The Reduced Basis Method Problem Set 1

for some unique choice of β j ∈ R, j = 1, . . . , N. (We implicitly assume that the ζ i , i = 1, . . . , N,
are linearly independent; it follows that W N is an N-dimensional subspace of X N .)
In the reduced-basis approach we look for an approximation u N (µ) to u N (µ) (which for our
purposes here we presume is arbitrarily close to u e (µ)) in W N ; in particular, we express u N (µ) as
u N (µ) =
N∑
u j N ζj ; (10)
j=1
we denote by u N (µ) ∈ R N the coefficient vector (u 1 N , . . . , uN N )T . The premise — or hope — is that
we should be able to accurately represent the solution at some new point in parameter space, µ, as
an appropriate linear combination of solutions previously computed at a small number of points in
parameter space (the µ i , i = 1, . . . , N). But how do we find this appropriate linear combination?
And how good is it? And how do we compute our approximation efficiently?
The energy principle is crucial here (though more generally the weak form would suffice). To
wit, we apply the classical Rayleigh-Ritz procedure to define
u N (µ) = arg
min J(w N ); (11)
w N ∈W N
alternatively we can apply Galerkin projection to obtain the equivalent statement
The output can then be calculated from
We now study this approximation in more detail.
α) Prove that, in the energy norm || · || ≡ (a(·, ·; µ)) 1/2 ,
a(u N (µ), v; µ) = l(v), ∀v ∈ W N . (12)
T root N (µ) = l O (u N (µ)). (13)
||u(µ) − u N (µ)|| ≤ ||u(µ) − w N ||, ∀w N ∈ W N . (14)
This inequality indicates that out of all the possible choices of w N in the space W N , the reducedbasis
method defined above will choose the “best one” (in the energy norm). Equivalently, we can
say that even if we knew the solution u(µ), we would not be able to find a better approximation to
u(µ) in W N — in the energy norm — than u N (µ).
β) Prove that
T root (µ) − T root N (µ) = ||u(µ) − u N (µ)|| 2 . (15)
γ) Show that u N (µ) as defined in (10–12) satisfies a set of N × N linear equations,
A N (µ)u N (µ) = F N ;
and that
T root N (µ) = L T N u N (µ).
Give expressions for A N (µ) ∈ R N×N in terms of A N (µ) and Z, F N ∈ R N in terms of F N and Z,
and L N ∈ R N in terms of L N and Z; here Z is an N × N matrix, the j th column of which is u N (µ j )
(the nodal values of u N (µ j )).
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