Post on 13-Apr-2017
Initial covariance matrix for the Kalman Filter
Alexander LitvinenkoGroup of Raul Tempone, SRI UQ, and Group of David Keyes,
Extreme Computing Research Center KAUST
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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 Matern class of covariance functions is defined as
C (r) := Cν,`(r) =2σ2
Γ(ν)
( r
2`
)νKν( r`
), (1)
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−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
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 20
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 UncertaintyQuantification
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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)
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Comparison
[q(x ,θ1), ..., q(x ,θnq)] ≈ ABT .
n rank k size, MB t, sec. ε maxi=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.
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Storage cost and computing time
k size, MB t, sec.
1 1548 332 1865 423 2181 504 2497 596 nem -
k size, MB t, sec.
4 463 118 850 22
12 1236 3216 1623 4320 nem -
Table : Dependence of the computing time and storage requirement onthe 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.
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Examples of H-matrix approximation
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Kullback-Leibler divergence (KLD)
DKL(P‖Q) is measure of the information lost when distribution Qis used to approximate P:
DKL(P‖Q) =∑i
P(i) lnP(i)
Q(i), DKL(P‖Q) =
∫ ∞−∞
p(x) lnp(x)
q(x)dx ,
where p, q densities of P and Q. For miltivariate normaldistributions (µ0,Σ0) and (µ1,Σ1)
2DKL(N0‖N1) = tr(Σ−11 Σ0)+(µ1−µ0)TΣ−11 (µ1−µ0)−k− ln
(det Σ0
det Σ1
)
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Convergence of KLD with increasing the rank k
k KLD ‖C− CH‖2 ‖C(CH)−1 − I‖2L = 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 4.8 636 0.34 1.6 9.4e-3 0.02 3.4 228 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 3.112 5.0e-4 2e-2 9.7e-5 5.6e-5 1.6e-2 0.515 1.0e-5 9e-4 2.0e-5 1.1e-5 8.0e-4 0.0220 4.5e-7 4.8e-5 6.5e-7 2.8e-7 2.1e-5 1.2e-350 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}.
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Convergence of KLD with increasing the rank k
k KLD ‖C− CH‖2 ‖C(CH)−1 − I‖2L = 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+2810 10 10e+17 4e-4 5.5e-4 276 1e+1915 3.7 1.8 1.1e-5 3e-6 112 4e+318 1.2 2.7 1.2e-6 7.4e-7 31 5e+220 0.12 2.7 5.3e-7 2e-7 4.5 7230 3.2e-5 0.4 1.3e-9 5e-10 4.8e-3 2040 6.5e-8 1e-2 1.5e-11 8e-12 7.4e-6 0.550 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}.
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Application of large covariance matrices
1. Kriging estimate s := CsyC−1yy y
2. Estimation of variance σ, is the diagonal of conditional cov.matrix Css|y = diag
(Css − CsyC−1yy Cys
),
3. Gestatistical optimal design ϕA := n−1traceCss|y ,
ϕC := cT(Css − CsyC−1yy Cys
)c ,
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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 − 1WcW
Tc ≈
1
Z − 1UkΣkΣT
k UTk . (6)
Cost is O(k2(Z + n)).Lemma: Let ‖W −Wk‖2 ≤ ε, and uk be a rank-k approximationof the mean u. Then a) ‖u− uk‖ ≤ ε√
Z,
b) ‖C− Ck‖ ≤ 1Z−1ε
2.
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