[CSC 435] Questions on Jacobi

[Andrew J. Pounds]

On 04/22/2018 06:20 PM, Andrew Devin Mcbryde wrote:
>
> My group and I are having some difficulty in understanding your
> expectation using the Jacobi algorithm to calculate the iterations
> that the simulation code uses. We do not understand what the Jacobi
> algorithm actually is and how it can be applied to the problem we are
> currently facing. Most of the information we find regarding the
> function references how the memory is referenced and not the actual
> math that calculates the final values.
>

Let's see if we can break this down a little... equation (2) on the
group project sheet (which shows how the concentrations of each cell
change with time)  should be replaced with something like equation 3.7
in the text.   In equation 3.7 the $\Phi$ values refer to the quantity
changing (in your case it is concentrations). 


You will, however, need to convert equation 3.7 to work in 3D.  To do
this you will need to add a $z$ component to each  concentration cell,
i.e. - $\Phi(x,y,z)$.  You will also need to add an additional term to
account for the change in the $z$ dimension. 


Since the diffusion is taking place through the faces only (areas), the
denominators in your 3D equation will remain squared terms, and note
cubed terms.


Now think about this -- your modified equation 3.7 will let you
determine how much any cell in your cube should change with a VERY
significant new feature -- you have BROKEN all data dependencies that
would inhibit parallelism (and in many cases those that would inhibit
vectorization).  Let's call $\Phi_c$ the copy of our diffusion cube.  I
can calculate the entities of $\Phi_c$ as...


$\Phi_c(x,y,z) = \Phi(x,y,z) + \frac{\delta\Phi(x,y,z)}{\delta t}$


With this technique you  can calculate ALL of the elements of $\Phi_c$
independently and then, when done, copy all of the elements of $\Phi_c
\rightarrow \Phi$, and then repeat the process until you reach equilibrium.


Once you develop a new algorithm,  test the serial version and try to
obtain the same results you got on your CSC 330 projects.


I STRONGLY encourage you to get the serial version working on hammer
first. 




-- 
Andrew J. Pounds, Ph.D.  (pounds_aj at mercer.edu)
Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627
http://faculty.mercer.edu/pounds_aj

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