {212, 568}. Center for Uncertainty Quantification tion Logo Lock-up 9 / 12
- 10. 4* Convergence of KLD with increasing the rank k k KLD C − CH 2 C(CH )−1 − I 2 L = 0.25 L = 0.75 L = 0.25 L = 0.75 L = 0.25 L = 0.75 5 nan nan 0.05 6e-2 2.1e+13 1e+28 10 10 10e+17 4e-4 5.5e-4 276 1e+19 15 3.7 1.8 1.1e-5 3e-6 112 4e+3 18 1.2 2.7 1.2e-6 7.4e-7 31 5e+2 20 0.12 2.7 5.3e-7 2e-7 4.5 72 30 3.2e-5 0.4 1.3e-9 5e-10 4.8e-3 20 40 6.5e-8 1e-2 1.5e-11 8e-12 7.4e-6 0.5 50 8.3e-10 3e-3 2.0e-13 1.5e-13 1.5e-7 0.1 Table : Dependence of KLD on the approximation H-matrix rank k, Matern covariance with parameters L = {0.25, 0.75} and ν = 1.5, domain G = [0, 1]2 , C(L=0.25,0.75) 2 = {720, 1068}. Center for Uncertainty Quantification tion Logo Lock-up 10 / 12
- 11. 4* Application of large covariance matrices 1. Kriging estimate ˆs := Csy C−1 yy y 2. Estimation of variance ˆσ, is the diagonal of conditional cov. matrix Css|y = diag Css − Csy C−1 yy Cys , 3. Gestatistical optimal design ϕA := n−1traceCss|y , ϕC := cT Css − Csy C−1 yy Cys c, Center for Uncertainty Quantification tion Logo Lock-up 11 / 12
- 12. 4* Mean and variance in the rank-k format u := 1 Z Z i=1 ui = 1 Z Z i=1 A · bi = Ab. (5) Cost is O(k(Z + n)). C = 1 Z − 1 WcWT c ≈ 1 Z − 1 UkΣkΣT k UT k . (6) Cost is O(k2(Z + n)). Lemma: Let W − Wk 2 ≤ ε, and uk be a rank-k approximation of the mean u. Then a) u − uk ≤ ε√ Z , b) C − Ck ≤ 1 Z−1ε2. Center for Uncertainty Quantification tion Logo Lock-up 12 / 12