Gigabyte GA-7PESH1 Review: A Dual Processor Motherboard through a Scientist’s Eyes
by Ian Cutress on January 5, 2013 10:00 AM EST- Posted in
- Motherboards
- Gigabyte
- C602
Two Dimensional Implicit Finite Difference
The ‘Finite Difference’ part of this computational grid solver means that the derivation of this method is similar to that shown in the Explicit Finite Difference method on the previous page. We are presented with the following equation which explains Fick’s first law of diffusion for mass transport in three dimensions:
[8]
The implicit method takes the view that the concentrations at time t+1 are a series of unknowns, and the equations are thus coupled into a series of simultaneous equations with an equal set of unknowns, which must be solved together:
[9]
[10]
The implicit method is algorithmically more complex than the explicit method, but does offer the advantage of unconditional stability with respect to time.
The Alternating Direction Implicit (ADI) Method
For a system in two dimensions (labelled r and z), such as a microdisk simulation, the linear system has to be solved in both directions using Fick’s Laws:
[11]
The alternating direction implicit (ADI) method is a straightforward solution to solving what are essentially two dimensional simultaneous equations whilst retaining a high degree of algorithm stability.
ADI splits equation [11] into two half time steps – by treating one dimension explicitly and the other dimension implicitly in the same half time step. Thus the explicit values known in one direction are fed into the series of simultaneous equations to solve the other direction. For example, using the r direction explicitly to solve the z direction implicitly:
[12]
[13]
By solving equation [13] for the concentrations in the z direction, the next half time step concentrations can be calculated for the r direction, and so on until the desired time in simulation is achieved. These time step equations are solved using the Thomas Algorithm for tri-diagonal matrices.
Application to this Review
For the purposes of this review, we generate an x-by-x grid of points with a relative concentration of 1, where the boundaries are fixed at a relative concentration of 0. The grid is a regular grid, simplifying calculations. The nature of the simulation allows that for each half-time step to focus on calculating in one dimension, for a simulation of x-by-x nodes we can spawn x threads as adjacent rows/columns (depending on direction) are independent. These threads, in comparison to the explicit finite difference, are substantially bulkier in terms of memory usage.
The code was written in Visual Studio C++ 2012 with OpenMP as the multithreaded source. The main function to do the calculations is as follows.
For our scores, we increase the size of the grid from a 2x2 until we hit 2GB memory usage. At each stage, the time taken to repeatedly process the grid over many time steps is calculated in terms of ‘million nodes per second’, and the peak value is used for our results.
Previously where the explicit 2D method was indifferent to HyperThreading and the explicit 3D method was very sensitive; the implicit 2D is a mix of both. There are still benefits to be had from enabling HyperThreading. Nevertheless, the line between single processor systems and dual processors is being blurred a little due to the different speeds of the SP results, but in terms of price/performance, the DP system is at the wrong end.
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toyotabedzrock - Saturday, January 5, 2013 - link
There is a large number of very smart people on Google+. You really should come join us.JlHADJOE - Tuesday, January 8, 2013 - link
Of course there are lots of smart people on G+! You're all google employees right? =PActivate: AMD - Saturday, January 5, 2013 - link
As a fellow chemist, I must say that you have to be some kind of nut to want to do computational/physical chemistry. If you need me, I'll be at the bench!Good article too!
engrpiman - Saturday, January 5, 2013 - link
I didn't read the article in full but what I did read was top notch. I found your simulations and mathematics very interesting. I took a Physics class which was focused in writing code to run mathematical simulations . Using the given java lib. I wrote my own code to calculate PI. When I returned from the gym the program had calculated 3.1 . I then re-wrote the program from scratch and ditched the built in libs. and reran. I had 20 decimals in 30 sec it was an epic improvement.All in all I think your article could be very useful to me.
Thanks for writing.
SodaAnt - Saturday, January 5, 2013 - link
THIS is why I read Anandtech. I'll admit that I wasn't quite in the mood to read all the equations (I'll have to do that later), but really, these kind of reviews make my day.Cardio - Saturday, January 5, 2013 - link
Wonderful review...as always. ThanksHakon - Saturday, January 5, 2013 - link
Hi Ian. Thanks for the nice article. I have one suggestion regarding the explicit finite difference code:You could try to reorder the loops such that the memory access is more cache friendly. Right now 'pos' is incremented by NX (or even NX*NY in 3D) which will generate a lot cache misses for large grids. If you switch the x and the y loop (in the 2D case) this can be avoided.
IanCutress - Saturday, January 5, 2013 - link
Either way I order the loops, each point has to read one up, one down, one left and one right. My current code tries to keep three as consecutive reads and jump once, keeping the old jump in local memory. If I adjusted the loops, I could keep the one dimension in local memory, but I'd have to jump outside twice (both likely cache misses) to get the other data. I couldn't cache those two values as I never use them again in the loop iteration.When I did this code on the GPU, one method was to load an XY block into memory and iterate in the Z-dimension, meaning that each thread per loop iteration only loaded one element, with a few of them loading another for the halo, but all cache aligned.
I hope that makes sense :)
Ian
Hakon - Saturday, January 5, 2013 - link
Yes, but when you access the array 'cA' at 'pos' the CPU will fetch the entire cache line (64 byte in case of your machine, i.e. 16 floats) of the corresponding memory address into the CPU cache. That means that subsequent accesses to say 'pos + 1', 'pos + 2' and so on will be served by the cache. Accessing an array in such a sequential manner is therefore fast.However, when you access an array in a nasty way, e.g. 'NX + x' -> '2*NX + x', -> '3*NX + x', then each such access implies a trip to main memory if NX happens to be large.
That you need to move up / down and sideways in memory does not matter. When you write down the accesses of the code with the reordered loops you will notice that they just access three "lines" in memory in a cache friendly way.
Not reusing the old values of the last iteration should not affect performance in a measurable way. Even if the compiler fails to see this optimization, the accesses will be served by the L1 cache.
Btw, did you allocate the array having NUMA in mind, i.e. did you initialize your memory in an OpenMP loop with the same access pattern as used in the algorithms? I am a bit surprised by the bad performance of your dual Xeon system.
IanCutress - Saturday, January 5, 2013 - link
Memory was allocated via the new command as it is 1D. When using a 2D array the program was much slower. I was unaware you could allocate memory in an OpenMP way, which thinking about it could make the 2D array quicker. I also tried writing the code using the PPL and lambdas, but that was also slower than a simple OpenMP loop.I'm coming at these algorithms from the point of view of a non-CompSci interested in hardware, and the others in the research group were chemists content to write single threaded code on multi-core machines. Transferring the OpenMP variations of that code from a 1P to a 2P, as the results show, give variable results depending on the algorithm.
There are always ways to improve the efficiency of the code (and many ways to make it unreadable), but for a large part moving to the 2P system all depends on how your code performs. Please understand that my examples being within my limits of knowledge and representative of the research I did :) I know that SSE2/SSE4/AVX would probably help, but I have never looked into those. More often than not, these environments are all about research throughput, so rather than spend a few week to improve efficiency by 10% (or less), they'd rather spend that money getting a faster system which theoretically increases the same code throughput 100%.
I'll have a look at switching the loops if I write an article similar to this in the future :)
Ian