non-linear partial differential equations

[Jose E. Roman]

Dear Nichols,

You can use SLEPc to solve the linear eigenproblems that arise within  
the self-consistency iteration for the KS equation. ScaLAPACK cannot  
be used because your matrices are not stored explicitly.

I know you are concerned about the diagonalization that is required  
within the iterative eigensolver, since you want to compute a large  
number of eigenpairs. In a previous communication, I said that we  
wanted to improve support in SLEPc for this case, and this is in fact  
done and included in version 3.0.0. Now you can control the growth of  
the subspace that is used internally by the eigensolver. See SLEPc  
users manual, section 2.6.4.

As an illustration, we have recently tried a real symmetric matrix  
coming from a computational chemistry application. The dimension of  
the matrix is about 65,000 and we compute 2,000 eigenpairs in chunks  
so that the internal diagonalization is of order 300 at most. This  
computation scales very well (we tried up to 500 processors). My bet  
is that you can reach matrices of order 10^7 and 10,000 eigenvalues  
without problems, provided that you have enough memory and processors.

We can provide support if necessary. Also, if you give us one of your  
matrices then we could do some tests. Contact us at the SLEPc  
maintainers email.

Best regards,
Jose E. Roman


On 13/06/2009, naromero at alcf.anl.gov wrote:

> Matt,
>
> Yes, it is a sparse eigenvalue problem. And yes, I have taken a look  
> at SLEPc before. For some of our
> very large problems, we may get up to 10,000 (out of 10^7)  
> eigenvalues and then SLEPc might need hooks
> into ScaLAPACK for the subspace diagonalization. Last time I checked  
> ScaLAPACK interface in SLEPc
> was not available.
>
>
> Nichols A. Romero, Ph.D.
> Argonne Leadership Computing Facility
> Argonne National Laboratory
> Building 360 Room L-146
> 9700 South Cass Avenue
> Argonne, IL 60490
> (630) 252-3441
>
>
> ----- Original Message -----
> From: "Matthew Knepley" <knepley at gmail.com>
> To: "PETSc users list" <petsc-users at mcs.anl.gov>
> Sent: Friday, June 12, 2009 11:21:08 AM GMT -06:00 US/Canada Central
> Subject: Re: non-linear partial differential equations
>
> You can solve matrix-free nonlinear equations with PETSc. If you are  
> actually
> solving an eigenproblem, I would recommend using SLEPc which has PETSc
> underneath.
>
> Matt
>
>
> On Fri, Jun 12, 2009 at 10:20 AM, < naromero at alcf.anl.gov > wrote:
>
>
> Hi,
>
> I would like to understand if the methods in PETSc are applicable to  
> my
> problem.
>
> I work in the area of density functional theory. The KS equation in
> real-space (G) is
>
> [-(1/2) (nabla)^2 + V_local(G) + V_nlocal(G) + V_H[rho(G)] psi_nG =  
> E_n*psi_nG
>
> rho(G) = \sum_n |psi_nG|^2
>
> n is the index on eigenvalues which correspond to the electron  
> energy levels.
>
> This KS equation is sparse in real-space and dense in fourier-space.  
> I think
> strictly speaking it is a non-linear partial differential equation.  
> V_nlocal(G)
> is an integral operator (short range though), so maybe it is  
> technically a
> non-linear integro-partial differential equation.
>
> I understand that PETSc is a sparse solvers. Does the non-linearity  
> in the
> partial differential equation make PETSc less applicable to this  
> problem?
>
> On one more technical note, we do not store the matrix in sparse  
> format. It is
> also matrix*vector based.
>
>
>
> Argonne Leadership Computing Facility
> Argonne National Laboratory
> Building 360 Room L-146
> 9700 South Cass Avenue
> Argonne, IL 60490
> (630) 252-3441
>
>
>
>
> -- 
> What most experimenters take for granted before they begin their  
> experiments is infinitely more interesting than any results to which  
> their experiments lead.
> -- Norbert Wiener