# Example file showing how the difference operators for sparse grid behave

by SeHyoun Ahn, July 2016

## Contents

Requires: <generate_spgrid.m> <gen_diff_op.m> <grid_evaluation.m>

Once the grid is built and the difference operators are computed, computing the approximation for the derivative values become straight forward. The procedure become

• Compute the Nodal values
• Change basis to Hierarchical Values
• Apply the Difference Operators (found for the given grid) to compute Derivative Values

## Grid Set-up

Grid parameters

n_dim = 2;
n_levels =7;

% Generate Grid
% by default the grid is set up over $[0,1]^n$
[spgrid,spspace] = generate_spgrid(n_dim,n_levels);

% Setup Change of basis from Hierachical to Nodal
hier_to_nodal = grid_evaluation(spgrid,spgrid,spspace);

% Difference Operator
[forward_diff_x,backward_diff_x] = gen_diff_op(1,spgrid,spspace,min(spspace(:,1)));
[forward_diff_y,backward_diff_y] = gen_diff_op(2,spgrid,spspace,min(spspace(:,2)));

% Second Order Difference Operator
% This is one difference scheme of many to get second order difference
fb = forward_diff_y*(hier_to_nodal\backward_diff_y);
bf = backward_diff_y*(hier_to_nodal\forward_diff_y);
second_deriv_y = (fb+bf)/2;
second_deriv_y(spgrid(:,2)==1,:) = fb(spgrid(:,2)==1,:);
second_deriv_y(spgrid(:,2)==0,:) = bf(spgrid(:,2)==0,:);

% Integration Weights
int_weight = (prod(spspace,2)./2.^sum(spspace==1,2))';


## Example 1: $x^2 + y^2$

close all;
% Compute Nodal Values
v=spgrid(:,1).^2+spgrid(:,2).^2;
% Change to Hierarchical Values
v_basis = hier_to_nodal\v;

figure(1);
% Use the Difference Operator to Compute forward Difference
scatter3(spgrid(:,1),spgrid(:,2),forward_diff_x*v_basis);
hold on;
scatter3(spgrid(:,1),spgrid(:,2),2*spgrid(:,1),'.');
title('D_x of x^2+y^2');
legend('approximation','actual','location','northoutside','Orientation','horizontal');

figure(2);
% Use Difference Operator to Compute Second Order Difference
scatter3(spgrid(:,1),spgrid(:,2),second_deriv_y*v_basis);
hold on;
scatter3(spgrid(:,1),spgrid(:,2),2.*ones(size(spgrid(:,1))),'.');
title('D_{yy} of x^2+y^2');
legend('approximation','actual','location','northoutside','Orientation','horizontal');


## Example 2: $e^{-2\cdot (x-0.5)^2-(y-0.6)^2}$

close all;
v = exp(-2*(spgrid(:,1)-0.5).^2-(spgrid(:,2)-0.6).^2);
v_basis = hier_to_nodal\v;
true_deriv = exp(-2*(spgrid(:,1)-0.5).^2-(spgrid(:,2)-0.6).^2).*(-4).*(spgrid(:,1)-0.5);

h = figure(1);
scatter3(spgrid(:,1),spgrid(:,2),v);
title('e^{-2*(x-0.5)^2-10*(y-0.6)^2}');
gif_maker_rotate('fig2_1.gif',h);

h = figure(2);
scatter3(spgrid(:,1),spgrid(:,2),forward_diff_x*v_basis);
hold on;
scatter3(spgrid(:,1),spgrid(:,2),true_deriv,'.');
title('D_x of e^{-2*(x-0.5)^2-10*(y-0.6)^2}');
legend('approximation','actual','location','northoutside','Orientation','horizontal');
gif_maker_rotate('fig2_2.gif',h);


## Example 3: $e^{-x-2\cdot y}$

close all;
v = exp(-spgrid(:,1)-2*spgrid(:,2));
v_basis = hier_to_nodal\v;
true_deriv = exp(-spgrid(:,1)-2*spgrid(:,2))*(-2);
true_2deriv = exp(-spgrid(:,1)-2*spgrid(:,2))*4;

h = figure(1);
scatter3(spgrid(:,1),spgrid(:,2),v);
title('e^{-x-2y}');
gif_maker_rotate('fig3_1.gif',h);

h = figure(2);
scatter3(spgrid(:,1),spgrid(:,2),forward_diff_y*v_basis);
hold on;
scatter3(spgrid(:,1),spgrid(:,2),true_deriv,'.');
title('D_{y} of e^{-x-2y}');
legend('approximation','actual','location','northoutside','Orientation','horizontal');
gif_maker_rotate('fig3_2.gif',h);

h = figure(3);
scatter3(spgrid(:,1),spgrid(:,2),second_deriv_y*v_basis);
hold on;
scatter3(spgrid(:,1),spgrid(:,2),true_2deriv,'.');
title('D_{yy} of e^{-x-2y}');
legend('approximation','actual','location','northoutside','Orientation','horizontal');
gif_maker_rotate('fig3_3.gif',h);