
A general memristor-based partial differential equation solver - godelmachine
http://www2.ece.rochester.edu/~xiguo/gomac15.pdf
======
petermcneeley
Not the original paper but this explains in detail how this works.
[http://www2.ece.rochester.edu/~xiguo/gomac15.pdf](http://www2.ece.rochester.edu/~xiguo/gomac15.pdf)

I would guess they are solving PDEs using systems of linear equations. The
approximate solutions to these linear equations are found via the analog
linear solvers.

~~~
dang
Thanks, we changed the URL to that from
[http://www.nature.com/articles/s41928-018-0100-6](http://www.nature.com/articles/s41928-018-0100-6)
since the latter is behind a paywall without workarounds. (This is explained
at
[https://news.ycombinator.com/newsfaq.html](https://news.ycombinator.com/newsfaq.html)
and
[https://news.ycombinator.com/item?id=10178989](https://news.ycombinator.com/item?id=10178989)).

~~~
tanderson92
With respect (I didn't downvote you), there is a workaround: sci-hub (I
checked that it is available there).

For others who want a comparison; the two papers are very different: the
Nature paper uses entirely analog techniques to achieve high precision results
while the newly-linked article is a mixed-precision approach. That is, the
newly-linked article only uses analog to obtain a low-precision estimate which
is fed back to a digital computing algorithm to produce the final high-
precision result. The hope is that the overall time to solution is still
faster because you do not need (hopefully eventually slower) digital
computation to achieve the low-accuracy result. I am not an expert in this
particular hardware field (I do applied math so I am used to solving the
problems they do here in digital computers), but here is what the authors of
the Nature article say:

> Recent theoretical and experimental studies have used memristor (including
> phase-change memory) arrays to generate an initial, low-precision guess
> (seed), and rely on an integrated high-precision digital solver to produce
> the required high-precision solutions from the seed solution. Such types of
> accelerators are certainly beneficial, as they reduce the number of
> iterations required by the digital solver. Determining whether memristor-
> based hardware can be used to directly perform high-precision computing
> tasks, however, will enable a better understanding of how broadly memristive
> hardware can be applied. Such knowledge will help pave the way to build more
> general memristor-based computing platforms, instead of special-purpose
> accelerators.

~~~
petermcneeley
I have done graduate computational physics (Multigrid and such) So im familiar
with the "digital" side of this but certainly not familiar with almost any
analog computations. (beyond those trivial op-amp integrators)

------
ChrisRackauckas
The title is incorrect. It's not a general partial differential equation
solver. The paper doesn't even claim it to be. The issue is that not all, not
even most, PDEs can be discretized to a linear system of equations. Normally
these need to be discretized to either a nonlinear rootfinding problem G(x)=0
find x, or a stiff ODE (normally solved by an implicit method, but not
necessarily).

However, both of those problems do include a subproblem which is solving
linear systems (when the ODE is solved implicitly), and so PDE solvers which
take these routes will generally be able to be accelerated by an improved
linear equation solver. Thus it's not an be all and end all algorithm, but
when combined with the tools that exist for PDEs in something like PETSc or
Julia then this can be used to give a more efficient algorithm for most PDEs
(though there are still other PDEs where linear solving is entirely avoided,
for example with exponential integrators or Runge-Kutta Chebyshev methods,
which can be more efficient depending on the problem).

~~~
tanderson92
The link was changed from the original Nature article which has the original
title. The new link is unrelated.

You're correct that the innovation is really about sparse linear algebra
(though not just solvers) done in hardware. However the advantages are
exploited not just by way of solving linear systems for implicit time-
stepping. They implemented a Jacobi iteration in hardware for solving a system
arising from discretization of the Poisson equation, but they also used an
efficient forward-map of the coefficient matrix to do explicit time-stepping
for a wave solver. So they solve both a time-independent and time-dependent
problem, but neither with implicit time-stepping (though there appears to be
no technical limitation here, it's just what they focused on).

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msla
Why isn't this PDF marked? Aren't they usually marked?

It even has a .pdf extension.

Accessibility fail.

------
msla
PDF

This crashed my browser.

------
politician
Should this be flagged due to the paywall?

