Adapted nested force-gradient integrators: the Schwinger model case
Dmitry Shcherbakov, Matthias Ehrhardt, Jacob Finkenrath, Michael Günther, Francesco Knechtli, Michael Peardon
AAdapted nested force-gradient integrators: the Schwingermodel case
Dmitry Shcherbakov ∗ , Matthias Ehrhardt , Jacob Finkenrath ,Michael G ¨unther , Francesco Knechtli and Michael Peardon Lehrstuhl f ¨ur Angewandte Mathematik und Numerische Analysis, Bergische Uni-versit¨at Wuppertal, Gaußstrasse 20, 42119 Wuppertal, Germany CaSToRC, CyI, 20 Constantinou Kavafi Street, 2121 Nicosia, Cyprus Theoretische Physik, Bergische Universit¨at Wuppertal, Gaußstrasse 20, 42119 Wup-pertal, Germany School of Mathematics, Trinity College, Dublin 2, Ireland
Abstract.
We study a novel class of numerical integrators, the adapted nested force-gradient schemes, used within the molecular dynamics step of the Hybrid Monte Carlo(HMC) algorithm. We test these methods in the Schwinger model on the lattice, a wellknown benchmark problem. We derive the analytical basis of nested force-gradienttype methods and demonstrate the advantage of the proposed approach, namely re-duced computational costs compared with other numerical integration schemes inHMC.
AMS subject classifications : 65P10 65L06 34C40
Key words : numerical geometric integration, decomposition methods, energy conservation, force-gradient, nested algorithms, multi-rate schemes, operator splitting, Schwinger model
For the Hybrid Monte Carlo algorithm (HMC) [5], often used to study quantum chromo-dynamics (QCD) on the lattice, one is interested in efficient numerical time integrationschemes which are optimal in terms of computational costs per trajectory for a givenacceptance rate. High order numerical methods allow the use of larger step sizes, butdemand a larger computational effort per step; low order schemes do not require suchlarge computational costs per step, but need more steps per trajectory. So there is a needto balance these opposing effects. ∗ Corresponding author.
Email addresses: [email protected] (D. Shcherbakov), [email protected] (M. Ehrhardt), [email protected] (J. Finkenrath), [email protected] (M. G ¨unther), [email protected] (F. Knechtli), [email protected] a r X i v : . [ m a t h . NA ] N ov Omelyan integration schemes [11] of a force-gradient type have proved to be an ef-ficient choice, since it is easy to obtain higher order schemes that demand a small ad-ditional computational effort. These schemes use higher-order information from force-gradient terms to both increase the convergence of the method and decrease the size ofthe leading error coefficient. Other ideas to achieve better efficiency for numerical timeintegrators are given by multirate or nested approaches. These schemes do not increasethe order but reduce the computational costs per path by recognizing the different dy-namical time-scales generated by different parts of the action. Slow forces, which areusually expensive to evaluate, need only to be sampled at low frequency while fast forceswhich are usually cheap to evaluate need to be sampled at a high frequency. A naturalway to inherit the advantages from both force-gradient type schemes and multirate ap-proaches would be to combine these two ideas.Previously, we studied the behavior of the adapted nested force-gradient scheme forthe example of the n -body problem [15] and would like to learn more about their useful-ness for lattice field theory calculations. Due to the huge computational effort requiredfor QCD simulations, it is natural to attempt an intermediate step first. We chose themodel problem of quantum electrodynamics (QED) in two dimensions, the Schwingermodel [12], since it is well-suited as a test case for new concepts and ideas which canbe subsequently applied to more computationally demanding problems [4]. As a latticequantum field theory, it has many of the properties of more sophisticated models such asQCD, for example the numerical cost is still dominated by the fermion part of the action.The fact that this model, with far fewer degrees of freedom, does not require such largecomputational effort makes it the perfect choice for testing purposes.We compare the behavior of numerical time integration schemes currently used forHMC [11] with the nested force-gradient integrator [3] and the adapted version intro-duced in [15]. We investigate the computational costs needed to perform numerical cal-culations, as well as the effort required to achieve a satisfactory acceptance rate duringthe HMC evolution. Our goal is to find a numerical scheme for the HMC algorithmwhich would provide a sufficiently high acceptance rate while not drastically increasingthe simulation time.The paper is organized as follows. In Section 2 we give a short overview of theHMC algorithm and numerical schemes for time integration, which are used in HMC.In Section 3 we present the 2-dimensional Schwinger model and introduce the idea ofthe force-gradient approach and the resulting novel class of numerical schemes. Section4 is devoted to the results of a comparison between widely used algorithms and the newapproach and Section 5 draws our conclusion. In this section we provide a general overview of the HMC algorithm [5] to introduceour novel integrator. We also present some standard numerical time integrating methods used in HMC, as well state-of-the-art numerical schemes, which we later compare byapplying them to the two-dimensional Schwinger model.
In the Hybrid Monte Carlo algorithm, the quantum lattice field theory is embedded in ahigher-dimensional classical system through the introduction of a fictitious (simulation)time [5]. The gauge field U is associated with its (fictitious) conjugate momenta P , andthe classical system is described by the Hamiltonian, H = A [ P ]+ B [ U ] , (2.1)where A [ P ] and B [ U ] represent the kinetic and potential energy respectively.For a given configuration U , a new configuration U (cid:48) is generated by performing anHMC update U → U (cid:48) , which consists of two steps: • Molecular Dynamics trajectory:
Evolve the gauge fields U , elements of a Lie group,and the momenta P , elements of the corresponding Lie algebra, in a fictitious com-puter time t according to Hamilton’s equations of motions˙ P = − ∂ H ∂ U = − F V ( U ) , ˙ U = PU . (2.2)Since analytical solutions are not available in general, numerical methods must beused to solve the system of Eqn. (2.2). The discrete updates of U and P with anintegration step h are e A h : U → U (cid:48) = exp ( iPh ) U e B h : P → P (cid:48) = P − ihF V ( U ) ,leading to a first-order approximation at time t + h . Since the momenta P are ele-ments of Lie algebra, we have an additive update of P . On the other hand, the links U must be elements of the Lie group, therefore an exponential update is used for U to preserve the underlying group structure. • Metropolis step:
Accept or reject the new configuration ( U (cid:48) , P (cid:48) ) with probability P ( U → U (cid:48) ) = min (cid:16) − ∆ H (cid:17) ,where ∆ H = H ( U (cid:48) , P (cid:48) ) − H ( U , P ) . In this paper we are concerned with numerical time integration schemes, which preservethe fundamental properties of geometric integration, time-reversibility and volume-pre-servation. All numerical schemes presented below possess these necessary properties.
Basic schemes:
Well-known, commonly used integration schemes in molecular dy-namics are given by • the leap-frog method, a 3-stage composition scheme of the discrete updates definedabove: ∆ ( h ) = e h ˆ B e h ˆ A e h ˆ B , (2.3) • and a 5-stage extension widely used in QCD computations: ∆ ( h ) = e h ˆ B e h ˆ A e h ˆ B e h ˆ A e h ˆ B . (2.4) Force gradient schemes:
Force-gradient schemes increase accuracy by using addi-tional information from the force gradient term C = {B , {A , B}} , with { , } defining Liebrackets. The 5-stage force-gradient scheme proposed by Omelyan et al [11] is the sim-plest; ∆ C ( h ) = e h ˆ B e h ˆ A e h ˆ B− h C e h ˆ A e h ˆ B . (2.5)Here we also test the modification of the force-gradient method (2.5) proposed in [17],where the force-gradient term C is approximated via a Taylor expansion. An extension isgiven by the 11-stage decomposition [11], recently implemented as the integrator in theopen source code openQCD as one of the standard options [10] ∆ ( h ) = e σ h ˆ B e η h ˆ A e λ h ˆ B e θ h ˆ A e ( − ( λ + σ )) h ˆ B e ( − ( θ + η )) h ˆ A e ( − ( λ + σ )) h ˆ B e θ h ˆ A e λ h ˆ B e η h ˆ A e σ h ˆ B , (2.6) where σ , θ , λ and η are parameters from equation (71) in Ref. [11]. Nested Schemes:
QED and QCD problems usually lead to Hamiltonians with thefollowing fine structure H = A [ P ]+ B [ U ]+ B [ U ] , (2.7)where the action of the system can be split into two parts: a fast action B such as thegauge action, and a slow part B , for example, the fermion action. This allows us toapply the idea of multirate schemes (an idea known as nested integration in physicsliterature) [13] in order to reduce the computational effort. At first we consider the nestedversion of the leap-frog method (2.3)ˆ ∆ ( h ) = e h ˆ B ∆ ( h ) M e h ˆ B , (2.8)where the inner cheaper system H = A [ P ]+ B [ U ] is solved by ∆ ( h ) M = (cid:16) e h M ˆ B e hM ˆ A e h M ˆ B (cid:17) M , with M being a number of iterations for the fast part of the action. Our main goal is tocompare the above-mentioned methods with more elaborated nested schemes: in [15], asimilar 5-stage decomposition scheme has been recently introduced:ˆ ∆ ( h ) = e h ˆ B ∆ (cid:18) h (cid:19) M e h ˆ B ∆ (cid:18) h (cid:19) M e h ˆ B . (2.9)A nested version of (2.5), which has been used in [1] readsˆ ∆ C ( h ) = e h ˆ B ∆ (cid:18) h (cid:19) M e h ˆ B + h C f ∆ (cid:18) h (cid:19) M e h ˆ B , (2.10)where ∆ ( h ) M = (cid:18) e M h ˆ B e M h ˆ A e M h ˆ B + ( hM ) C g e M h ˆ A e M h ˆ B (cid:19) M ,with C g = {B , {A , B }} and C f = {B , {A , B }} . In the limit M → ∞ we have ∆ = ∆ . Notethat this approach uses force-gradient information at all levels, i.e., the high computa-tional cost of high order schemes appears at all levels.One may overcome this problem by using schemes of different order at the differentlevels without losing the effective high order of the overall multirate scheme. For thelatter, we include appropriate force gradient information as we explain in the followingfor the case of two time levels, where the gauge action plays the role of the fast andcheap part, and the fermionic action plays the role of the slow and expensive part. Thereasoning is as follows: if one uses the 5-stage Sexton-Weingarten integrator of secondorder for the slow action, and approximates the fast action by m Leap-frog steps of stepsize h / ( m ) , the error of the overall multirate scheme will be of order O ( h )+ O (( hm ) )+ O ( h ) . With the use of force gradient information only at the slowest level it is possibleto cancel the leading error term of order O ( h ) . As m ≥ h usually holds in the multiratesetting, the overall order is then given by the leading error term of order O ( h ) , i.e., thescheme has an effective order of four. One example for such a scheme for problems oftype (2.7) is given by the 5-stage nested force-gradient scheme introduced in [15]˜ ∆ C ( h ) = e h ˆ B ∆ (cid:18) h (cid:19) M e h ˆ B + h C f ∆ (cid:18) h (cid:19) M e h ˆ B . (2.11)To summarize, the adapted scheme (2.11) differs from the original one (2.10) in twoperspectives: • The force gradient scheme for the fast action is replaced by a leap-frog scheme. • Only the part {B , {A , B }} of the full force gradient {B + B , {A , B + B }} is neededto gain the effective order of four. The numerical schemes (2.3)-(2.4) and (2.8)-(2.9) are second order convergent schemes.Methods (2.5)-(2.6) and (2.10) - (2.11) have the fourth order of convergence. We do notconsider integrators of higher order than four since the computational costs are too high.The schemes of the same convergence order differ from each other by the number ofstages (updates of momenta and links per time step). Usually methods with more stageshave smaller leading error coefficients and therefore have better accuracy, but highercomputational costs. We would like to determine which integrator would represent thebest compromise between high accuracy and computational efficiency.We will apply all these numerical integration schemes (2.3)–(2.11) to the two-dimensionalSchwinger model. The most challenging task from the theoretical point of view is to de-rive the force-gradient term C . In the next section we introduce the Schwinger model andexplain how to obtain the force-gradient term. The 2 dimensional Schwinger model is defined by the following Hamiltonian function H = V ,2 ∑ n = µ = p n , µ + S f ull [ U ] = V ,2 ∑ n = µ = p n , µ + S G [ U ]+ S F [ U ] . (3.1)with V = L × T the volume of the lattice. Unlike QCD, where U ∈ SU ( ) and p n , µ ∈ su ( ) ,for this QED problem (3.1), the links U are the elements of the Lie group U ( ) and the mo-menta p n , µ belong to R , which represents the Lie algebra of the group U ( ) . This makesthis test example (3.1) very cheap in terms of the computational time. This together withthe fact that the Schwinger model also shares many of the features of QCD simulations,makes the Schwinger model an excellent test example when considering numerical in-tegrators: a fast dynamics given by the computationally cheap gauge part S G [ U ] of theaction demanding small step sizes, and a slow dynamics given by the computationallyexpensive fermion part S F [ U ] allowing large step sizes.The pure gauge part of the action S G sums up over all plaquettes P ( n ) in the two-dimensional lattice with P ( n ) = U ( n ) U ( n + ˆ1 ) U †1 ( n + ˆ2 ) U †2 ( n ) ,and is given by S G = β V ∑ n = ( − Re P ( n )) . (3.2)The links U can be written in the form U µ ( n ) = e iq µ ( n ) ∈ U ( ) and connect the sites n and n + ˆ µ on the lattice; q µ ( n ) ∈ [ − π , π ] , µ , ν ∈ { x , t } are respectively space and time directionsand β is a coupling constant. Note that from now on we will set the lattice spacing a = The fermion part of the action S F is given by S F = η † (cid:16) D † D (cid:17) − η , (3.3)where η is a complex pseudofermion field. Here, D denotes the Wilson–Dirac operatorgiven by D n , m = ( + m ) δ n , m − ∑ µ = (cid:16) ( − σ µ ) U µ ( n ) δ n , m − ˆ µ +( + σ µ ) U † µ ( n − ˆ µ ) δ n , m + ˆ µ (cid:17) ,where σ µ are the Pauli matrices σ = (cid:18) (cid:19) and σ = (cid:18) − ii (cid:19) . m is the mass parameter and the Kronecker delta δ n , m acts on the pseudofermion field by ∑ Vm = δ n , m η ( m )= η ( n ) with η ( n ) the pseudofermion field, a vector in the two-dimensionalspinor space taking values at each lattice point n . In order to proceed with the numericalintegration we need to obtain the force F and the force gradient term C . The force term F ( n , µ ) with respect to the link U µ ( n ) is given by the first derivative of the action S f ull and can be written as F ( n , µ ) = F S G ( n , µ )+ F S F ( n , µ ) = ∂ S G ∂ q µ ( n ) + ∂ S F ∂ q µ ( n ) . (3.4)Since the numerical schemes (2.9)–(2.11) use the multi-rate approach, the shifts in themomenta updates are split on F S G and F S F and we can consider them separately. Theforce terms F S G and F S F are obtained by differentiation over U ( ) group elements, whichfor the Schwinger model is the standard differentiation.The force associated with link U µ ( n ) from the gauge action is given by β g ( n , µ ) : = F S G ( n , µ ) = β Im ( P ( n ) − P ( n − ˆ ν )) | µ (cid:54) = ν . (3.5)The force term of the fermion part is given by f ( n , µ ) : = F S F = − Im (cid:104) χ † ( n )( − σ µ ) U µ ( n ) ξ ( n + ˆ µ ) − χ † ( n + ˆ µ )( + σ µ ) U † µ ( n ) ξ ( n ) (cid:105) (3.6) where vectors χ and ξ are given χ = D † − η , ξ = D − D † − η . (3.7)For the numerical methods (2.5) and (2.10) we need to find the force gradient term C ( n , µ ) with respect to the link U µ ( n ) . In case of the Schwinger model (2.1) this termreads C ( n , µ ) = V ,2 ∑ m = ν = ∂ S f ull ∂ q ν ( m ) ∂ S f ull ∂ q ν ( m ) ∂ q µ ( n ) . (3.8) For simplicity we decompose the force gradient term (3.8) in four parts C GG = V ,2 ∑ m = ν = ∂ S G ∂ q ν ( m ) ∂ S G ∂ q ν ( m ) ∂ q µ ( n ) , C FG = V ,2 ∑ m = ν = ∂ S F ∂ q ν ( m ) ∂ S G ∂ q ν ( m ) ∂ q µ ( n ) , C GF = V ,2 ∑ m = ν = ∂ S G ∂ q ν ( m ) ∂ S F ∂ q ν ( m ) ∂ q µ ( n ) , C FF = V ,2 ∑ m = ν = ∂ S F ∂ q ν ( m ) ∂ S F ∂ q ν ( m ) ∂ q µ ( n ) . (3.9)This decomposition is also useful since the numerical integrator (2.10) only uses the term C FF by construction. As shown in [15], to obtain the fourth order convergent scheme(2.10) from the second order convergent method (2.9) we must eliminate the leading errorterm, which is exactly represented by C FF . For completeness we discuss all 4 parts below.The C GG part of the force-gradient term is C GG = β [ Im ( P ( n , µ ) − P ( n , µ ) − P ( n , µ ) − P ( n , µ ) − P ( n , µ )) Re ( P ( n , µ )) − Im ( P ( n , µ ) − P ( n , µ ) − P ( n , µ ) − P ( n , µ ) − P ( n , µ )) Re ( P ( n , µ ))] with the set of plaquettes P ( n , µ ) = U µ ( n ) U ν ( n + ˆ µ ) U † µ ( n + ˆ ν ) U † ν ( n ) , P ( n , µ ) = U µ ( n − ˆ ν ) U ν ( n − ˆ ν + ˆ µ ) U † µ ( n ) U † ν ( n − ˆ ν ) , P ( n , µ ) = U µ ( n + ˆ µ ) U ν ( n + µ ) U † µ ( n + ˆ ν + ˆ µ ) U † ν ( n + ˆ µ ) , P ( n , µ ) = U µ ( n + ˆ ν ) U ν ( n + ˆ µ + ˆ ν ) U † µ ( n + ν ) U † ν ( n + ˆ ν ) , P ( n , µ ) = U µ ( n − ˆ µ ) U ν ( n ) U † µ ( n + ˆ ν − ˆ µ ) U † ν ( n − ˆ µ ) , P ( n , µ ) = U µ ( n − ˆ µ − ˆ ν ) U ν ( n − ˆ ν ) U † µ ( n − ˆ µ ) U † ν ( n − ˆ µ − ˆ ν ) , P ( n , µ ) = U µ ( n − ν ) U ν ( n − ν + ˆ µ ) U † µ ( n − ˆ ν ) U † ν ( n − ν ) , P ( n , µ ) = U µ ( n − ˆ ν + ˆ µ ) U ν ( n − ˆ ν + µ ) U † µ ( n + ˆ µ ) U † ν ( n − ˆ ν + ˆ µ ) .Then by using the vectors f ( n , µ ) defined in (3.6) we obtain the C FG piece of the force-gradient term given by C FG ( n , µ )= β [( f ( n , µ )+ f ( n + ˆ µ , ν ) − f ( n + ˆ ν , µ ) − f ( n , ν )) Re ( P )+( f ( n , µ ) − f ( n + ˆ µ − ˆ ν , ν ) − f ( n − ˆ ν , µ )+ f ( n − ˆ ν , ν )) Re ( P )] . The second derivative of the fermion action is ∂ S F ∂ q ν ( m ) ∂ q µ ( n ) = χ † (cid:20) ∂ D ∂ q ν ( m ) D − ∂ D ∂ q µ ( n ) + ∂ D ∂ q µ ( n ) D − ∂ D ∂ q ν ( m ) − ∂ D ∂ q ν ( m ) ∂ q µ ( n ) (cid:21) ξ + χ † ∂ D ∂ q µ ( n ) ( D † D ) − ∂ D † ∂ q ν ( m ) χ , = (cid:20) z †1, m , ν ∂ D ∂ q µ ( n ) ξ + χ † ∂ D ∂ q µ ( n ) D − w m , ν − χ † ∂ D ∂ q ν ( m ) ∂ q µ ( n ) ξ + χ † ∂ D ∂ q µ ( n ) D − z m , ν (cid:21) = (cid:20) z †1, m , ν w n , µ + w †1, n , µ z m , ν − χ † ∂ D ∂ q ν ( m ) ∂ q µ ( n ) ξ (cid:21) (3.10)in terms of the vectors χ and ξ defined in (3.7). Now the fields z m , ν and z m , ν are givenby z m , ν : = D † − ∂ D † ∂ q ν ( m ) χ = D † − w m , ν z m , ν : = D − ( ∂ D ∂ q ν ( m ) ξ + z m , ν ) = D − ( w m , ν + z m , ν ) with w m , ν ( n ) : = ∑ n (cid:48) ∂ D † n , n (cid:48) ∂ q ν ( m ) χ ( n (cid:48) ) = δ n , m + ˆ ν i ( − σ ν ) U † ν ( m ) χ ( m ) − δ n , m i ( + σ ν ) U ν ( m ) χ ( m + ˆ ν ) , w m , ν ( n ) : = ∑ n (cid:48) ∂ D n , n (cid:48) ∂ q ν ( m ) ξ ( n (cid:48) ) = − δ n , m i ( − σ ν ) U ν ( m ) ξ ( m + ˆ ν )+ δ n , m + ˆ ν i ( + σ ν ) U † ν ( m ) ξ ( m ) .In order to calculate C GF and C FF it is possible to perform the summation of ∑ m , ν before the inversions of D and D † to get z and z which save O ( V ) additional inversionsfor the force gradient terms. It follows for the force gradient term C FF C FF ( n , µ ) = (cid:20) Z †1 w n , µ + w †1, n , µ Z − χ † ∂ D ∂ q µ ( n ) ∂ q µ ( n ) ξ · f ( n , µ ) (cid:21) (3.11)with Z : = D † − V ,2 ∑ m = ν = ( w m , ν · f ( m , ν )) , Z : = D − (cid:32) V ,2 ∑ m = ν = [ w m , ν · f ( m , ν )]+ Z (cid:33) . (3.12)The expression for C GF can be obtained from the one for C FF by replacing in (3.11) and(3.12) the vector f with β g defined in (3.5). It is important to mention that the computationally most demanding part of the nu-merical integration of the Schwinger model and quantum field theory in general is theinverse of the Dirac operator D − . Every momenta update, which includes fermion ac-tion (3.6) requires 2 inversions of the Dirac operator, the addition of the force-gradientterm C requires 4 more inversions. Therefore leap-frog based methods (2.3) and (2.8)need 4 computations of D − per time step; schemes (2.4) and (2.9) 6 times; force-gradientbased methods 8 for (2.10) and (2.11), 10 for (2.5) and the 11 stage method (2.6) has 12inversions of the Dirac operator. Since we use the multi-rate approach for schemes (2.9),(2.10) and (2.11), which leads generally to fewer macro time steps needed than for thestandard schemes we expect the integrator (2.11) will be the most efficient choice amongthe methods considered. In the next section we present numerical tests of this prediction. In this section we apply the numerical integrators (2.3) – (2.11) to compute the moleculardynamics step for the Schwinger model (3.1) when studied with the HMC algorithm. Weconsider a 32 by 32 lattice with a coupling constant β = m = − z = τ = | ∆ H | and estimate its statistical error from thestandard deviation. Also the parameter M is chosen in such a way to make micro stepsize to be 10 times smaller than the macro step size h . step-size, h | " H | -4 -3 -2 -1 h h leapnested leap5-stagenested 5-stageforce-gradientapprox. force-gradientad nested force-gradientnested force-gradient11-stage Figure 1: Absolute error for different numerical integrators. Figure 1 presents the comparison between the numerical integrators (2.3) – (2.11). Itshows the absolute error | ∆ H | versus the step-size of the numerical scheme. Here themulti-rate schemes (2.8), (2.9), (2.10) and (2.11) outperform their standard versions asexpected. Also it is easy to see that the scheme (2.6) has the best accuracy and the nestedforce-gradient method (2.10) just slightly edges the adapted nested force-gradient scheme(2.11). cpu time, s | " H | -3 -2 -1 leap Figure 2: Computational costs for different methodsFigure 2 presents the CPU time, required for the proposed integrators (2.4)–(2.11),versus the achieved accuracy. We can observe that the nested force-gradient method(2.10) and adapted nested force- gradient method (2.11) show much better results in termsof a computational efficiency than the integrators (2.9) and (2.5); and even compared tothe 11 stage scheme (2.6). Here we can see that the modification of (2.5) proposed in [17]also performs better than its original version. It shows almost similar computationalcosts as nested versions of the force-gradient approach (2.10) -(2.11), since it has the samenumber of D − (see Table 1). But it is less efficient because the proposed nested approachis more precise.Integrator: step size h M D − per step D − per trajectory5 stage method 0.0294 - 6 420nested 5 stage method 0.0286 700 6 4085 stage force-gradient 0.0550 - 10 370approx. force-gradient [17] 0.0540 - 8 290nested force-gradient 0.0560 450 8 285adapted nested force-gradient 0.0560 450 8 28511 stage method 0.0625 - 12 384 Table 1: Step-sizes and number of inversions of D per step and per trajectory for acceptance rate of 90% Table 1 shows the number of inversions of the Dirac operator D , which is needed toreach 90% acceptance rate of the HMC. Since D − is the most computationally demand-ing part it is important to see how many of these inversions are required per each trajec-tory. From Table 1 it easy to see that the adapted nested force-gradient method (2.11) andnested force-gradient method (2.10) need the least number of D − per trajectory to reachthe chosen acceptance rate ≈ D is much more significant. We presented the nested force-gradient approach (2.10) and its adapted version (2.11)applied to a model problem in quantum field theory, the two-dimensional Schwingermodel. The derivation of the force-gradient terms was given and the Schwinger modelwas introduced. Nested force-gradient schemes seem to be an optimal choice with rela-tively high convergence order and low computational effort. Also it would be possible toimprove the algorithm (2.11) by measuring the Poisson brackets of the shadow Hamilto-nian of the proposed integrator and then tuning the set of optimal parameters, e. g. microand macro step sizes.In future work we will apply this approach to the HMC algorithm for numerical in-tegration in Lattice QCD. Here we expect the adapted nested-force gradient scheme tooutperform the original one, if we further partition the action into more than two parts,by using techniques to factorize the fermion determinant: less force-gradient informationis needed for the most expensive action, and only leap-frog steps are needed for the highfrequency parts of the action.
Acknowledgments
This work is part of project B5 within the SFB/Transregio 55
Hadronenphysik mit Gitter-QCD funded by DFG (Deutsche Forschungsgemeinschaft).
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