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
Explicit Finite Difference
For a grid of points or nodes over a simulation space, each point in the space can describe a number of factors relating to the simulation – concentration, electric field, temperature, and so on. In terms of concentration, the material balance gradients are approximated to the differences in the concentrations of surrounding points. Consider the concentration gradient of species C at point x in one dimension, where [C]x describes the concentration of C at x:
[1]
The second derivative around point x is determined by combining the half-differences from the adjacent half-points next to x:
[2]
Equations [1] and [2] can be applied to the partial differential equation [3] below to reveal a set of linear equations which can be solved.
Fick’s first law for the rate of diffusional mass transport can be applied in three dimensions:
[3]
where D is the diffusion coefficient of the chemical species, and t is the time. For the dimensional analyses used in this work, the Laplacian is split over the Cartesian dimensions x, y and z.
Dimension transformations are often employed to these simulations to relieve the simulation against scaling factors. The expansion of equation [3] given specific dimension transforms not mentioned here give equation [4] to be solved.
[4]
The expansion of equation [4] using coefficient collation and expansion by finite differences mentioned in equations [1] and [2] lead to equation [5]:
[5]
Equation [5] represents a series of concentrations that can be calculated independently from each other – each concentration can now be solved for each timestep (t) by an explicit algorithm for t+1 from t.
The explicit algorithm is uncommonly used in electrochemical simulation, often due to stability constraints and time taken to simulate. However, it offers complete parallelisation and low thread density – ideal for multi-processor systems and graphics cards – such that these issues should be overcome.
The explicit algorithm is stable when, for equation [4], the Courant–Friedrichs–Lewy condition [6] holds.
[6]
Thus the upper bound on the time step is fixed given the minimum grid spacing used.
There are variations of the explicit algorithm to improve this stability, such as the Dufort-Frankel method of considering the concentration at each point as the linear function of time, such that:
[7]
However, this method requires knowledge of the concentrations of two steps before the current step, therefore doubling memory usage.
Application to this Review
For the purposes of this review, we generate an x-by-x / x-by-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. As each node in the grid is independently calculable from each other, it would be ideal to spawn as many threads as there are nodes. However, each node has to load the data of the nodes of the previous time step around it, thus we restrict parallelization in one dimension and use an iterative loop to restrict memory loading and increase simulation throughput.
The code was written in Visual Studio C++ 2012 with OpenMP as the multithreaded source. The main functions to do the calculations are as follows.
For 2D:
For 3D:
For our scores, we increase the size of the grid from a 2x2 or 2x2x2 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.
Firstly, the results for our 2D and 3D Explicit Finite Difference simulations are shocking. In 3D, the dual processor system gets beaten by a mobile i7-2670QM (!). In both 2D and 3D, it seems being able to quickly access main memory and L3 caches is top priority. Each processor in the DP machine has to go out to memory when they need a value not in cache, whereas in a single processor machine it can access the L3 cache if the simulation fits wholly in there and the processor can predict which numbers are required next.
Also it should be noted that enabling HyperThreading in 3D gave a better than 50% increase in throughput on the dual processor machine.
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Hakon - Saturday, January 5, 2013 - link
Thank you for the detailed answer. I very much appreciate your article and hope to see more stuff like this on Anandtech.What I meant regarding to NUMA is the following. When you have a dual socket Xeon you have two memory controllers. The first time you 'touch' a memory location it is assigned to the memory controller of the CPU that runs the current thread. This assignment is in general permanent and all further memory read/writes to that location will be served by that memory controller.
If you first-touch (e.g. initialize the array to zero) using one thread, then the whole array is assigned to one of the two memory controllers. When you then run the multi-threaded code on that array one memory controller is idle while the other is oversubscribed since it has to serve both CPUs.
In contrast, if you first-touch your array in an OpenMP loop and use the same access pattern as in the algorithm, you will benefit from both memory controllers later on. In this case your large array is correctly 'distributed' over both memory controllers.
This kind of memory layout optimization becomes extremely important when you deal with quad socket Opterons. You then have eight memory controllers. A NUMA aware code is therefore up to eight times as fast since it utilizes all memory controllers.
toyotabedzrock - Saturday, January 5, 2013 - link
You should go ask the people on the assembly boards for help with making your code faster.They are very friendly compared to a Linux kernel devs, I think they just enjoy the acknowledgement that they still exist and are useful.
snajpa - Saturday, January 5, 2013 - link
Blame the scheduler. Neither Windows nor linux can effectively handle larger NUMA systems. It randomly moves the process across the physical hardware.psyq321 - Sunday, January 6, 2013 - link
Hmm, this is definitely not true at least for Windows Server 2008 R2 / Windows 7, and I am sure it holds true for some versions of Linux (I am not a Linux expert).Windows Server 2008 R2 / Windows 7 scheduler will try to match the memory allocations (even if they are not tagged for a specific NUMA node) with the NUMA node the process/thread resides on, and they will not move a thread to a foreign NUMA node unless if that has been explicitly requested by the application (by setting the thread affinity)
Of course, without explicit NUMA node tagging when doing allocations, application code is the main culprit for not respecting the NUMA layout (e.g. creating bunch of threads, allocating memory from one of them - and then pinning the threads to different CPUs - you will have lots of LLC requests from remote DRAM because memory was a-priori allocated on one node).
For this - some sane coding helps a lot, here:
http://www.dimkovic.com/node/15
I describe how I extracted more than double performance by careful memory allocation (NUMA-aware) - please note that neither Windows nor Linux scheduler is able to cope with code which is not written to be NUMA aware and it is using large number of threads that are supposed to run on all CPUs.. Simply put, application writer will have to manage memory allocation and usage in the way so that there are as little remote DRAM requests as possible.
snajpa - Sunday, January 6, 2013 - link
About Windows scheduler - I only worked with Windows XP, now I don't have any reason to work with Win anymore, so what you say probably is really true. As for the linux versions - well, long story short, CFS sucks and everyone knows it - this is particularly noticeable if you have fully virtualized VMs which appear as one single process at the host system - the process is randomly swapped between CPU cores and even CPU dies.... sad story. That's why people have to pin their CPUs to their tasks manually.psyq321 - Sunday, January 6, 2013 - link
Ah, XP - that explains it. True, XP did not care about NUMA at all.Windows Server 2008 / Vista introduced NUMA-aware memory allocations, and changed their CPU scheduler so it does not move the thread across NUMA nodes. They will also try to allocate the memory from the thread's own NUMA node when legacy VirtualAlloc etc. APIs are used.
Windows Server 2008 R2 / Windows 7 introduced the concept of CPU groups - allowing more than 64 CPUs. This does require some adaptation of the application, as old threading APIs only work with 64-bit affinity bitmask which only allowed recognizing 64 CPUs. Now, there is a new set of APIs that work with GROUP_AFFINITY structure, allowing control of CPU groups, too. However, this needs explicit change of the legacy process/threading APIs to the new ones.
Furthermore, none of the above can replace some manual intervention*- while Windows scheduler will, indeed, respect NUMA node boundaries and not try to mess around with moving threads across them - it still does not know what the underlying algorithm wants to do.
* There is no need to set the thread affinity to one specific CPU anymore - this prevents running the thread on any other CPU completely. Instead, there is an API called SetThreadIdealProcessor(Ex) which signals Windows scheduler that thread >should< run on that particular CPU - but, under certain circumstances the scheduler can move the thread somewhere else - if the CPU is completely taken away by some other thread/process. Scheduler will try to move the thread as close as possible - to the next core in the socket, for example - or to the next core in the group (group is always contained within a NUMA node).
You can, however, absolutely forbid Windows scheduler from passing the thread to another NUMA node under any circumstances by simply getting the said NUMA node affinity mask (GetNumaNodeProcessorMask(Ex)) and setting this affinity as a thread affinity. This + setting the "ideal" processor still gives Windows scheduler some headroom to move the thread to another core if it is found to be better in a given moment, but it will not even attempt to cross the NUMA boundary in any case whatsoever.
lmcd - Monday, January 7, 2013 - link
While I haven't personally researched them, there are tons of other schedulers that have been written for Linux and I'm certain *at least* one of them is more fitting to this line of work. I've heard of alternatives like BFS and the Linux kernel is so widely used I'm sure there's a gem out there for this application.toyotabedzrock - Saturday, January 5, 2013 - link
Have you ever tried the Intel Math Kernel Library? It might speed up some of the equations. It also hands off work to the Intel MIC card if it thinks it will speed it up.http://software.intel.com/en-us/intel-mkl/
KAlmquist - Saturday, January 5, 2013 - link
The GA-7PESH1 motherboard is $855, and the CPU's are $2020 each, which adds up to $4895. On tasks which don't parallelize well, you can get similar performance from the i7-3770K, which costs an order of magnitude less. (Prices: i7-3770K $320, ASRock Z77 Extreme6 motherboard $152, total from motherboard and CPU $472.) On tasks which parallelize well enough that they can be run on a GPU, the system with the GA-7PESH1 will beat the i7-3770K, but will be crushed by a midrange GPU. So the price/performance of this system is pretty bad unless you throw just the right workload at it.The motherboard price from super-laptop-parts dot com, and the other prices are from a major online retailer that I won't name in order to get around the spam filter.
Death666Angel - Saturday, January 5, 2013 - link
So, your 3770K has ECC memory or VT-D, TXT etc.?