Euclidean space

All the errors of updated models with the Kriging model are below 10%, while the error 2-norm of the Kriging model is higher than GA.

According to the definition of the 2-norm, we have [parallel][I.sub.n] - B[B.sub.+][parallel] = 1 and

Of course, the proximal map of the 2-norm is easily computed by other means; our purpose here is to use this as a building block for more useful penalties such as Example 3.1, which involves the sum-of-norms function shown in (3.3).

Owing to the fact that {[??]} is scaled by its 2-norm, {[??]} can be rewritten as

where [[parallel]*[parallel].sub.2] denotes matrix 2-norm.

Where B = ([b.sub.1], ..., [b.sub.m]) = H[([H.sup.T]H).sup.-]1 [H.sub.T]-I,B' = ([b.sub.i1], ...,[b.sub.ik]),a = [([a.sub.i1], ...,[a.sub.ik]).sup.T], and K is a vector whose 2-Norm is less than [[tau].sub.a].

In Equation (3) the operator [parallel][parallel] indicates the 2-norm of the matrix, and inv is the inverse matrix operator.

is called an 2-norm on X and the pair (X, [parallel].,.[parallel]) is called an 2-normed space.

4) Select the number of modes (POD basis vectors) (P) to be employed in the reconstruction by considering the relative "energy" (measured by the 2-norm) captured by the ith basis vector ([[lambda].sub.i] / [[summation].sup.M.sub.i=1] [[lambda].sub.i]).

p-norm distance = [([n.summation over (i=1)][[absolute value of [x.sub.i] - [y.sub.i]].sup.p]).sup.1/p] In the Euclidean space [R.sup.n], the distance between two points is usually given by the Euclidean distance (2-norm distance).

In the Euclidean space, the distance between two points is given by the Euclidean distance (2-norm distance).

Then [parallel].,.[parallel] is called a 2-norm on X and the pair (X,[parallel].,.[parallel]) is called a 2-normed space.

Two performance measures, the normalized 2-norm and infinite-norm, were applied to the model prediction errors in nozzle pressure [P.sub.2] and screw position x.

It is well known that [[Sigma].sub.[Lambda]] is the 2-norm of the perturbation matrix E of smallest 2-norm for which [Lambda] is an exact eigenvalue of H + E.