228
9 High-Dimension Model Representation via Sparse GridTechniques
• Global Grids: suitable for globally smooth functions. Quadrature is based on
a number of rules, including Clenshaw-Curtis, and interpolation is based on
global Lagrange polynomials. Nodal point selection follows the same rules as
quadrature. These grids are most suitable for our use, and will be discussed in
more detail below.
• Local Polynomial Grids: suitable for non-smooth functions with locally sharp
behavior. Interpolation is based on hierarchical piecewise polynomials with local
support and user-specified order. These grids are suitable for local refinement.
• Wavelet Grids: are similar to local polynomials, except that it is assumed that
the order is either 1 or 3. When coupled with local refinement, these grids often
provide the same accuracy with fewer abscissas.
Lagrange Polynomial Interpolation Given nodal values, f (x i ), the LP interpolator is given by
f (x) =
N
i=0
l i (x)f (x i ) ,
(9.20)
where
l i (x) =
N
j =0
j =i
x − x j
x i − x j
, i = 0, . . . , N .
(9.21)
The interpolating polynomials, {l i (x)}, satisfy l i (x j ) = δ ij . An example for N = 3
is given here:
l 0 (x) =
x − x 1
x 0 − x 1
x − x 2
x 0 − x 2
x − x 3
x 0 − x 3
l 1 (x) =
x − x 0
x 1 − x 0
x − x 2
x 1 − x 2
x − x 3
x 1 − x 3
l 2 (x) =
x − x 0
x 2 − x 0
x − x 1
x 2 − x 1
x − x 3
x 2 − x 3
l 3 (x) =
x − x 0
x 3 − x 0
x − x 1
x 3 − x 1
x − x 2
x 3 − x 2
.
(9.22)
The nodal points, or knots, are located at the extrema (maxima or minima) of
Chebyshev polynomials. If m i > 1 is the number of knots at the ith level of
approximation in a given dimension, then the knots, over the interval [−1, +1],
are given by
x
i
j = − cos
π(j − 1)
m i − 1
, j = 1, . . . , m i ,
(9.23)
9 High-Dimension Model Representation via Sparse GridTechniques
• Global Grids: suitable for globally smooth functions. Quadrature is based on
a number of rules, including Clenshaw-Curtis, and interpolation is based on
global Lagrange polynomials. Nodal point selection follows the same rules as
quadrature. These grids are most suitable for our use, and will be discussed in
more detail below.
• Local Polynomial Grids: suitable for non-smooth functions with locally sharp
behavior. Interpolation is based on hierarchical piecewise polynomials with local
support and user-specified order. These grids are suitable for local refinement.
• Wavelet Grids: are similar to local polynomials, except that it is assumed that
the order is either 1 or 3. When coupled with local refinement, these grids often
provide the same accuracy with fewer abscissas.
Lagrange Polynomial Interpolation Given nodal values, f (x i ), the LP interpolator is given by
f (x) =
N
i=0
l i (x)f (x i ) ,
(9.20)
where
l i (x) =
N
j =0
j =i
x − x j
x i − x j
, i = 0, . . . , N .
(9.21)
The interpolating polynomials, {l i (x)}, satisfy l i (x j ) = δ ij . An example for N = 3
is given here:
l 0 (x) =
x − x 1
x 0 − x 1
x − x 2
x 0 − x 2
x − x 3
x 0 − x 3
l 1 (x) =
x − x 0
x 1 − x 0
x − x 2
x 1 − x 2
x − x 3
x 1 − x 3
l 2 (x) =
x − x 0
x 2 − x 0
x − x 1
x 2 − x 1
x − x 3
x 2 − x 3
l 3 (x) =
x − x 0
x 3 − x 0
x − x 1
x 3 − x 1
x − x 2
x 3 − x 2
.
(9.22)
The nodal points, or knots, are located at the extrema (maxima or minima) of
Chebyshev polynomials. If m i > 1 is the number of knots at the ith level of
approximation in a given dimension, then the knots, over the interval [−1, +1],
are given by
x
i
j = − cos
π(j − 1)
m i − 1
, j = 1, . . . , m i ,
(9.23)