[Trilinos-Users] XYCE math formulation article

Eric Keiter erkeite at sandia.gov
Wed Jul 18 14:31:56 MDT 2007

Hi Neal,

I think I received an email from your boss a few months ago.  Sorry I didn't
reply - I was on travel at the time.

Anyway, I'm the original author of the math document.  The thing you are
doing in SPICE to produce the J*dx = -f system should work, as long as the
J*x you are subtracting is based on the correct x.  I actually considered
doing that at one point in time, (mostly for doing exact comparisons between
Xyce and SPICE) but never got around to it.

The dx that the linear solver is solving for needs to be the total dx - ie
the sum of the dx from the Newton step and also the dx due to voltage
limiting.  SPICE applies the dx_{voltlim} in an implicit way, so SPICE's
unmodified system is in fact:

  J * x^{k+1} = -f + J * x_{voltlim}

Where x_{voltlim} is the solution after voltage limiters have been applied
to it.

If you add -J*x^{k} to both sides you get:

  J* dx_{total} = -f + J * dx_{voltlim}

  where:  dx_{voltlim} = x_{voltlim}-x^{k}
          dx_{total} = dx_{Newton} + dx_{voltlim}

SPICE applies the voltlim terms internally to each device load, by applying
the limiter functions to local variables that are derived from what is in
x_{k}.  Then, when it calculates J and f, it uses the "limited" values, and
essentially throws away x_{k}.

Anyway, this is all a long way of saying that what you are doing should
work.  And, I wouldn't call it "cheating" ;  doing it your way is a lot
faster than rewriting every device model!

Also regarding your question on numerical errors;  there will of course be
roundoff differences between what you are now doing and SPICE's original
form.  However, those roundoff differences won't necessarily be more or less
accurate.  In particular, if you want to use iterative linear solvers (you
probably don't...), iterative solvers will generally be happier if you are
solving for dx, as the values in dx will tend to be small, especially as the
solution gets near convergence.

The main reason we used the DX formulation, however, is that this is what
most nonlinear solvers do, and we planned to use  the NOX solver library.

By the way, the math document is mostly written for folks who have numerical
backgrounds but haven't ever tried to understand circuit simulation before;
it doesn't have much in it that would be pertinent to making a circuit
simulator faster or more robust.

Eric Keite
erkeite at sandia.gov

On 7/18/07 11:59 AM, "Neal Parsons" <nparsons at umich.edu> wrote:

> Hi Robert (etc.),
> I have a few questions/ observations about the xyce math formulation article.
> First, in the Xyce Math document, it describes reformulating the
> equations in xyce to solve for delta X instead of X.  I'm wondering
> "how" you do that internally in the code and if our approach is sound.
> We are solving for delta X instead of X, but we aren't doing in each
> devices' load routine.  We are doing it by "cheating".  Specifically,
> our strategy is to generate J and the RHS with Spice's unmodified
> device Load routines, then we are subtracting J * X[k] from the RHS to
> get "f".  So the first question I have is whether this approach is
> inferior to modifying each device's DevLoad routine to directly place
> "f" (or is it "-f") in the RHS.
> I'm concerned about mathematical errors related to the magnitudes of
> the terms being subtracted (e.g. 5.000000000000000001 - 5.0 = 0).
> Second, regarding voltage limiting, the math article (p. 27) suggests
> that we need to add J * (delta X hat)[k+1] to "-f".  But with our
> "cheat", this modification to "-f" seems to be unnecessary.  Perhaps
> this is because when the RHS is calculated by spice in the "Spice" way,
> it already takes into account the voltage limiting?  I'm not sure about
> that.  But if you have any insight into these issues, I'd appreciate
> hearing it.
> -- 
> Neal Parsons (and Jon Engelbert, my boss)
> University of Michigan
> Mechanical Engineering
> Cell: 517-896-2171
> Quoting Neal Parsons <nparsons at umich.edu>:
>> Thank you all for your responses. Since our SPICE package generally
>> deals with smaller circuits, it sounds like Amesos/KLU solver would
>> be better than the iterative AztecOO solver. I'm working for a very
>> small software company for the summer that currently uses the Berkely
>> SPICE engine, which fails roughly 1/3 of our NCSU benchmark test
>> suite. The motivation for integrating Trilinos libraries comes from
>> reading about XYCE's use of Trilinos.  We are hoping to get better
>> convergence and faster results.
>> Our solvers were updated to solve J*deltaX(k+1) = -f, as Xyce does.
>> We initially used AztecOO Line Search Based solver with ILU
>> preconditioning for our solver. This gave accurate operating point
>> results, but as noted, was generally slow for transient computations.
>> After wrestling with AztecOO for several weeks to no avail and given
>> the advice here, it appears that my next step would be to change-over
>> from AztecOO to Amesos/KLU. I just want to make sure that this
>> implementation will be beneficial to our application. Also, any other
>> suggestions might be helpful as well. Again, thanks for your help.
>> Neal Parsons
>> Quoting "Hoekstra, Robert J" <rjhoeks at sandia.gov>:
>>> As Mike mentioned, KLU (by Tim Davis, also the originator of the
>>> aforementioned UMFPack)
>>> which is the default solver in the Trilinos/Amesos package has specific
>>> optimizations for
>>> circuit problems and should outperform the more generic tools (UMFPack
>>> and SuperLU) for
>>> sparse direct solves.  As mentioned by both Mike and Randy, it is very
>>> difficult to get
>>> iterative sparse methods to perform well on this class of problems.  For
>>> the Xyce simulator,
>>> (Sandia's in house tool) we use GMRES with ILU(k) preconditioning only
>>> for very large scale
>>> problems (approaching a million or more devices).  If you are bent on
>>> demonstrating an
>>> iterative method rather than performance our default usage is along the
>>> lines of:
>>> AztecOO GMRES
>>> EpetraExt Singleton Filtering
>>> Ifpack ILU(k) w/ RCM Reordering
>>> Just using AztecOO's ILU(t) may do the trick but I am not sure if you
>>> will get the RCM
>>> reordering by default and it's bandwidth minimization is critical to the
>>> performance of
>>> the preconditioner.  You can probably neglect the singleton filter
>>> unless you have a very
>>> large distributed problem but not the bandwidth minimizing reorder
>>> (approx. minimum degree
>>> works even better than RCM and is what Tim Davis uses in KLU but it is
>>> not available in Ifpack
>>> or AztecOO).
>>> Robert Hoekstra
>>> ____________________________
>>> Electrical & Microsystems Modeling
>>> Sandia National Laboratories
>>> P.O. Box 5800 / MS 0316
>>> Albuquerque, NM 87185
>>> phone: 505-844-7627
>>> fax:      505-284-5451
>>> e-mail: rjhoeks at sandia.gov
>>> web: http://www.cs.sandia.gov
>>> -----Original Message-----
>>> From: trilinos-users-bounces at software.sandia.gov
>>> [mailto:trilinos-users-bounces at software.sandia.gov] On Behalf Of Randall
>>> Bramley
>>> Sent: Thursday, July 12, 2007 12:37 PM
>>> To: Heroux, Michael A
>>> Cc: trilinos-users at software.sandia.gov
>>> Subject: Re: [Trilinos-Users] Slow AztecOO solve time
>>>>> I'm a student, and a beginner with Trilinos trying to use the AztecOO
>>>>> solver in a SPICE package.   I am getting relatively slow results,
>>> Circuit simulation problems are notoriously difficult for iterative
>>> solvers with non-specialized preconditioners like ILUt, etc. It's not
>>> just a matter of a bad condition number, but the overall eigenvalue
>>> distributions tend to give worst-case properties for Krylov solvers.
>>> That would explain having a large number of iterations, but not
>>> necessarily the time spent in memory management - unless it is used on
>>> every iteration, or you have an lax dropping parameter set in the
>>> preconditioner (and hence many fill-in locations). So this may not be of
>>> help on your problem but be aware that just about any iterative solver
>>> package (e.g., Hypre, PETsC, ...) will in general perform slowly for
>>> SPICE.
>>> There may be some specialized application-specific preconditioners
>>> available now for circuit simulation, but I've been out of the game for
>>> too long to know. Still, that is something I'd recommend hunting for -
>>> it would well repay a couple of hours on Google if such a magic
>>> preconditioner has been developed. And if you do find such a thing,
>>> please post it back to this mail list - a lot of us would be interested
>>> and grateful.
>>> Way back when *I* was a student (shortly after the War of 1812) we
>>> typically had to bite the bullet and use a direct sparse solver rather
>>> than iterative methods. Or more accurately, a complete LU decomposition
>>> followed by a few steps of iterative refinement. Ain't no such thing as
>>> purely direct or purely iterative solvers nowadays and all practical
>>> solvers like Trilinos are essentially hybrids.
>>> Mike's recommendation is good, but I'd go further. You can get a quick
>>> estimate of the number of nonzeros (and sparsity pattern) of the LU
>>> factors when using SuperLU or UMFPACK. Take that number, multiply by 8
>>> to get the number of bytes, and see if that fits inside the amount of
>>> memory you have on the target machine. For parallel solves like
>>> SuperLUdist you'd want to do the same on a per process basis.  Using
>>> memory size can also be used to determine how much fill-in would be
>>> optimal for a solver, modulo adding in 3-10 vectors for the iterative
>>> solver.
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