Hello...
I am a retired theoretical physical chemist with a long association with computers and computing.
As briefly as possible, my interests are in the behavior of fluids at a phase boundary, such as a real gas at a solid
surface: the attractive forces of the solid cause an increased concentration (density) of the gas in the region near the surface,
a measureable phenomenon called "adsorption". Thermodynamics requires that, at equilibrium at a constant temperature and
pressure, all parts of the system have the same "chemical potential" and that the distribution of densitys must also yield the minimum value
of free energy for the system.
A fairly recent way of treating such problems is by "Density Functional Theory". The integral equation can be expressed as a series of functions, each of which is a function of the density at the system coordinates (hence a functional).
One of the terms in the relationship is a type of "N-Body" problem, in which it is necessary to calculate the potential (a scalar quantity) at each of N
volume elements due to all other N volume elements in the system; an (O)N^2 problem. Usually, the Lennard-Jones potential is used.
Since the fluid (gas) is inhomogenious, every volume element is characterized by its own density. The solution to the problem
is the distribution of densities that minimize the free energy of the system. This can be done by a numerical iterative method, but convergence is slow
hence the iterations required are many. Naturally, the Lennard-Jones term uses way over 98% of the time. A reasonable value for N in a small (nanoscale) 3 dim model will be a grid with a few million points.
The fastest CPU system I have at my disposal is an Intel 5960x (8 cores/16 threads) and x99 chipset running happily at 4.3 Ghz. (lspci says its a Xeon E5 v3/core I7) Using Intel's latest C tools, and after a lot of tuning, the inhomogeneous L-J problem I have outlined maxes
out at 342 seconds for 1e6 points ( 1e12 interactions ). I estimate about 30 double precision ops/interaction,
so roughly 87 Gflop/sec. (This is on Scientific Linux 6.6 platform) The system monitor shows all threads at 100%.
This would not really be intolerable for exploring a small, but meaningful model ... I'm retired, after all.
But it is frustratingly slow for finishing the task, as developing the heuristics to get efficient and reliable convergence takes
a lot of experimentation. As far as I know, a fully 3D non-local DFT has never been explored.
Briefly, I rasterize the 3-D grid into an array of struct of float x,y,z coordinates (call it vecA) and similar vector with an additional member, float 'dens'.(vecB). The calculations consist of a nested loop, each element of A looks at each element of B,caclulates a function of their separation times the density at B and sums it into a temp variable. After all B, the sum is stored in a third (C) vector element identified with index of A. So no data is changed during the calculation. The result is stored in vecC. Only the densities ar changed between calls to the function.
So my questions are,
Do you think I'm getting about the max from my present CPU? (Already about 92x an unoptimized single thread)
How much more could I expect to get from a Xeon Phi, and which one? (I'd really like a factor of 10...)
J. P. Olivier
