Numerical Algorithms in LQCD

Here I will attempt to explain some of the more technical numerical aspects as to the algorithms being used in tmLQCD and some other interesting things about QCD. A computationally significant part of Lattice Quantum Chromodynamics is the solving of the linear system .

The obvious approach would be to simply invert the matrix, however M, acting as the Dirac operator at every lattice point, is extremely large and so this is extremely expensive and unfeasible. As M is highly sparse there are far more efficient ways. Sparse matrices are matrices in which most of the entries are zero.

A very popular approach to solving a system such as this is the Conjugate Gradient algorithm. Given M is a NxN symmetric positive definite matrix, the conjugate gradient algorithm guarantees convergence in most N steps. However this approach is completely dependent on the fixed , so we need to completely redo it for every different right-hand-side.

In order to make solving for multiple ‘s more efficient, one approach is to make use of the Arnoldi algorithm, implemented in ARPACK to solve for some of the eigenvectors of the system and make use of these in the Conjugate Gradient algorithm. This is known as deflation. Solving for a number of eigenvectors which correspond to the smallest eigenvalues of the system we remove, or deflate, these degrees of freedom from the system.

Another technique is even-odd preconditioning. For this you decompose the lattice into odd and even sites, in a chequered way, such as that of a chessboard. Due to each site only being dependent on its nearest neighbour, we can also decompose the matrix in this way. Approaching the odd and even parts separately you can, usually by a factor 2, improve your run times.

A lot of people with scientific backgrounds will have heard about the nuclear strong force in relation to nuclear physics and will have seen me and others blog about the strong force from QCD, and it might be fair to think they are the same but in fact that is untrue. The nuclear strong force is in fact a consequence of the QCD strong force and is known as the residual strong force. Its force carrier is not the gluon as with the QCD strong force but the virtual light mesons, such as virtual pions.

A pion is shown above, in order to give an idea of the structure of this particular boson. It is made up of an up quark and an anti-down quark.