Perspectives of wavelet bases in simulation of lattice theories 1 Mikhail V. Altaisky Space Research Institute RAS, Profsoyuznaya 84/32, Moscow, 117997, Russia altaisky@mx.iki.rssi.ru seminar on Theory of hadronic matter under extreme conditions 1 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Abstract We consider the perspectives of using orthogonal wavelet expan- sion with Daubechies wavelets for lattice theories. The discrete wavelet transform have been already applied to simulate the Landau- Ginzburg/Φ 4 theory with the assumption that the wavelet coeffi- cients of the order parameter Φ( x ) are delta-correlated Gaussian processes in the scale-position space. This reduces the autocor- relation time of simulation, and is not the only merit of wavelet transform. By construction the wavelet transform represents the snapshot of a field at a given scale, and therefore can be used as a tool to study the correlations between fluctuations of different scales. For the same reason the relation of wavelet transform to the renormalization group are considered. We also discuss the prospec- tive of wavelet transform to improve the Metropolis algorithm and the simulated annealing procedure. 2 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
References: This talk is based on O.P. Le Maitre, H.N.Hajm, P.P.Pebay, R.G.Khanem and O.M.Knio. Multi-resolution-analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29(2007)864 A.E.Ismail, G.C.Rutledge and G.Stephanopoulos. Multiresolution analysis in statistical mechanics: I. J. Chem. Phys. 118(2003)4414; II. ibid. 118(2003)4424 A.E.Cho, J.D.Doll and D.L.Freeman. Wavelet formulation of path integral Monte Carlo. J. Chem. Phys. 117(2002)5971 C.Best. Wavelet-induced renormalization group for Landau-Ginzburg model. NPB (P.S.) 83(2000)848 I.G.Halliday and P.Suranyi. Simulation of field theories in wavelet representation. NPB 436(1995)414 C.Best, A.Sch¨ afer and W.Greiner. Wavelets as a variational basis of the XY model. NPB (P.S.) 34(1994)780 T.Drapper and C.McNeil. An investigation into a wavelet accelerated gauge fixing. arxiv.org:hep-lat/9312044 3 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
References ...and also related author’s works: M.V.Altaisky and N.E.Kaputkina. Continuous wavelet transform in quantum field theory. Phys. Rev. D 88(2013)025015 M.V.Altaisky and N.E.Kaputkina. Quantum hierarchic models for information processing. Int. J. Quant. Inf. 10(2012)1250026 M.V.Altaisky. On quantum kinetic equation for hierarchic systems. Phys. Lett. A 374(2010)522 M.V.Altaisky. Quantum field theory without divergences. Phys. Rev. D 81(2010) 125003 M.V.Altaisky, E.A.Popova, D.Yu.Saraev. Application of orthogonal wavelets for the stochastic wavelet-Galerkin solution of the Kraichnan-Orszag system. pp.25-31 in Proc. Int. Conf. Days on Diffraction 2009 , SPb. 4 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Subjects Continuous and discrete wavelet transform 5 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Subjects Continuous and discrete wavelet transform Wavelet transform in quantum field theory 5 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Subjects Continuous and discrete wavelet transform Wavelet transform in quantum field theory Resolution-dependent fields 5 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Subjects Continuous and discrete wavelet transform Wavelet transform in quantum field theory Resolution-dependent fields Wavelets and Monte Carlo 5 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Subjects Continuous and discrete wavelet transform Wavelet transform in quantum field theory Resolution-dependent fields Wavelets and Monte Carlo Sampling in amplitude vs. sampling in space 5 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Subjects Continuous and discrete wavelet transform Wavelet transform in quantum field theory Resolution-dependent fields Wavelets and Monte Carlo Sampling in amplitude vs. sampling in space Gauge theories 5 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Continuous Wavelet Transform CWT in L 2 -norm: φ a ( b ) dad d b 1 1 � x − b � � φ ( x ) = a d / 2 g a d +1 , C g a � 1 � x − b � φ ( x ) d d x , φ a ( b ) = a d / 2 g a For isotropic wavelets g the normalization constant C ψ is readily evaluated using Fourier transform: � ∞ g ( k ) | 2 d d k g ( ak ) | 2 da � C g = | ˜ a = | ˜ S d | k | < ∞ , 0 where S d = 2 π d / 2 Γ( d / 2) is the area of unit sphere in R d . d µ ( a , b ) = dad d b � x − b � G : x ′ = ax + b , U ( a , b ) g ( x ) = a − d / 2 g , a d +1 a 6 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Basic wavelets for CWT Examples Vanishing Momenta Family: wavelets g1 - g4 1 g1(x) g2(x) g3(x) g4(x) 0.5 0 dx n e − x 2 g n ( x ) = ( − 1) n +1 d n g(x) 2 -0.5 -1 -1.5 -4 -2 0 2 4 x 7 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Discrete Wavelet Transform Wavelet transform on a sublattice m=2 a = a m 0 , b = nb 0 a m 0 , n , m ∈ Z m=1 often choice a 0 = 2 m=0 − m b 0 ψ m a − m � � n ( x ) = a 2 ψ − nb 0 0 0 Wavelet coefficients � − m ¯ d m n = � ψ m ψ ( a − m n | f � ≡ a 2 x − nb 0 ) f ( x ) dx 0 0 Reconstruction � ˜ ψ m n ( x ) d m f ( x ) = n + error term 8 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Multi-Resolution Analysis Consider a Hilbert space of L 2 ( R ) functions, then the Mallat mul- tiresolution analysis (MRA), is an increasing sequence of subspaces { V j } j ∈ Z , V j ∈ L 2 ( R ), such that 1 . . . ⊂ V 1 ⊂ V 0 ⊂ V − 1 ⊂ V − 2 ⊂ . . . 2 clos ∪ j ∈ Z V j = L 2 ( R ) 3 ∩ j ∈ Z V j = ∅ 4 The spaces V j and V j − 1 are similar in a sense that f ( x ) ∈ V j ⇔ f (2 x ) ∈ V j − 1 , j ∈ Z . 5 V j = linear span { φ j φ 0 k ( x ) , j , k ∈ Z } , k ( x ) = φ ( x − k ) Since V j and V j +1 are different in resolution, some details are lost in projection f ∈ V N on a ladder of spaces V N +1 , V N +2 , . . . . The details can be stored in orthogonal complements W j = V j − 1 \ V j , Q m = P m − 1 − P m . ψ m n is a basis in W m Explicitly: V 0 = V 1 ⊕ W 1 , V 1 = V 2 ⊕ W 2 , . . . Hence V 0 = W 1 ⊕ W 2 ⊕ W 3 ⊕ . . . . ⊕ V N 9 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Fast Wavelet Transform The numerical implementation of the decomposition of a function f ∈ L 2 ([0 , 1]) is based on the truncation of the Mallat sequence with certain finest resolution level V 0 . The unit interval in N = 2 , 4 , 8 , 16 , 32 , . . . points. The initial data vector is then denoted as s 0 = ( s 0 0 , . . . , s 0 N − 1 ) ∈ V 0 . The projections onto the spaces V 1 , W 1 , V 2 , W 2 , . . . are sequentially performed s 0 h ⇓ g ց s 1 d 1 s j i = � N − 1 k =0 h k s j − 1 h ⇓ g ց k +2 i , s 2 d 2 d j i = � N − 1 k =0 g k s j − 1 k +2 i , h ⇓ g ց s 3 d 3 . . . where N denotes the size of current data vector. 10 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Haar wavelet algorithm Scaling function φ ( x ) = χ [0 , 1) ( x ) Low-pass filter is a pair-averaging, high-pass filter is a difference 1 √ h 1 = h 2 = 2 Decomposition k = s j − 1 + s j − 1 k = s j − 1 − s j − 1 s j 2 k 2 k +1 d j 2 k 2 k +1 √ , √ 2 2 Reconstruction = s j k + d j 2 k +1 = s j k − d j s j − 1 s j − 1 k k √ , √ 2 k 2 2 11 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Daubechies wavelets Daubechies, I. Comm. Pure. Appl. Math. 41(1988)909 Orthogonal wavelets with compact support - the Daubechies wavelets - are given not explicitly, but recursively, by functional scal- ing equation: N − 1 √ � φ ( x ) = 2 h k φ (2 x − k ) k =0 The coefficients h k give complete definition of the wavelet N − 1 √ � ψ ( x ) = 2 g k φ (2 x − k ) k =0 The coefficients h k and g k are referred to as low- and high-pass filter coefficients. They are related by g k = ( − 1) k h N − 1 − k , 0 ≤ k < N 12 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Scaling and wavelet functions for DAUB4 wavelet 2 "grf.dat" u 1:2 "grf.dat" u 1:3 1.5 1 0.5 0 -0.5 -1 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Graphs of φ ( x ) and ψ ( x ) obtained at recursion level 8 for DAUB4 wavelet 13 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
2 and more dimensions from M.V.Altaisky, Wavelets:Theory,Implementation,Applications, 2005 ... K 3 K 2 hg hg K 3 K 3 gg gh K 1 hg K 2 K 2 gh gg K 1 K 1 gg gh 14 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Two ways of use Coordinate resolution: a microscope at a given point � d j k ( ξ ) ψ j SACI φ ( x , ξ ) = k ( x ) 55 "saci.int" jk 50 45 40 35 30 25 100 200 300 400 500 600 700 days 15 Mikhail V. Altaisky Perspectives of wavelet bases in simulation of lattice theories
Recommend
More recommend
