Alternatives to conventional Monte Carlo

Transcription

1 Alternatives to conventional Monte Carlo Recursive umerical Integration & Julia Volmer Andreas Ammon, Alan enz, Tobias Hartung, Karl Jansen, Hernan Leövey DESY Zeuthen 16. September 216 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

2 Monte Carlo importance sampling O = D d dx O[x] e S[x] D d dx e S[x], prob. density [x] e S[x] p(x) = dx e D S[x] d O = 1 i=1 O p [x] S[x 1, x 2 ] > O[x 1, x 2 ] O 1/ x 2 x 1 x 2 x 1 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

3 Monte Carlo importance sampling O = D d dx O[x] e S[x] D d dx e S[x], prob. density [x] e S[x] p(x) = dx e D S[x] d O = 1 i=1 O p [x] S[x 1, x 2 ] is complex Re( S[x 1, x 2 ] ) Re( O[x 1, x 2 ] ) O 1.1 x 2 x 1 x 2 x 1 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

4 Solutions Take specific integration points O S > 1/ Recursive umerical Integration auss quadrature points O S C 1.1 Spherical t-designs Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

5 Toplogical Oscillator Recursive umerical Integration Topological Oscillator φ i I S(φ) = T I a dt I 2 ( ) φ 2 φ [, 2π) t 3 (1 cos(φ i+1 φ i )) i=1 t i t 1 a t 2 t 3 T = 2π Partition function Z = dφ 1 dφ 2 dφ 3 e S[φ 1,φ 2,φ 3 ] [,2π) 3 3 = dφ 1 dφ 2 dφ 3 e a I (1 cos(φ i+1 φ i )) [,2π) }{{} 3 i=1 f (φ i,φ i+1 ) 2π 2π dφ 1 dφ 2 f (φ 1, φ 2 ) dφ 3 f (φ 2, φ 3 ) f (φ 3, φ 1 ) Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

6 Toplogical Oscillator Recursive umerical Integration auss quadrature points g auss x 1 x 2 x x 1 1 g(x) P 2 1 dx g(x) = w r g(x r )+O r=1 x r, w r : L (x) Legendre polynoms ( 1 ) (2)! Z = 2π w t t=1 dφ 1 2π 2π dφ 2 f (φ 1, φ 2 ) dφ 3 f (φ 2, φ 3 ) f (φ 3, φ 1 ) w s f (φ s, φ t ) s=1 w r f (φ s, φ r ) f (φ r, φ t ) r=1 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

7 Toplogical Oscillator Recursive umerical Integration Truncation Error Scaling φ i I t i Topological Charge Q(φ) = 1 2π T dt ( ) φ t Error χ i = χ i χ( = 56) Constants I =.25 a =.4, T = 2 Topological Susceptibility χ = Q2 (φ) T Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

8 Toplogical Oscillator Recursive umerical Integration RI - Comparison with MCMC Error χ i,auss = χ i χ( = 4) χ i,cluster : 1 runs Constants I =.25 a =.1, T = 2 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

9 1d-QCD Ψ e µ U 1 e µ U 2 e µ 1 Ψ 2 Ψ 3 U 3 x 1 x 2 x Ψ : mass m 3 S[U, Ψ, Ψ] = mψ i Ψ i + e µ Ψ i U i Ψ i+1 + e µ Ψ i 1 Ui Ψ i i = Ψ D[U] Ψ, U i, e.g. U(), SU() partition function Z[U] = dh (U 1 ) dh (U 2 ) dh (U 3 ) dψ dψ e S[U,Ψ,Ψ], = dh 3 3 (U) det D[U] ( ( ) 3 = dh 3 3 (U) c(m)+ 2 3 e 3µ U j j=1 ( ) 3 ) +( 1) e 3µ U j j=1 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

10 1d-QCD Ψ e µ 1 e µ 1 e µ 1 Ψ 2 Ψ 3 U x 1 x 2 x Ψ : mass m 3 S[U, Ψ, Ψ] = mψ i Ψ i + e µ Ψ i U i Ψ i+1 + e µ Ψ i 1 Ui Ψ i i = Ψ D[U] Ψ, U i, e.g. U(), SU() partition function Z[U] = dh (U 1 ) dh (U 2 ) dh (U 3 ) dψ dψ e S[U,Ψ,Ψ], = dh 3 3 (U) det D[U] ( = dh (U) c(m)+ 2 3 e 3µ U ) +( 1) e 3µ U Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

11 Why difficult for MC? Z[U] = dh (U) det ( c(m) e 3µ U + ( 1) e 3µ U ) Z[U] = U(1) du U = 2π dθ e iθ = e i 5 4 π ei 3 5 π e i 1 3 π 1 MC 1 ( 1 + e i 1 3 π + e i 5 3 π + e i 5 π) i 4 MC = O(1) O() Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

12 Why difficult for MC? Z[U] = dh (U) det ( c(m) e 3µ U + ( 1) e 3µ U ) Z[U] = U(1) du U = 2π dθ e iθ = 1 i 1 MC 1 ( 1 + e i 1 3 π + e i 5 3 π + e i 5 π) i 4 MC = O(1) O() i sym 1 (1 + i + ( i) + ( 1)) = 4 Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

13 Problem of the chemical potential Z[U] = U(1) du ( c(m) e 3µ U + ( 1) e 3µ U ) imaginary exact MC µ = MC µ >> 1 real U(1) du U = µ > du e 3µ U = U(1) Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

14 Spherical t-designs = S n, f (U) P t dh (U) f (U) 1 t+1 t + 1 f (U k ) k=1 rules for S 1, S 3 and S 5 [enz 23] use isomorphisms to get rules for U(1), U(2), U(3), SU(2), SU(3) [Ammon 216] Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14

15 Result Partition Function Z[U] = U(1) du ( c(m) e 3µ U + ( 1) e 3µ U ) Z quadrature Z analytic Z analytic = U(1) double prec µ = lattice points 1-12 MCMC poly. exact Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14 m

16 Result Chiral Condensate ΨΨ = m ln Z = dh m det D dh det D 1 ΨΨ quadrature ΨΨ analytic ΨΨ analytic 1-4 = U(2) double prec µ = lattice points 1-16 MCMC poly. exact Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14 m

17 Result Chiral Condensate ΨΨ = m ln Z = dh m det D dh det D ΨΨ quadrature ΨΨ analytic ΨΨ analytic = U(2) 124 bit prec µ = 1 8 lattice points MCMC poly. exact Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14 m

18 Conclusions O S > 1/ [ ] topol. osci. O 1 4 S C 1 [ ] 1d-QCD ΨΨ MCMC poly. exact m Julia Volmer (DESY Zeuthen) MC Alternatives 16. September / 14