Download - Lecture 7 Backus-Gilbert Generalized Inverse and the Trade Off of Resolution and Variance
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Lecture 7
Backus-Gilbert Generalized Inverse and the Trade Off of Resolution and
Variance
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SyllabusLecture 01 Describing Inverse ProblemsLecture 02 Probability and Measurement Error, Part 1Lecture 03 Probability and Measurement Error, Part 2 Lecture 04 The L2 Norm and Simple Least SquaresLecture 05 A Priori Information and Weighted Least SquaredLecture 06 Resolution and Generalized InversesLecture 07 Backus-Gilbert Inverse and the Trade Off of Resolution and VarianceLecture 08 The Principle of Maximum LikelihoodLecture 09 Inexact TheoriesLecture 10 Nonuniqueness and Localized AveragesLecture 11 Vector Spaces and Singular Value DecompositionLecture 12 Equality and Inequality ConstraintsLecture 13 L1 , L∞ Norm Problems and Linear ProgrammingLecture 14 Nonlinear Problems: Grid and Monte Carlo Searches Lecture 15 Nonlinear Problems: Newton’s Method Lecture 16 Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals Lecture 17 Factor AnalysisLecture 18 Varimax Factors, Empircal Orthogonal FunctionsLecture 19 Backus-Gilbert Theory for Continuous Problems; Radon’s ProblemLecture 20 Linear Operators and Their AdjointsLecture 21 Fréchet DerivativesLecture 22 Exemplary Inverse Problems, incl. Filter DesignLecture 23 Exemplary Inverse Problems, incl. Earthquake LocationLecture 24 Exemplary Inverse Problems, incl. Vibrational Problems
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Purpose of the Lecture
Introduce a new way to quantify the spread of resolution
Find the Generalized Inverse that minimizes this spread
Include minimization of the size of variance
Discuss how resolution and variance trade off
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Part 1
A new way to quantify thespread of resolution
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criticism of Direchlet spread() functions
when m represents m(x)is that they don’t capture the sense of
“being localized” very well
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These two rows of the model resolution matrix have the same Direchlet spread …
Rij Rijindex, j index, ji i… but the left case is better “localized”
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old way
new way
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old way
new way
add this function
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old way
new wayBackus-Gilbert
Spread Function
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grows with physical distance between model
parameters
zero when i=j
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if w(i,i)=0 then
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for one spatial dimensionm is discretized version of m(x)m = [m(Δx), m(2 Δx), … m(M Δx) ]T
w(i,j) = (i-j)2 would work fine
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for two spatial dimensionm is discretized version of m(x,y)on K⨉L grid
m = [m(x1, y1), m(x1, y2), … m(xK, yL) ]Tw(i,j) = (xi-xj) 2 + (yi-yj) 2
would work fine
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Part 2
Constructing a Generalized Inverse
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under-determined problem
find G-g that minimizes spread(R)
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under-determined problem
find G-g that minimizes spread(R)with the constraint
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under-determined problem
find G-g that minimizes spread(R)with the constraint
(since Rii is not constrained by
spread function)
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once again, solve for each row of G-g separately
spread of kth row of resolution matrix R
with
symmetric in i,j
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for the constraint
with
[1]k=
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Lagrange Multiplier Equation
now set ∂Φ/∂G-gkp
+
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S g(k) – λu = 0with g(k)T the kth row of G-g
solve simultaneously withuT g(k) = 1
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putting the two equations together
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construct inverse
A, b, c unknown
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construct inverse
so
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=g(k) =g(k)T = uT
kth row of G-g
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Backus-Gilbert Generalized Inverse(analogous to Minimum Length Generalized Inverse)
written in terms of components
with
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=
=
i
j
j
i0 2 4 6 8 10
-0.2
0
0.2
0.4
0.6
0.8
z
m
0 2 4 6 8 10-0.2
0
0.2
0.4
0.6
0.8
z
m
(A) (B)
(C) (D)
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Part 3
Include minimization of the size of variance
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minimize
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new version of Jk
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with
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with
same
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so adding size of varianceis just a small modification to S
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Backus-Gilbert Generalized Inverse(analogous to Damped Minimum Length)
written in terms of components
with
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In MatLab
GMG = zeros(M,N);
u = G*ones(M,1);
for k = [1:M]
S = G * diag(([1:M]-k).^2) * G';
Sp = alpha*S + (1-alpha)*eye(N,N);
uSpinv = u'/Sp;
GMG(k,:) = uSpinv / (uSpinv*u);
end
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$20 Reward!
to the first person who sends me MatLab code that computes the BG generalized
inverse without a for loop
(but no creation of huge 3-indexed quantities, please. Memory requirements
need to be similar to my code)
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The Direchlet analog of the
Backus-Gilbert Generalized Inverse
is the
Damped Minimum LengthGeneralized Inverse
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0
special case #2
1 ε2
G-g[GGT+ε2I] =GTG-g=GT [GGT+ε2I]-1
I
damped minimum length
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Part 4
the trade-off of resolution and variance
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the value ofa localized average with small spread
is controlled by few dataand so has large variance
the value ofa localized average with large spread
is controlled by many dataand so has small variance
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(A) (B)
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small spreadlarge variance
large spreadsmall variance
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α near 1small spread
large variance
α near 0 large spread
small variance
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2 2.5 3 3.5
-2
0
2
4
log10 spread(R)
log1
0 si
ze(C
m)
90 95 100-3
-2
-1
0
1
2
3
4
spread(R)
size
(Cm
)
(A) (B)
α=0
α=1
α=½
α=0
α=1
α=½
log10 spread of model resolution spread of model resolution
log 10
siz
e of
cov
aria
nce
log 10
siz
e of
cov
aria
nce
Trade-Off Curves