Changes between Version 1 and Version 2 of Working Groups/TAM/Reference Manual/testing/tangent


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Timestamp:
2010-02-21T15:07:17+01:00 (11 years ago)
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vigilant
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  • Working Groups/TAM/Reference Manual/testing/tangent

    v1 v2  
    11 
    2 Still to write... 
     2Last edited [[Timestamp]] 
     3= ''' Tangent Linear Validation ''' = 
     4 
     5[[PageOutline]] 
     6 
     7== Principle == 
     8The 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: 
     9 
     10{{{ 
     11M(X0+p.dX) = M(X0) + L(p.dX) + e 
     12}}} 
     13with, 
     14{{{ 
     15X0: initial state vector 
     16dX: perturbation state vector 
     17p: scale factor 
     18e: residu  
     19}}} 
     20when the factor '''p''' tends to zero we have: 
     21{{{ 
     22-1- M(X0+p.dX) - M(X0) tends to L(p.dX) 
     23and 
     24-2- e is equivalent to O(p^2), means Norm(e,.) <= K.p^2 (with K a constant) 
     25}}} 
     26 
     27This 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 p^2. 
     28 
     29=== First order validation === 
     30Let's note  
     31{{{ 
     32Np = M(X0+p.dX) - M(X0) 
     33Lp = L(p.dX) 
     34}}} 
     35the first order validation checks: 
     36{{{ 
     37for a given p0, for all p>p0: 
     38Ep = Norm(Np,.)/Norm(Lp,.) tends to 1 
     39}}} 
     40 
     41=== Second order validation === 
     42the second order validation ensure that the residu is controlled. We check then: 
     43{{{ 
     44for a given p0, for all p>p0: 
     45Rp = Norm(Np - Lp,.) behaves as p^2 
     46}}} 
     47 
     48note that we can also perform the following equivalent tests: 
     49|| test || expression             || behaviour || 
     50||b     ||Norm((Np - Lp)/Lp,.)    || p         || 
     51||c     ||Norm((Ep - 1,.)/p       || constant  || 
     52||d     ||Norm((Ep - Lp/(p.Lp),.) || constant  || 
     53 
     54== Implementation == 
     55 
     56... work in progress ...