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Tangent Linear Validation
Principle
The tangent validation test checks that the Tangent Linear model L is a first order approximation of the model M. We can then write a second order Taylor expansion as follows:
M(X0+p.dX) = M(X0) + L(p.dX) + e
with,
X0: initial state vector dX: perturbation state vector p: scale factor e: residu
when the factor p tends to zero we have:
-1- M(X0+p.dX) - M(X0) tends to L(p.dX) and -2- e is equivalent to O(p^2), means Norm(e,.) <= K.p^2 (with K a constant)
This means that for a given p0, for p smaller than p0 the error of the Linear Tangent with respect to the model is decreasing as p2.
First order validation
Let's note
Np = M(X0+p.dX) - M(X0) Lp = L(p.dX)
the first order validation checks:
for a given p0, for all p>p0: Ep = Norm(Np,.)/Norm(Lp,.) tends to 1
Second order validation
the second order validation ensure that the residu is controlled. We check then:
for a given p0, for all p>p0: Rp = Norm(Np - Lp,.) behaves as p^2
note that we can also perform the following equivalent tests:
test | expression | behaviour |
b | Norm((Np - Lp)/Lp,.) | p |
c | Norm((Ep - 1,.)/p | constant |
d | Norm((Ep - Lp/(p.Lp),.) | constant |
Implementation
... work in progress ...