Matlab slow performance using closures

Question

I'm coding a solution for Poisson equation on a 2d rectangle using finite elements. In order to simplify the code I store handles to the basis functions in an array and then loop over these basis functions to create my matrix and right hand side. The problem with this is that even for very coarse grids it is prohibitively slow. For a 9x9 grid (using Dirichlet BC, there are 49 nodes to solve for) it takes around 20 seconds. Using the profile I've noticed that around half the time is spent accessing (not executing) my basis functions.

The profiler says matrix_assembly>@(x,y)bilinearBasisFunction(x,y,xc(k-1),xc(k),xc(k+1),yc(j-1),yc(j),yc(j+1)) (156800 calls, 11.558 sec), the self time (not executing the bilinear basis code) is over 9 seconds. Any ideas as to why this might be so slow?

Here's some of the code, I can post more if needed:

%% setting up the basis functions, storing them in cell array
basisFunctions = cell(nu, 1); %nu is #unknowns 
i = 1;
for j = 2:length(yc) - 1
for k = 2:length(xc) - 1
    basisFunctions{i} = @(x,y) bilinearBasisFunction(x,y, xc(k-1), xc(k),...
        xc(k+1), yc(j-1), yc(j), yc(j+1)); %my code for bilinear basis functions
    i = i+1;
end
end

%% Assemble matrices and RHS
M = zeros(nu,nu);
S = zeros(nu,nu);
F = zeros(nu, 1);

for iE = 1:ne
for iBF = 1:nu
    [z1, dx1, dy1] = basisFunctions{iBF}(qx(iE), qy(iE));

    F(iBF) = F(iBF) + z1*forcing_handle(qx(iE),qy(iE))/ae(iE);

    for jBF = 1:nu
        [z2, dx2, dy2] = basisFunctions{jBF}(qx(iE), qy(iE));

        %M(iBF,jBF) = M(iBF,jBF) + z1*z2/ae(iE);
        S(iBF,jBF) = S(iBF, jBF) + (dx1*dx2 + dy1*dy2)/ae(iE);
    end        
end
end

Answer 1

Try to change basisFunctions from being a cell array to being a regular array.

You can also try to inline the direct call to bilinearBasisFunctionwithin your jBF loop, rather than using basisFunctions. Creating and later using anonymous functions in Matlab is always slower than directly using the target function. The code may be slightly more verbose this way, but will be faster.