Variants of HMC

In the preceding sections of this chapter, we summarized a selection of different tech-niques which are used in current lattice QCD codes. Indeed, the list is not intended to be exhaustive (see [23–25]). Since the first suggestion of the HMC algorithm, some variants of it have been arisen which make use of different algorithmic techniques to overcome numerical problems and/or to gain performance. ThePolynomialHybridMonte Carlo (PHMC) which was proposed byFrezotti andJansen [91, 92] combines the multiboson technique with the HMC algorithm. The inverse of the fermion matrix is approximated by a polynomial written in powers of the fermion matrix. In the case of two degenerated Wilson quarks (Qis defined as in section 4.8), the partition function

Z =

Z DUDφ^†DφDη^†Dη Wexp{−S_G−SP−Sη} (4.73) whereS_G is Wilson’s gauge action and

S_P=φ^†P_n,(Q²)φ, (4.74)

S_η =η^†η (4.75)

contains a correction factorW which is given by W = exp

The description (4.73) of the partition function with (4.74),(4.75) and (4.76) is still exact due to the trivial identity

det(Q²) = det(Q²P_n,(Q²)) det(P_n,(Q²)⁻¹)

which was used above. The polynomial Pn,(Q²) approximates the inverse (Q²)⁻¹ for all eigenvalues λof Q² with λ= [,1]. Quantities measured on configurations sampled

4.9 Variants of HMC by the exponential in (4.73) have to be reweighted with hWi⁻¹_P to obtain the correct average.

hOi=hWi⁻¹_P hWOi_P. (4.77)

h. . .i_P denotes the average evaluated with the action S_G+S_P+S_η. As the authors quote, the above description of the partition function allows to split up the eigenvalue spectrum ofQ² smoothly by choosing a suitable polynomialPn,(Q²) in a part which is included in the update procedure and a second part which is handled with the correction factor. The parameter in Pn,(Q²) serves as an infrared cutoff and controls the very low-lying eigenvalues of Q². Since the minimal eigenvalue of Q² is dictated by , it is expected that the expense of simulation decreases. Due to the infrared cutoff, the sampling in configuration space is different than with the conventional HMC algorithm.

The modes lower than do not “occur”. Therefore extra care has to be taken for observables which receive large contributions from low-lying modes (λ(Q²)< ) [91]. A comparable performance of PHMC to HMC can be achieved easily by choosing n and according to some estimation which are given in [92], but for a gain of performance, special tunings of nandare needed. The authors tested the performance of PHMC in comparison to HMC on a 16×8³ lattice with two flavors ofO(a) improved Wilson quarks with Schrödinger functional boundary conditions, even-/odd- preconditioning and the Chebyschev polynomials as an approximation for (Q²)⁻¹ as suggested byLüscher [90].

They could show a gain of a factor of around two in performance with some extra tuning of n and and quoted that this gain increases if one simulates on larger volumes. A further advantage of PHMC is that simulations with odd numbers of flavors are possible [80]. Furthermore a non-Hermitian Polynomial Hybrid Monte Carlo (NPHMC) has been studied [93] in which the inverse of the non-Hermitian-Wilson-Dirac operator was approximated by Chebyshev polynomials. Numerical tests showed that the performance of the NPHMC algorithm is comparable with conventional HMC algorithm but under certain conditions a slight gain was achieved.

Instead of approximating the fermion determinant by a polynomial, Clark and Ken-nedy proposed a rational approximation (RHMC) [94, 95]

det(M^†M)^α =

Z Dφ^†Dφexpⁿ−φ^†(M^†M)^−αφ^o (4.78)

≈

Z Dφ^†Dφexpⁿ−φ^†r²(M^†M)φ^o (4.79) where r(x) = x^−α/2. Equation (4.78) implies that one can define a theory with arbi-trary numbers of flavors N_f if one allows non-integer values for the parameter α. The conventional HMC algorithm would fail in this case because there is no method to eval-uate the action for fractional α. This is different in the case of RHMC. r(x) is usually written as a partial fraction r(x) =^Pm_k=1x+β^αk

k which can be evaluated very efficiently using a multi-shift solver [96]. For the cases of interest |α| < 1, all poles and roots of the rational approximation are positive and real, and the coefficients α_k of the partial fractions appear as positive. This leads to a numerically stable algorithm. The reason

4 Algorithmic improvements

for the positivity of the poles, roots and the coefficientsα_k is still not understood (and called therefore as miracles in [58]). The force calculation with the pseudo-fermion ac-tion in (4.79) would need a double inversion due to the square of r(x). To avoid this problem, a second rational approximation is used in the molecular dynamics [95]. The costs of RHMC as described above is comparable with HMC but RHMC permits the easy introduction of a single quark flavor [95] and is therefore suitable for simulations with odd numbers of flavors. An increasing of the step size, and thus a speedup, can be achieved by using then^th root trick in RHMC. However, as discussed before in sec-tion 4.7, the timescale for all pseudo-fermions in the n^th root trick is the same due to the same magnitude of pseudo-fermion forces and therefore one cannot derive benefit from the multiple timescale integration. A comparison of the performance of HMC with Hasenbusch preconditioning and RHMC with multiple pseudo-fermion fields (n^th root trick) revealed no significant difference between them. But it is reported [97] that the use of higher order integrator inn^th root trick leads to an improved volume scaling. As discussed in [95], the rational approximation has much better convergence properties than the polynomial approximation.

The last variant of HMC which we want to mention here is the DD-HMC provided by Lüscher. DD-HMC stands for Domain Decomposed Hybrid Monte Carlo and is a combination of domain decomposition methods [74] and the conventional HMC algorithm [68]. Lüscher discusses in [84, 85, 98] the applicability of domain decomposition methods to lattice QCD algorithms and describes the implementation of the Schwarz-alternating procedure as a preconditioner for HMC. Main features of Schwarz-preconditioning have already been discussed in section 4.5. In [87], he demonstrates very impressively how the DD-HMC code can be accelerated by applyingdeflation techniques.