Parallel adaptive methods for Feynman loop integrals. Conference on Computational Physics (CCP 2011)

Transcription

1 Outline Elise de Doncker 1 Fukuko Yuasa 2 1 Department of Computer Science, Western Michigan University, Kalamazoo MI 49008, U. S. 2 High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki , Japan Conference on Computational Physics (CCP 2011)

2 Outline Outline 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

3 Outline Outline 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

4 Outline Outline 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

5 Outline Outline 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

6 Outline Outline 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

7 Outline Outline 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

8 Introduction Automatic numerical integration Obtain an approximation Q(f) to an integral If = f( x) d x D in order to satisfy a specified accuracy requirement for the error Ef = Qf If (and an error estimate Ef ) such that: Qf If Ef max { t a, t r If } for given integrand function f, region D and (absolute/relative) error tolerances t a and t r.

9 Iterated integral Introduction Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface Integration over a product region D = D 1... D l, Iterated integral If = d x (1)... d x (l) f( x (1),..., x (l) ), D 1 D l implemented using lower-dimensional code across successive groups of dimensions, j = 1,...,l. E.g., 1D integration code (such as DQAGE from Quadpack[15]) can be used for 1D levels; or a combination of 1D and multivariate methods (such as DCUHRE [4]) across levels.

10 Outline Introduction Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

11 Priority driven adaptive algorithm Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface Sample problem Algorithm on each iterated level Evaluate initial region & update results Initialize priority queue to empty while (evaluation limit not reached and estimated error too large) Retrieve region from priority queue Split region Evaluate subregions & update results Insert subregions into priority queue (2D) k w kf(x k, y k ) (Subregion approx.) (1D 1D) i u i j v jf(x i, y j ) 0 dy 2δy = (x+y 1) 2 +δ 2 ] 2δy (x+y 1) 2 +δ 2 Figure: (for δ = 0.1) 1 0 dx dx [ 1 0 dy

12 Priority driven adaptive algorithm Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface Sample problem Algorithm on each iterated level Evaluate initial region & update results Initialize priority queue to empty while (evaluation limit not reached and estimated error too large) Retrieve region from priority queue Split region Evaluate subregions & update results Insert subregions into priority queue (2D) k w kf(x k, y k ) (Subregion approx.) (1D 1D) i u i j v jf(x i, y j ) 0 dy 2δy = (x+y 1) 2 +δ 2 ] 2δy (x+y 1) 2 +δ 2 Figure: (for δ = 0.1) 1 0 dx dx [ 1 0 dy

13 Outline Introduction Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

14 Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface Iterated vs. standard multivariate integration (a) (b) Figure: (a) Standard subdivision; (b) Iterated adaptive strategy for singularity on diagonal

15 Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface Iterated vs. standard multivariate integration (a) (b) Figure: (a) Standard subdivision; (b) Iterated adaptive Strategy for corner singularity

16 Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface Iterated vs. standard multivariate integration 1 0 dx 1 0 dy 2δy (x+y 1) 2 +δ 2, δ = 10 p DQAGE DQAGE DCUHRE p ABS. ERR. # EVAL. ABS. ERR. # EVAL e e e e e e e e e e e e e e e e e e e e e e

17 Outline Introduction Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

18 Modifications Elise de Doncker for relative and Fukuko error Yuasa tolerances Parallel adaptive also methods result. for Introduction Error tolerance interface Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface For double integration, If = I 1 = b 1 a 1 dx b 2 a 2 dy f(x, y), an error is incurred at both inner and outer integration [14], Q I b1 a 1 dx E 2 (x) + E 1 t (2) a b 2 a 2 + t (1) a. The error control can be implemented using estimated errors [8]. Extension to general l times iterated integral: Q I t (1) a + D 1 t (2) a where D j is the volume of region D j. l 1 j = 1 D j t (l) a

19 (1) On the rule or points level: As in non-adaptive algorithms, e.g., in Monte-Carlo algorithms and in composite rules using grid or lattice points, If = D f k w kf( x k ) : computation of the f( x k ) evaluation points in parallel; (2) On the region level: As in adaptive methods, task pool algorithms, load balancing (distributed memory systems); or maintaining shared priority queue structure; (3) On the iterated integration level: we compute inner integrals in parallel (within adaptive algorithm), e.g. (in 2D): over subregion S (inner region D 2 ) S F(x)dx k w kf(x k ), with F(x k ) = D 2 f(x k, y)dy; (4) Distributed computation of integrals from reductions.

20 (1) On the rule or points level: As in non-adaptive algorithms, e.g., in Monte-Carlo algorithms and in composite rules using grid or lattice points, If = D f k w kf( x k ) : computation of the f( x k ) evaluation points in parallel; (2) On the region level: As in adaptive methods, task pool algorithms, load balancing (distributed memory systems); or maintaining shared priority queue structure; (3) On the iterated integration level: we compute inner integrals in parallel (within adaptive algorithm), e.g. (in 2D): over subregion S (inner region D 2 ) S F(x)dx k w kf(x k ), with F(x k ) = D 2 f(x k, y)dy; (4) Distributed computation of integrals from reductions.

21 (1) On the rule or points level: As in non-adaptive algorithms, e.g., in Monte-Carlo algorithms and in composite rules using grid or lattice points, If = D f k w kf( x k ) : computation of the f( x k ) evaluation points in parallel; (2) On the region level: As in adaptive methods, task pool algorithms, load balancing (distributed memory systems); or maintaining shared priority queue structure; (3) On the iterated integration level: we compute inner integrals in parallel (within adaptive algorithm), e.g. (in 2D): over subregion S (inner region D 2 ) S F(x)dx k w kf(x k ), with F(x k ) = D 2 f(x k, y)dy; (4) Distributed computation of integrals from reductions.

22 (1) On the rule or points level: As in non-adaptive algorithms, e.g., in Monte-Carlo algorithms and in composite rules using grid or lattice points, If = D f k w kf( x k ) : computation of the f( x k ) evaluation points in parallel; (2) On the region level: As in adaptive methods, task pool algorithms, load balancing (distributed memory systems); or maintaining shared priority queue structure; (3) On the iterated integration level: we compute inner integrals in parallel (within adaptive algorithm), e.g. (in 2D): over subregion S (inner region D 2 ) S F(x)dx k w kf(x k ), with F(x k ) = D 2 f(x k, y)dy; (4) Distributed computation of integrals from reductions.

23 Outline Introduction 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

24 Higher order corrections (in addition to the lowest order/tree level) are required for accurate theoretical predictions of cross-sections for particle interactions. Loop diagrams need to be taken into account, necessitating the evaluation of loop integrals. L-loop integral with N internal lines I = Γ `N nl Z 1 NY 2 ( 1) N dx (4π) nl/2 i δ(1 X C N n(l+1)/2 x i ) (D iδc) N nl/2 0 i=1 C and D are polynomials determined by the topology of the corresponding diagram and physical parameters. The integral only exists in the limit as δ 0.

25 Higher order corrections (in addition to the lowest order/tree level) are required for accurate theoretical predictions of cross-sections for particle interactions. Loop diagrams need to be taken into account, necessitating the evaluation of loop integrals. L-loop integral with N internal lines I = Γ `N nl Z 1 NY 2 ( 1) N dx (4π) nl/2 i δ(1 X C N n(l+1)/2 x i ) (D iδc) N nl/2 0 i=1 C and D are polynomials determined by the topology of the corresponding diagram and physical parameters. The integral only exists in the limit as δ 0.

26 Higher order corrections (in addition to the lowest order/tree level) are required for accurate theoretical predictions of cross-sections for particle interactions. Loop diagrams need to be taken into account, necessitating the evaluation of loop integrals. L-loop integral with N internal lines I = Γ `N nl Z 1 NY 2 ( 1) N dx (4π) nl/2 i δ(1 X C N n(l+1)/2 x i ) (D iδc) N nl/2 0 i=1 C and D are polynomials determined by the topology of the corresponding diagram and physical parameters. The integral only exists in the limit as δ 0.

27 Sample loop diagrams m, (p2) e, - (p1) t, (p4) m 2 γ, (p1) m M m 4 m 3 m m, (p3) e, + (p2) m 1 t, (p3) produced by GRACEFIG produced by GRACEFIG g b e, - (p1) m1 e, - (p5) m5 g g t W t W H b e, + (p2) m2 m3 m4 Z, (p4) e, + (p3) produced by GRACEFIG produced by GRACEFIG Figure: One-loop vertex; Box e e + t t; Diagram for g g b bh; Pentagon for e e + e e + Z [7]

28 One-loop integral example Sample box integral (real part): 1 0 dx 1 x dy 1 x y dz g(x,y) (d 2 δ 2 ) [10] 0 0 (d 2 +δ 2 ) 2 Figure: Surface where (quadratic) d(x, y, z) = 0

29 1 0 dx 1 x 0 dy 1 x y 0 dz f(x, y, z) Figure: Plot of inner integral (as a fn. of x, y) for sample box function

30 Outline Introduction 1 Introduction 2 Iterated Integration Priority driven adaptive algorithm Iterated vs. standard multivariate integration Error tolerance interface 3 4 Feynman loop diagrams 5 6

31 Introduction Automatic packages are available for one-loop [17, 1, 2, 11, 16, 13, 3, 5, 12]. However, many diagrams are required for an interaction; analytic integration not possible in general (for higher order and for general mass configurations). Numerical Direct Computation (DCM), (e.g., [9, 18, 6, 7]): numerical iterated integration + extrapolation/ sequence acceleration for limit calculation (as δ 0) Reduction + DCM: reduction applied for one-loop through hexagon, DCM for resulting 2D (triangle) and 3D (box) integral Motivation for DCM on multi-core, particularly for 3D and higher dimensions

32 Introduction Automatic packages are available for one-loop [17, 1, 2, 11, 16, 13, 3, 5, 12]. However, many diagrams are required for an interaction; analytic integration not possible in general (for higher order and for general mass configurations). Numerical Direct Computation (DCM), (e.g., [9, 18, 6, 7]): numerical iterated integration + extrapolation/ sequence acceleration for limit calculation (as δ 0) Reduction + DCM: reduction applied for one-loop through hexagon, DCM for resulting 2D (triangle) and 3D (box) integral Motivation for DCM on multi-core, particularly for 3D and higher dimensions

33 Introduction Automatic packages are available for one-loop [17, 1, 2, 11, 16, 13, 3, 5, 12]. However, many diagrams are required for an interaction; analytic integration not possible in general (for higher order and for general mass configurations). Numerical Direct Computation (DCM), (e.g., [9, 18, 6, 7]): numerical iterated integration + extrapolation/ sequence acceleration for limit calculation (as δ 0) Reduction + DCM: reduction applied for one-loop through hexagon, DCM for resulting 2D (triangle) and 3D (box) integral Motivation for DCM on multi-core, particularly for 3D and higher dimensions

34 Introduction Automatic packages are available for one-loop [17, 1, 2, 11, 16, 13, 3, 5, 12]. However, many diagrams are required for an interaction; analytic integration not possible in general (for higher order and for general mass configurations). Numerical Direct Computation (DCM), (e.g., [9, 18, 6, 7]): numerical iterated integration + extrapolation/ sequence acceleration for limit calculation (as δ 0) Reduction + DCM: reduction applied for one-loop through hexagon, DCM for resulting 2D (triangle) and 3D (box) integral Motivation for DCM on multi-core, particularly for 3D and higher dimensions

35 Introduction Automatic packages are available for one-loop [17, 1, 2, 11, 16, 13, 3, 5, 12]. However, many diagrams are required for an interaction; analytic integration not possible in general (for higher order and for general mass configurations). Numerical Direct Computation (DCM), (e.g., [9, 18, 6, 7]): numerical iterated integration + extrapolation/ sequence acceleration for limit calculation (as δ 0) Reduction + DCM: reduction applied for one-loop through hexagon, DCM for resulting 2D (triangle) and 3D (box) integral Motivation for DCM on multi-core, particularly for 3D and higher dimensions

36 Hexagon reduction Reduction n-dimensional N-point function set of box and triangle functions distribution of work load Representation: I n N = d n k iπ n/2 1 N l=1 ((k r l) 2 m 2 l ) with external momenta p j and r l = l j=1 p j. The n-dimensional hexagon, pentagon and box functions (N = 6, 5, 4) are expressed in terms of n-dimensional triangle and n + 2-dimensional box functions. In non-exceptional kinematic conditions, N-point functions with N 6 can be expressed in terms of pentagon functions.

37 Hexagon reduction Reduction n-dimensional N-point function set of box and triangle functions distribution of work load Representation: I n N = d n k iπ n/2 1 N l=1 ((k r l) 2 m 2 l ) with external momenta p j and r l = l j=1 p j. The n-dimensional hexagon, pentagon and box functions (N = 6, 5, 4) are expressed in terms of n-dimensional triangle and n + 2-dimensional box functions. In non-exceptional kinematic conditions, N-point functions with N 6 can be expressed in terms of pentagon functions.

38 Hexagon reduction Reduction n-dimensional N-point function set of box and triangle functions distribution of work load Representation: I n N = d n k iπ n/2 1 N l=1 ((k r l) 2 m 2 l ) with external momenta p j and r l = l j=1 p j. The n-dimensional hexagon, pentagon and box functions (N = 6, 5, 4) are expressed in terms of n-dimensional triangle and n + 2-dimensional box functions. In non-exceptional kinematic conditions, N-point functions with N 6 can be expressed in terms of pentagon functions.

39 Hexagon reduction Reduction n-dimensional N-point function set of box and triangle functions distribution of work load Representation: I n N = d n k iπ n/2 1 N l=1 ((k r l) 2 m 2 l ) with external momenta p j and r l = l j=1 p j. The n-dimensional hexagon, pentagon and box functions (N = 6, 5, 4) are expressed in terms of n-dimensional triangle and n + 2-dimensional box functions. In non-exceptional kinematic conditions, N-point functions with N 6 can be expressed in terms of pentagon functions.

40 Reduction n-dimensional N-point function Reduction I n N = N κ=1 B κi N 1,κ + (N n 1) det(g) det(s) In+2 N, det(s) 0, G is the Gram matrix, rank(g) = min{4, N 1} and B κ = N λ=1 S 1 κλ, S κλ = (r λ r κ ) 2 + mλ 2 + m2 κ, 1 κ,λ N hexagon I6 n = lin. combination of six pentagon In 5 functions, pentagon I5 n = lin. combination of five box In 4 fncs. + O(ε), box I4 n = lin. combination of four triangle In 3 and a box In+2 4 Infrared singularities emerge in the box and triangle functions through poles in 1 ε = 2 4 n and can be handled through sector decomposition.

41 Reduction n-dimensional N-point function Reduction I n N = N κ=1 B κi N 1,κ + (N n 1) det(g) det(s) In+2 N, det(s) 0, G is the Gram matrix, rank(g) = min{4, N 1} and B κ = N λ=1 S 1 κλ, S κλ = (r λ r κ ) 2 + mλ 2 + m2 κ, 1 κ,λ N hexagon I6 n = lin. combination of six pentagon In 5 functions, pentagon I5 n = lin. combination of five box In 4 fncs. + O(ε), box I4 n = lin. combination of four triangle In 3 and a box In+2 4 Infrared singularities emerge in the box and triangle functions through poles in 1 ε = 2 4 n and can be handled through sector decomposition.

42 Reduction n-dimensional N-point function Reduction I n N = N κ=1 B κi N 1,κ + (N n 1) det(g) det(s) In+2 N, det(s) 0, G is the Gram matrix, rank(g) = min{4, N 1} and B κ = N λ=1 S 1 κλ, S κλ = (r λ r κ ) 2 + mλ 2 + m2 κ, 1 κ,λ N hexagon I6 n = lin. combination of six pentagon In 5 functions, pentagon I5 n = lin. combination of five box In 4 fncs. + O(ε), box I4 n = lin. combination of four triangle In 3 and a box In+2 4 Infrared singularities emerge in the box and triangle functions through poles in 1 ε = 2 4 n and can be handled through sector decomposition.

43 Iterated integral parallelization (DCM for 3D) Timing results Figure: Time (s) for singular function

44 Iterated integral parallelization Figure: Time (s) for non-scalar loop integral (box) [10]

45 Iterated integral parallelization Figure: Time (s) for high degree polynomial

46 The two-level integration was tested with various schemes for the error control of inner and outer integrations. Extensions to a small number of levels have proved efficient with good results for several applications, e.g., in high energy physics and for some problems in biometrical modeling (higher dimensions). Computation of the integration rule can be parallelized on successive levels. Further extensions/generalizations: hybrid systems (cluster/distributed), multi/many-core nodes; more complex, higher-dimensional DCM problems.

47 The two-level integration was tested with various schemes for the error control of inner and outer integrations. Extensions to a small number of levels have proved efficient with good results for several applications, e.g., in high energy physics and for some problems in biometrical modeling (higher dimensions). Computation of the integration rule can be parallelized on successive levels. Further extensions/generalizations: hybrid systems (cluster/distributed), multi/many-core nodes; more complex, higher-dimensional DCM problems.

48 The two-level integration was tested with various schemes for the error control of inner and outer integrations. Extensions to a small number of levels have proved efficient with good results for several applications, e.g., in high energy physics and for some problems in biometrical modeling (higher dimensions). Computation of the integration rule can be parallelized on successive levels. Further extensions/generalizations: hybrid systems (cluster/distributed), multi/many-core nodes; more complex, higher-dimensional DCM problems.

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