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y
i
=
j
c
j
Φ
(|x
i
-x
j
|),
(2)
where y
i
=f(x
i
). The solution can be found by direct inversion of the moderately sized
Nexp*Nexp system matrix
(|x
i
-x
j
|). The result can be written in a form of
weighted sum f(x)=
i
w
i
(x)y
i
, with the weights
Φ
ij
=
Φ
-1
w
i
(x)=
j
Φ
Φ
(|x-x
j
|).
(3)
ij
A suitable choice for the RBF, providing non-degeneracy of
-matrix for all finite
datasets of distinct points and all dimensions n, is the multi-quadric function [5]
Φ
Φ
(r)=(b
2
+r
2
)
1/2
, where b is a constant defining smoothness of the function near data
point x=x
i
. RBF interpolation can also be combined with polynomial detrending,
adding a polynomial part p():
f(x)=
i=1..Nexp
c
i
Φ
(|x-x
i
|)+p(x).
(4)
This allows reconstructing exactly polynomial (including linear) dependencies and
generally improving precision of interpolation. The precision can be controlled via the
following cross-validation procedure: the data point is removed, data are interpolated
to this point and compared with the actual value at this point. For an RBF metamodel
this procedure leads to a direct formula [13-15]
(5)
-1
err
i
= f
interpol
(x
i
)-f
actual
(x
i
) = -c
i
/(
Φ
)
ii
.
Specifics of Bulky Data:
although RBF metamodel is directly applicable for
interpolation of multidimensional data, it contains one matrix-vector multiplication
f(x)=yw(x), comprising O(mNexp) floating point operations per every interpolation.
Here y
ij
, i=1..m, j=1..Nexp is the data matrix, where every column forms one
experiment, every row forms a data item varied in experiments.
Dimensional reduction technique applicable for acceleration of this computation is
provided by principal component analysis (PCA). At first, an average value is row-
wise subtracted, forming centered data matrix dy
ij
= y
ij
- <y
i
>. For this matrix a
singular value decomposition (SVD) is written: dy=U
V
T
, where
is a diagonal
matrix of size Nexp*Nexp, U is a column-orthogonal matrix of size m*Nexp, V an
orthogonal square matrix of size Nexp*Nexp:
U
T
U=1, V
T
V=VV
T
=1.
Λ
Λ
(6)
A computationally efficient method [14] for this decomposition in the case m>>Nexp
is to find Gram matrix G=dy
T
dy, to perform its spectral decomposition G=V
2
V
T
,
Λ
-1
and to compute the remaining U-matrix with post-multiplication U=dyV
Λ
. The
Λ
values are non-negative and sorted in non-ascending order. If all these values in the
range k>Nmod are omitted (set to zero), the resulting reconstruction of y-matrix will
have a deviation
δ
y. L
2
-norm of this deviation gives
err
2
=
ij
y
ij
2
=
k>Nmod
2
δ
Λ
(7)
k
(Parseval's criterion). This formula allows controlling precision of reconstructed y-
matrix. Usually
Λ
k
rapidly decreases with k, and a few first
Λ
values give sufficient
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