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average energy and its derivatives and the eld correlation functions, from which one can determine the masses of the particles The core of the algorithm that updates the eld values is a conjugate gradient computation that inverts a very large matrix, which is spread out over many nodes Its size is proportional to V V , where V is the number of lattice sites For a typical calculation on a 4D lattice with 32 sites per dimension, this matrix is approximately 324 324 or about 106 106 Inverting such a matrix would be impossible if the matrix was dense However, since the interactions are local, the matrix is sparse and typically has entries only for nearest neighbor points on the 4D grid Because of the sparse nature of the matrix, the conjugate gradient algorithm only requires the multiplication of a column vector by this matrix.

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This operation needs to be performed for every iteration of the conjugate gradient algorithm and forms the innermost kernel of the QCD algorithm On most computers and algorithms, it consumes about 90% of the CPU compute cycles The routine that performs this calculations is usually called the QCD kernel or the lattice Dirac operator, or simply D-slash Optimizing D-slash for a given computer is the heart of the effort of implementing the QCD algorithm and will be discussed in detail in Section 86 Because the D-slash matrix involves only nearest neighbor entries, it maps well on any massively parallel supercomputer that provides nearest neighbor communication between computing nodes One simply splits the lattice into smaller sublattices and assigns each sublattice to a computing node.

For example, if the massively parallel supercomputer is wired as a connected 4D grid with 84 processors, then the original problem on a 324 lattice can be implemented by putting 44 sites on each node For illustration purposes, Figure 82 shows a two-dimensional 8 8 square lattice split into a 2 2 grid of compute nodes Each node contains a 4 4 sublattice The sublattice surface data are communicated to the nearest neighbor nodes via the supercomputers communication links (shadowed arrows) If the massively parallel machine is not a four-dimensional grid of compute nodes, one would have to map the problem appropriately However, most interestingly, the design of parallel supercomputers is also constrained by the requirement of locality The cables that connect compute nodes need to be as short and as few as possible The particular map needed for QCD on BG/L is discussed in Section 85.

Although the lattice formulation makes it possible to simulate a relativistic quantum eld theory like QCD, what we are really after is the continuum properties of the theory These are obtained by taking the limit a 0; that is, by reducing the lattice spacing a while keeping a physical scale, such as the con nement scale, xed However, a cannot be made too small because this would result in too many lattice points In actual practice, one monitors how physical quantities change as a decreases and the results are extrapolated to a = 0 For a simulation of a region with a given physical size, the lattice spacing is carefully controlled so that it is not too small (this would mean more grid points and too much computing), or too large (this would make the nuclear particles smaller than the lattice spacing).

s(2 - 4) s(2 - 5 ) ~ (- i 6)

Because of these limitations, one must compute variables on many different lattice sizes and then extrapolate to a = 0 Early simulations used rather modest lattices 44 84 Today, lattice sizes are in the range of 324 to 644 As anecdotal reference has it that Wilson made an early.

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