Linear and Quadratic Taylor ExpansionsΒΆ
The approximation of the model response
around a specific set
of input
parameters may be of interest. One may then substitute
for
its Taylor expansion at point
. Hence
is replaced with a first or second-order polynomial
whose evaluation is inexpensive, allowing the
analyst to apply the uncertainty propagation methods.
We consider the first and second order Taylor expansions around
.
Introducing a vector notation, the previous equation rewrites:
where:
is the vector model response evaluated at
;
is the current set of input parameters;
is the transposed Jacobian matrix evaluated at
.
Introducing a vector notation, the previous equation rewrites:
where
is the transposed Hessian matrix.