MATLAB Examples

Demo2: Demonstrate Constrained Solution Spaces

This script demonstrates the generation of a particular solution and the determination of a set of homogeneously constrained basis functions which fulfil constraints. The particular solution together with the constrained basis functions define the space of all possible solutions to the constrained problem. This space can then be used to solve differential equations, perform least square fitting etc.

They can be used as admissible functions for discrete implementation of a discrete Rayleigh Ritz method for the solution of boundary value problems. See:

for their use in such problems.

The constrained basis functions are orthonormal, i.e., Bh^T Bh = I. This property simplifies the solution of many problems, such as least squares approximation.

(c) 2013 Paul O'Leary and Matthew Harker Institute for Automation University of Leoben A-8700 Leoben Austria

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close all;
clear all;
% Set some defaults
FontSize = 12;
set(0,'DefaultTextInterpreter', 'latex');

Define the Number of Basis Functions and x

nrBfs = 10;
% Generate the vector of x values
nrPts = 200;
x = dopNodes( nrPts, 'Gramends');

Definig the Triplets for the Constraints

Each triplet ti = [n, a, b] defined a constraint of the form

D^(n) y(a) = b

Value constraints y(a) = b

t2 = [0, x(end), 2];
t3 = [0, x(1), 1];
% Derivative constraints D y(a) = b
t1 = [1, x(1), 0];
t4 = [1, x(end), 0];
% Concatinate the triplets to form an array of triplest which define all
% the constraints.
T = [t1; t2; t3; t4];

Call dopGenConstrained

The function dopGenConstrained computes a particular solution y_p and determine a set of basis functions Bh suche that the homogeneous solution y_h = Bh * \beta. In this manner the possible solutions to y are of the form: y = y_p + y_h = y_p + Bh * \beta.

[ypMD, Bh, S] = dopGenConstrained( x, nrBfs, T );

Extract some additional information from the structure:

1) The derivatives of the homogeneously constrained basis functions.

dBh = S.dBh;

2) The minimum norm particular solution

ypMN = S.ypMN;

3) The structuring matrix R such that Bh = B * R

R = S.R;

Display the Homogeneously Constrained Basis Functions

fig1 = figure;
plot( x, Bh, 'k');
xlabel( 'Support' );
ylabel( '$$B_h(x)$$' );
grid on;

Display the Derivitives of Bh

fig2 = figure;
plot( x, dBh, 'k');
xlabel( 'Support' );
ylabel( '$$\frac{d B_h(x)}{d x}$$' );
grid on;

Display Two Different particular Solutions

Here the minimum degree and minimum norm psrticular solutions are displayed.

fig3 = figure;
plot(x, ypMD, 'b');
hold on;
plot(x, ypMN, 'r');
xlabel( 'Support' );
ylabel( '$$y_p(x)$$' );
grid on;
legend( 'Min degree', 'Min norm', 'Location', 'NorthWest');

Plot the positions of the constraints. The value constraints are drawn at their position and value with white filled circles. All other constraints i.e. differential constrainst, are marked as a position on the x axis.

[nt, mt] = size( T );
for k = 1:nt
    if T(k,1) == 0
        plot( T(k,2), T(k,3), 'ko', 'MarkerFaceColor', 'w');
        plot( T(k,2), 0, 'ko', 'MarkerFaceColor', 'k');