October 2004 - Seminar for Applied Mathematics

October 2004 - Seminar for Applied Mathematics

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14 Largest Eigenvalues of exp(−|x−x′|
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δ=0.25, predicted
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δ=0.50, predicted
δ=0.75, observed
δ=0.75, predicted
observed
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1st Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain
6th Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain
−0.1
1
−0.2
−0.3
0.5
−0.4
0
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−0.6
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−1
11th Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain
−1
16th Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain
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1.5
1
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0.5
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0
0
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1st Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain − clustering
6th Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain − clustering
−0.1
1.5
−0.2
1
−0.3
−0.4
0.5
−0.5
−0.6
0
−0.7
−0.8
−0.5
−0.9
−1
1
−1
1
1
0.5
1
0.5
0.5
0
0.5
0
0
−0.5
0
−0.5
−0.5
−1
−0.5
−1
−1
11th Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain − clustering
−1
16th Eigenfunction of C(x,x′) = exp(−4|x−x′|2) on L−shaped Domain − clustering
1.5
2
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
1
−2
1
1
0.5
1
0.5
0.5
0
0.5
0
0
−0.5
0
−0.5
−0.5
−0.5
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m
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eigenvalues decay of C (x,x′)=exp(−100|x−x′| ), x∈ L−shaped
adaptive polynomial degree r for C (x,x′)=exp(−100|x−x′| ), x∈ L−shaped
a
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0.031
10
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9
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0.029
7
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0
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fBf
deterministic solution on same mesh
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
pointwise (expected - deterministic)/deterministic
0.25
0.2
0.15
0.1
0.05
0
-0.1
1
1
0.5
-1
-0.5
x
0
0.5
0
-0.5
1
0.5
-1
y
-0.5
x
1 -1
0
0.5
0
-0.5
0.5
-1
y
-0.5
x
1 -1
0
0.5
0
-0.5
y
1 -1
-4*|x-x’|2
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expected value
Ea=3+x1, Va=e
, f=2*sin(x1+x2), l = 5 (2048 El.), r = 2,1,1,1,1,1,1,1,1,1,1,1,1 (12288 det. Prob.)
min and max values
autocorrelation
0.1
standard deviation
0.01
0.008
0.006
0.004
0.002
0
0.05
0
-0.05
-0.1
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
1
1
0.5
-1
-0.5
x
0
0.5
0
-0.5
1 -1
y
1
0.5
-1
-0.5
x
0
0.5
0
-0.5
1 -1
y
0.5
-1
-0.5
x
0
0.5
0
-0.5
1 -1
y
hˆf
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expected value
deterministic solution on same mesh
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0.04
0.02
0
-0.02
-0.04
0
0
0.2
0.4
x 0.6 0.8
1
0.8
0.6
0.4 y
0.2
pointwise (expected - deterministic)/deterministic
0
0.2
1 0
0.4
x 0.6 0.8
1
0.8
0.6
0.4 y
0.2
0
0.2
1 0
0.4
x 0.6 0.8
1
0.8
0.6
0.4 y
0.2
1 0
-|x-x’|2
Ea = 3+sin(pi*x_1), Va = e
, f = 1, l = 5 (2048 Elements), r = 4,2,2,1,1,1,1,1,1,1,1,1,1 (46080 det. Prob.)
autocorrelation
min and max values
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
standard deviation
0.0005
0.00045
0.0004
0.00035
0.0003
0.00025
0.0002
0.00015
0.0001
5e-05
0
0
0.2
0.4
x 0.6 0.8
1
0.8
0.6
0.4 y
0.2
1 0
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
0.2
0.4
x 0.6 0.8
1
0.8
0.6
0.4 y
0.2
1 0
0
0.2
0.4
x 0.6 0.8
1
0.8
0.6
0.4 y
0.2
1 0
pointwise (expected - overkill)/overkill
pointwise (expected - overkill)/overkill
M = 3 r = 1,1,1 γ = 0.2
M = 8 r = 1,1,1,1,1,1,1,1 γ = 0.2
-0.009
-0.002
-0.0025
-0.003
-0.0035
-0.004
-0.0045
-0.005
-0.0055
-0.006
-0.0065
-0.007
-0.0075
-0.01
-0.011
-0.012
-0.013
-0.014
-0.015
-0.016
-0.017
-0.018
-0.019
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0.3
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
0.7
0.9
1
1
0.9
0.8
0.7
0.6
0.5
x
y
0.4
0.3
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
0.7
0.9
x
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