parallel coding: how to solve linear system?

Hi everyone! I need to solve a (very large) linear system in my code, so I'm trying to use a parallel session.

Basically my original local code does the following:

function U = myfun(U0,J0,nx,S0,Mbcs,Bbcs)
A = spdiags(funA(J0),0,nx,nx);
B = spdiags(funB(J0),0,nx,nx);
C = spdiags(funC(J0),0,nx,nx);
D = spdiags(funD(J0),0,nx,nx);

K = [A B; C D];
V = S0;

K([1,nx,nx+1,2*nx],:)=0;
V([1,nx,nx+1,2*nx])=0;

K = K + Mbcs;
V = V + Bbcs;

U = K\V;

The use of sparse array speeds up a lot the solver time. Now I tried to launch a parallel session:

parpool('local2');

And the code was modified as follows for distributing the matrices to the workers:

K = distributed(K);
V = distributed(V);

But at the end I get this error:

Error using \ (line 48)

Sparse input arguments are not supported.

Error in distributed/wrapRemoteCall>iInnerWrapper (line 83)

[varargout{:}] = fcnH( varargin{:} );

Error in spmd_feval_fcn>get_f/body (line 78)

[outCell{:}] = fcnH( inCell{:} );

Where is the error? I don't know much about parallel programming, but from the examples I found, it seems to me that I only need do distribute the matrix to the workers.

Thanks in advance for all your suggestions.

Pietro

Answer 1

[Edric Ellis]

Unfortunately, as the error message states, you cannot use the \ operator with sparse distributed arrays at this time.

[Pietro Pollaci]

So, which is the solution? Using dense arrays and allocating more memory? I'll try to use diag()! Thanks

[John D'Errico]

Suppose you gain a 10-1 speedup by the use of sparse matrices. This would not be at all surprising. If you go back to dense matrices so that you can use a pool of workers to solve it, then you would need to use a pool with at least 10 workers to get the same speedup, and that just gets you to break-even.

[Pietro Pollaci]

Then, the use of a pool is not (always) convenient, right? Let us assume, as you wrote, a 10:1 gain then, if I well understood, at least "11" workers are needed in the parallel session in order to get significant advantages. The questions are:

- is it always true?

- does it depend on the size of the problem?

- when does a parallel session beat a sparse-array approach?

[Edric Ellis]

Unfortunately, the best options you have are either:

1. Use sparse on your local machine
2. Use dense distributed arrays on a cluster
3. Solve a smaller problem

Note that distributed arrays on a single machine almost never offer any benefit since most linear algebra operations are already fairly efficiently multi-threaded by MATLAB, and using a multi-process approach cannot usually compete.