### Master degree course on

### Approximation Theory and Applications,

Lab exercises

Prof. Stefano De Marchi A.Y. 2017-18

1. Construct the Dubiner points on the square [−1, 1]^{2} and on the disk x^{2}+ y^{2}= 1.

2. At the CAA Research Group you find the software for making the so-called (Weakly) Admissible Meshes, (W)AM on various 1,2 and 3 dimensional domains (see the web page http://www.math.

unipd.it/~marcov/wam.html).

• On [0, 1]^{d}, d = 1, 2, 3, interpolate on the corresponding (W)AM the function
f (x) = 4^{d}

d

Y

k=1

x_{k}(1 − x_{k}), x = (x_{1}, . . . , x_{d}) ∈ [0, 1]^{d}

• Compute the Lebesgue constant for the WAMs of the previous point.

3. Consider the Franke function in 2 and 3 dimensions. The Matlab codes to define them are

% Franke function in 2d

term1 = 0.75 * exp(-(9*x1-2)^2/4 - (9*x2-2)^2/4);

term2 = 0.75 * exp(-(9*x1+1)^2/49 - (9*x2+1)/10);

term3 = 0.5 * exp(-(9*x1-7)^2/4 - (9*x2-3)^2/4);

term4 = -0.2 * exp(-(9*x1-4)^2 - (9*x2-7)^2);

y = term1 + term2 + term3 + term4;

or simply use the built-in function z=franke(x,y), with x,y two-dimensional Matlab grids.

% A version of Franke function in 3d

term1 = 0.75 * exp(-(9*x1-2)^2/4 - (9*x2-2)^2/4- (9*x3-2)^2/4);

term2 = 0.75 * exp(-(9*x1+1)^2/49 - (9*x2+1)/10-(9*x3+1)^2/10);

term3 = 0.5 * exp(-(9*x1-7)^2/4 - (9*x2-3)^2/4 - (9*x3-5)^2/4);

term4 = -0.2 * exp(-(9*x1-4)^2 - (9*x2-7)^2- (9*x3-5)^2);

y = term1 + term2 + term3 + term4;

Then, compute for the Franke function f its integral Z

[0,1]^{d}

f (x)dx, d = 2, 3 ,

using the Monte Carlo (random points) and the Quasi Monte Carlo (Halton and Hammersly points) for different increasing values of the number of points. Compare the results with the built-in functions quad2d and quad3d.

About the Hammersly points, you can produce them by the corresponding function found in the Matlab File Exchange, toolbox SNTO (Sequential Number Theoretical Optimization package).

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4. Plot the following strictly positive definite RBF, centered at the origin

• Gaussian - Laguerre for n = 1, 2 and s = 1, 2

s n = 1 n = 2

1 (3/2 − x^{2})e^{−x}^{2} (15/8 − 5/2 x^{2}+ 1/2x^{4})e^{−x}^{2}
2 (2 − kxk^{2})e^{−kxk}^{2} (3 − 3kxk^{2}+ 1/2kxk^{4})e^{−kxk}^{2}
for s = 1, x ∈ [−1, 1] while for s = 2, x ∈ [−1, 1]^{2}.

• Poisson functions for s = 2, 3, 4 in [−1, 1]^{2} using the shape parameter = 10 (to be introduced
in the below definitions)

s = 2 s = 3 s = 4

J0(kxk) r2

π

sin(kxk) kxk

J1(kxk) kxk

where J_{p} is the Bessel function of first kind and order p (in Matlab besselj(p,z) where z is an
evaluation vector of points).

• Mat´ern functions in [−1, 1]^{2}, for three values of β (again with = 10)
β1 = ^{s+1}_{2} β2= ^{s+3}_{2} β3= ^{s+5}_{2}

e^{−kxk} (1 + kxk) e^{−kxk} 3 + 3kxk + kxk^{2}
e^{−kxk}

Notice that Mat´ern for β1 is not differantiable at the origin. For β2 is C^{2}(R^{s}) and for β3is C^{4}(R^{s}).

• Generalized inverse multiquadrics Φ(x) = (1 + kxk^{2})^{−β}, s < 2β, in [−1, 1]^{2} (always using = 5)
for β = 1/2 (Hardy) and β = 1 (inverse quadrics).

• Truncated powers Φ(x) = (1 − kxk)^{l}_{+} when l = 2, 4 (in [−1, 1]^{2}).

• Whittaker’s potentials in [−1, 1]^{2}, for the following α, k e β

α k = 2 k = 3

0 β − kxk + kxke^{−β/kxk}
β^{2}

β^{2}− 2βkxk + 2kxk^{2}− 2kxk^{2}e^{−β/kxk}
β^{3}

1 β − 2kxk + (β + 2kxk)e^{−β/kxk}
β^{3}

β^{2}− 4βkxk + 6kxk^{2}− (2βkxk + 6kxk^{2})e^{−β/kxk}
β^{4}

Note: For the plots take β = 1.

5. Construct the RBF interpolant of the Franke functions on a grid 20 × 20 of Chebyshev points with Poisson and Mat´ern RBF. Compute also the RMSE (Root Mean Square Error)

6. Plot the most important Conditionally Positive Definite RBF

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• Generalized multiquadrics

Φ(x) = (1 + kxk^{2})^{β}, x ∈ R^{s}, β ∈ R\N0

with β = 1/2 (Hardy) we get SCDP of order 1 while with β = 5/2 we get a SCDP of order 3.

• Power functions (shape parameter free!)

Φ(x) = kxk^{β}, x ∈ R^{s}, 0 < β /∈ 2N

For β = 3 we get a SCPD of order 2 while for β = 5 a SCPD of order 3.

• thin-plate splines (shape parameter free!)

Φ(x) = kxk^{2β}log kxk, x ∈ R^{s}, β ∈ N

The “classic” one is for β = 1 (SCPD of order 2), while for β = 2 we get a function SCPD of order 3. Verify that also these functions are shape parameter free.

• The Matlab functions necessary for interpolation can be dowloaded at the link http://www.math.unipd.it/∼demarchi/TAA2010.

7. Plot the power function for two kernels, one PD and one CPD, say Φ_{1}, Φ_{2} and scattered centers X on
2 and 3 dimensions.

8. Implement the Shepard’s method with orders 1 up to 3, in 1 and 2 variables.

9. Complete the function PU.m of the toolbox PUM (available at the website of the course) and then run MAIN PU.m with different shape parameters.

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