The Nullspace of Linear Operators

Nick Hale & Stefan Güttel, 12th December 2011

Simple Example #1
Incomplete boundary conditions
An application
Exotic constraints
References

We've recently introduced some new functionality in Chebfun for computing the nullspace of differential operators. Let's explore this with a couple of simple examples!

Simple Example #1

Let's start as simply as we can, and take

   u"(x) = 0, for x in [-1 1]

L = chebop(@(u) diff(u,2));

Clearly the nullspace, that is the nontrivial functions v for which

   L(v) = 0,

for this operator is spanned the two functions

v = [1 chebfun('x')];
norm(L*v)

ans =
     0

Supposing we didn't know this, we could compute the space with the new NULL function.

V = null(L)
plot(V)
V'*V
norm(L*V)

V = 
   chebfun column 1 (1 smooth piece)
       interval       length   endpoint values   
[      -1,       1]        7     -1.4     0.44   
vertical scale = 1.4 
   chebfun column 2 (1 smooth piece)
       interval       length   endpoint values   
[      -1,       1]        7     0.29     -1.3   
vertical scale = 1.3 
ans =
    1.0000    0.0000
    0.0000    1.0000
ans =
   7.3130e-12

where we find that V'V = I and LV ~ 0 as required.

Clearly V doesn't correspond directly to 1 and x, since there is some freedom in how we orthogonalise the basis. However, we can check that V and {1,x} correspond to the the same spaces by computing the angle between the spaces with the SUBSPACE command.

subspace(v,V)

ans =
   5.4439e-13

Incomplete boundary conditions

Now let's consider the more complicated 2nd-order operator

   Lu = u'' + .1*(1-x.^2)u' - sin(x)u, x in [-pi pi]                (*)

dom = [-pi pi];
L = chebop(@(x,u) diff(u,2) + .1*x.*(1-x.^2).*diff(u) + sin(x).*u, dom);
x = chebfun('x',dom);

As before, it has a nullspace of rank 2.

V = null(L)
plot(V)
V'*V
norm(L(x,V(:,1)))
norm(L(x,V(:,2)))

V = 
   chebfun column 1 (1 smooth piece)
       interval       length   endpoint values   
[    -3.1,     3.1]       39     -1.8   -0.044   
vertical scale = 1.8 
   chebfun column 2 (1 smooth piece)
       interval       length   endpoint values   
[    -3.1,     3.1]       40   -0.011      1.3   
vertical scale = 1.3 
ans =
    1.0000    0.0000
    0.0000    1.0000
ans =
   1.1447e-10
ans =
   4.9739e-11

However, now suppose we impose ONE boundary condition, say Dirichlet at the left. This removes ONE degree of freedom, and we are now left with a rank 1 nullspace.

L.lbc = 0;
L.rbc = [];
v = null(L)
plot(v), shg
v'*v
norm(L(x,v))

v = 
   chebfun column (1 smooth piece)
       interval       length   endpoint values   
[    -3.1,     3.1]       40 -3.4e-15      1.3   
vertical scale = 1.3 
ans =
    1.0000
ans =
   4.9698e-11

Clearly this null vector must satisfy the given condition v(-pi) = 0.

An application

Where might this be useful? Well, suppose we were interested in equation (*) with a homogeneous Dirichlet condition at the left, and wanted to know what inhomogenous Dirichlet condition gave the minimal 2-norm of the solution to Lu = 1.

Rather than solving the linear system for a number of different boundary conditions (which would be computationally expensive) we could simply solve for one, say again homogenous Dirichlet

L.rbc = 0;
u = L\1;
hold on, plot(u,'--r'), hold off;

and compute the rest by adding a scalar multiple of the null-function v.

E = chebfun(@(c) norm(u+c*v,2),'vectorise',[-10 10],'splitting','on');
plot(E)

We compute the 2-norm as a chebfun in the unknown variable c, which we can then minimise to obtain the minimal energy solution

[minE c_star] = min(E)
u_star = u + c_star*v
plot(u_star)

minE =
    4.1220
c_star =
    3.1438
u_star = 
   chebfun column (1 smooth piece)
       interval       length   endpoint values   
[    -3.1,     3.1]       44 -5.7e-15        4   
vertical scale =   4

So the condition we require is that u(pi) = bc_star, where

bc_star = u_star(pi)

bc_star =
    3.9894

Exotic constraints

The NULL function can also handle the exotic types of boundary conditions that can be enforced in Chebfun (see [1]). For example, suppose we wish to again compute the null-space of the 3rd-order piecewise-smoooth ODE

     Lu := 0.001u''' + sign(x)*u'' + u, x in [-1 1]

with the 'boundary' condition that

           sum(u) = u(0).

dom = [-1 1];
L = chebop(@(x,u) 1e-2*diff(u,3) + sign(x).*diff(u,2) + u);
L.lbc = []; L.rbc = [];
L.bc = @(u) sum(u)-u(0);
x = chebfun('x',dom);

Here NULL has no problems!

V = null(L)
plot(V), shg
V'*V

V = 
   chebfun column 1 (2 smooth pieces)
       interval       length   endpoint values   
[      -1,       0]       65     0.84     0.98   
[       0,       1]       64     0.98    -0.75   
Total length = 129   vertical scale = 0.98 
   chebfun column 2 (2 smooth pieces)
       interval       length   endpoint values   
[      -1,       0]       65       -1     0.72   
[       0,       1]       65     0.72     0.84   
Total length = 130   vertical scale =   1 
ans =
    1.0000   -0.0000
   -0.0000    1.0000

sum(V(:,1))-V(0,1)
norm(L(x,V(:,1)),1)
sum(V(:,2))-V(0,2)
norm(L(x,V(:,2)),1)

ans =
   1.4380e-12
ans =
   4.8847e-09
ans =
  -7.6439e-13
ans =
   4.5108e-09

References