Multivariate S2Fun

Structural conventions of the input and output of multivariate S2FunHarmonic

In this part we deal with multivariate functions of the form

$$f\colon \bf{S}^2\to \bf{R}^n$$ .

For example we got four nodes $v_1, v_2, v_3$ and $v_4$ and six functions $f_1, f_2, f_3, f_4, f_5$ and $f_6$, which we want to store in a 3x2 array, then the following scheme applies to function evaluations:

$$F(:, :, 1) = \pmatrix{f_1(v_1) & f_2(v_1) & f_3(v_1) \cr f_1(v_2) & f_2(v_2) & f_3(v_2) \cr f_1(v_3) & f_2(v_3) & f_3(v_3) \cr f_1(v_4) & f_2(v_4) & f_3(v_4)} \quad\mathrm{and}\quad F(:, :, 2) = \pmatrix{f_4(v_1) & f_5(v_1) & f_6(v_1) \cr f_4(v_2) & f_5(v_2) & f_6(v_2) \cr f_4(v_3) & f_5(v_3) & f_6(v_3) \cr f_4(v_4) & f_5(v_4) & f_6(v_4)}.$$

For the intern Fourier-coefficient matrix the first dimension is reserved for the Fourier-coefficients of a single function; the dimension of the functions itself begins again with the second dimension.

If $\bf{\hat f}_1, \bf{\hat f}_2, \bf{\hat f}_3, \bf{\hat f}_4, \bf{\hat f}_5$ and $\bf{\hat f}_6$ would be the column vectors of the Fourier-coefficients of the functions above, internally they would be stored in $\hat F$ as follows. $$\hat F(:, :, 1) = \pmatrix{\bf{\hat f}_1 & \bf{\hat f}_2 & \bf{\hat f}_3} \quad\mathrm{and}\quad \hat F(:, :, 2) = \pmatrix{\bf{\hat f}_4 & \bf{\hat f}_5 & \bf{\hat f}_6}.$$

Defining a multivariate S2FunHarmonic

Definition via function values

At first we need some vertices

nodes = equispacedS2Grid('points', 1e5);
nodes = nodes(:);

Next we define function values for the vertices

y = [S2Fun.smiley(nodes), (nodes.x.*nodes.y).^(1/4)];

Now the actual command to get a 2x1 sF1 of type S2FunHarmonic

sF1 = S2FunHarmonic.interpolate(nodes, y)

sF1 = S2FunHarmonic (y↑→x)
  size: 2 x 1
  bandwidth: 224

Definition via function handle

If we have a function handle for the function we could create a S2FunHarmonic via quadrature. At first let us define a function handle which takes vector3d as an argument and returns double:

f = @(v) [exp(v.x+v.y+v.z)+50*(v.y-cos(pi/3)).^3.*(v.y-cos(pi/3) > 0), v.x, v.y, v.z];

Next we convert this function handle into a S2FunHarmonic of size $4 \times 1$

sF2 = S2FunHarmonic(f, 'bandwidth', 50)

sF2 = S2FunHarmonic (y↑→x)
  size: 4 x 1
  bandwidth: 50
  isReal: true

Definition via Fourier-coefficients

If we already know the Fourier-coefficients, we can simply hand them in the format above to the constructor of S2FunHarmonic.

sF3 = S2FunHarmonic(eye(9))

sF3 = S2FunHarmonic (y↑→x)
  size: 9 x 1
  bandwidth: 2

Operations which differ from an univariate S2FunHarmonic

Some default matrix and vector operations

You can concatenate and refer to functions as MATLAB does with vectors and matrices

sF4 = [sF1; sF2];
sF4(2:3);

You can conjugate the Fourier-coefficients and transpose/ctranspose the multivariate S2FunHarmonic.

conj(sF1);
sF1.';
sF1';

Some other operations

length(sF1);
size(sF2);
sF3 = reshape(sF3, 3, []);

sum and mean

If we do not specify further options to sum or mean they give we the integral or the mean value back for each function. You could also calculate the conventional sum or the mean value over a dimension of a multivariate S2FunHarmonic.

sum(sF1, 1);
sum(sF3, 2);

min/max

If the min or max command gets a multivariate S2FunHarmonic the pointwise minimum or maximum can calculated along the dimension specified as third argument.

% this computes the minimum along the first dimension
min(sF3,[],1);

Remark on the matrix product

At this point the matrix product is implemented per element and not as the usual matrix product.

Visualization of multivariate S2FunHarmonic

The same plot commands as for univariate S2FunHarmonic work on multivariate as well. The difference is that, now, each component is plotted next to one another.