A flat histogram stochastic growth algorithm
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1 A flat histogram stochastic growth algorithm (with applicatio to polymers ad proteis) Thomas Prellberg TU Clausthal Joit work with Jaroslaw Krawczyk, TU Clausthal Thomas Prellberg, Göttige, Ja 4, 24 p./24
2 Itroductio Thomas Prellberg, Göttige, Ja 4, 24 p.2/24
3 Modellig of Polymers i Solutio Polymers: log chais of moomers Coarse-Graiig : beads o a chai Excluded Volume : miimal distace Cotact with solvet: effective short-rage iteractio Good/bad solvet: repellig/attractig iteractio Cosequece: chais clump together Eight polymers i a bad solvet (Grassberger, FZ Jülich) Thomas Prellberg, Göttige, Ja 4, 24 p.3/24
4 Lattice Model Self-Avoidig Walks with Iteractios Physical space lattice Z 3 (or Z d ) Polymer i solutio radom walk with self-avoidace Quality of solvet short-rage iteractio ɛ Properties of moomers i, j iteractio ɛ = ɛ i,j Thomas Prellberg, Göttige, Ja 4, 24 p.4/24
5 Lattice Model Self-Avoidig Walks with Iteractios Physical space lattice Z 3 (or Z d ) Polymer i solutio radom walk with self-avoidace Quality of solvet short-rage iteractio ɛ Properties of moomers i, j iteractio ɛ = ɛ i,j Two examples: ISAW model: iteractio ɛ i,j = of iterest: thermodyamic limit (V = ad ) HP model: two types of moomers H ad P ɛ HH =, ɛ HP = ɛ P H = ɛ P P = of iterest: fixed fiite sequece, desity of states Thomas Prellberg, Göttige, Ja 4, 24 p.4/24
6 Lattice Model: ISAW = 26 m = 7 2 e βɛ i,j Boltzma weight ISAW model: iteractio ɛ i,j =, ω = e β Partitio fuctio: Z (ω) = m C,mω m C,m umber of SAW with legth ad m iteractios Thomas Prellberg, Göttige, Ja 4, 24 p.5/24
7 Lattice Model: HP model HP model: iteractio ɛhh =, ɛhp = ɛp H = ɛp P = Hydrophobic Polar Groudstate of sequece with 85 moomers (d = 2) Thomas Prellberg, Göttige, Ja 4, 24 p.6/24
8 Why Simulatios? ISAW model: Tricritical phase trasitio, d u = 3 I priciple uderstood, however surprisig details e.g. pseudo-first-order trasitio for d > 3 No good uderstadig of collapsed regime Thomas Prellberg, Göttige, Ja 4, 24 p.7/24
9 Why Simulatios? ISAW model: Tricritical phase trasitio, d u = 3 I priciple uderstood, however surprisig details e.g. pseudo-first-order trasitio for d > 3 No good uderstadig of collapsed regime HP model: Toy model for proteis Desig of sequeces with specific groud state structure Desity of states foldig dyamics Thomas Prellberg, Göttige, Ja 4, 24 p.7/24
10 Why Simulatios? ISAW model: Tricritical phase trasitio, d u = 3 I priciple uderstood, however surprisig details e.g. pseudo-first-order trasitio for d > 3 No good uderstadig of collapsed regime HP model: Toy model for proteis Desig of sequeces with specific groud state structure Desity of states foldig dyamics Most iterestig ope questios i collapsed regime Collapsed regime is otoriously difficult to simulate Thomas Prellberg, Göttige, Ja 4, 24 p.7/24
11 Stochastic Growth Algorithms Thomas Prellberg, Göttige, Ja 4, 24 p.8/24
12 PERM: Go With The Wiers PERM = Prued ad Eriched Rosebluth Method Rosebluth Method: kietic growth Grassberger, Phys Rev E 56 (997) 3682 /3 /2 trapped Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
13 PERM: Go With The Wiers PERM = Prued ad Eriched Rosebluth Method Rosebluth Method: kietic growth Grassberger, Phys Rev E 56 (997) 3682 /3 /2 trapped Erichmet: weight too large make copies of cofiguratio Pruig: weight too small remove cofiguratio occasioally Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
14 PERM: Go With The Wiers PERM = Prued ad Eriched Rosebluth Method Rosebluth Method: kietic growth Grassberger, Phys Rev E 56 (997) 3682 /3 /2 trapped Erichmet: weight too large make copies of cofiguratio Pruig: weight too small remove cofiguratio occasioally Observatio: kietic growth weights ad iteractios balace each other at suitable temperatures (i the collapse regio) Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
15 PERM cotiued PERM well suited for simulatio of collapsig polymers Erichmet creates correlated tours = = max Error estimatio a priori impossible, a posteriori difficult Thomas Prellberg, Göttige, Ja 4, 24 p./24
16 PERM cotiued PERM well suited for simulatio of collapsig polymers Erichmet creates correlated tours = = max Error estimatio a priori impossible, a posteriori difficult PERM is blid - it beefits from visual aids, such as Markovia aticipatio (learig from experiece) Thomas Prellberg, Göttige, Ja 4, 24 p./24
17 PERM further developmets A sigificat improvemet: PERM = ew PERM Eforce distict erichmet steps Crossover to exact eumeratio Hsu et al, J Chem Phys 8 (23) 444 Thomas Prellberg, Göttige, Ja 4, 24 p./24
18 PERM further developmets A sigificat improvemet: PERM = ew PERM Eforce distict erichmet steps Crossover to exact eumeratio Curret work: flatperm = flat histogram PERM Hsu et al, J Chem Phys 8 (23) 444 TP ad JK, cod-mat/32253, PRL (24) flatperm samples a geeralised multicaoical esemble Covers the whole temperature rage i oe simulatio! Related: multicaoical chai growth algorithm Bachma ad Jake, PRL (23) Thomas Prellberg, Göttige, Ja 4, 24 p./24
19 PERM further developmets A sigificat improvemet: PERM = ew PERM Eforce distict erichmet steps Crossover to exact eumeratio Curret work: flatperm = flat histogram PERM Hsu et al, J Chem Phys 8 (23) 444 TP ad JK, cod-mat/32253, PRL (24) flatperm samples a geeralised multicaoical esemble Covers the whole temperature rage i oe simulatio! Related: multicaoical chai growth algorithm Bachma ad Jake, PRL (23) Applicatios: liear ad brached polymers, proteis, percolatio,... Thomas Prellberg, Göttige, Ja 4, 24 p./24
20 Algorithm details - kietic growth View kietic growth as approximate eumeratio Thomas Prellberg, Göttige, Ja 4, 24 p.2/24
21 Algorithm details - kietic growth View kietic growth as approximate eumeratio Exact eumeratio: choose all a cotiuatios with equal weight Kietic growth: chose oe cotiuatio with a-fold weight (a may be zero). Thomas Prellberg, Göttige, Ja 4, 24 p.2/24
22 Algorithm details - kietic growth View kietic growth as approximate eumeratio Exact eumeratio: choose all a cotiuatios with equal weight Kietic growth: chose oe cotiuatio with a-fold weight (a may be zero). A step cofiguratio gets assiged a weight W = k= S growth chais with weights W (i) a k give estimate C est = W = S i W (i) Thomas Prellberg, Göttige, Ja 4, 24 p.2/24
23 Algorithm details - pruig/erichmet W (i) is estimate of C Cosider ratio r = W (i) /C est r > erichmet step: make c = mi( r, a ) distict copies with weight c W (i) r < pruig step: cotiue growig with probability r ad weight C est Thomas Prellberg, Göttige, Ja 4, 24 p.3/24
24 Algorithm details - pruig/erichmet W (i) is estimate of C Cosider ratio r = W (i) /C est r > erichmet step: make c = mi( r, a ) distict copies with weight c W (i) r < pruig step: cotiue growig with probability r ad weight C est Cosequeces Number of samples geerated for each is roughly costat Ideally, ubiased radom walk i cofiguratio size We have a flat histogram algorithm Thomas Prellberg, Göttige, Ja 4, 24 p.3/24
25 From PERM to flatperm PERM: estimate umber of cofiguratios C C est = W r = W (i) /C est Thomas Prellberg, Göttige, Ja 4, 24 p.4/24
26 From PERM to flatperm PERM: estimate umber of cofiguratios C C est = W r = W (i) /C est Cosider eergy E m = ɛm, temperature β = /k B T thermal PERM: estimate partitio fuctio Z (β) Z est (β) = W exp( βe) r = W (i) exp( βe m (i) )/Z est (β) Thomas Prellberg, Göttige, Ja 4, 24 p.4/24
27 From PERM to flatperm PERM: estimate umber of cofiguratios C C est = W r = W (i) /C est Cosider eergy E m = ɛm, temperature β = /k B T thermal PERM: estimate partitio fuctio Z (β) Z est (β) = W exp( βe) r = W (i) exp( βe m (i) )/Z est (β) flatperm: estimate desity of states C,m C est,m = W,m r = W (i),m/c est,m Thomas Prellberg, Göttige, Ja 4, 24 p.4/24
28 From PERM to flatperm PERM: estimate umber of cofiguratios C C est = W r = W (i) /C est Cosider eergy E m = ɛm, temperature β = /k B T thermal PERM: estimate partitio fuctio Z (β) Z est (β) = W exp( βe) r = W (i) exp( βe m (i) )/Z est (β) flatperm: estimate desity of states C,m C est,m = W,m r = W (i),m/c est,m Geeralizatio to more microcaoical parameters possible Thomas Prellberg, Göttige, Ja 4, 24 p.4/24
29 flatperm details Implemetatio details Parameter free implemetatio Equilibrate by slowly icreasig maximal size of cofiguratios Reduce correlatios by cosiderig effective sample size Thomas Prellberg, Göttige, Ja 4, 24 p.5/24
30 flatperm details Implemetatio details Parameter free implemetatio Equilibrate by slowly icreasig maximal size of cofiguratios Reduce correlatios by cosiderig effective sample size What if we do t eed the whole histogram? Restrictio of histogram possible e.g. impose iteral eergy restrictio m/ < u cutoff Thomas Prellberg, Göttige, Ja 4, 24 p.5/24
31 flatperm details Implemetatio details Parameter free implemetatio Equilibrate by slowly icreasig maximal size of cofiguratios Reduce correlatios by cosiderig effective sample size What if we do t eed the whole histogram? Restrictio of histogram possible e.g. impose iteral eergy restrictio m/ < u cutoff What if we do t wat a flat histogram? Sample profile ca be tued Selectively perform pruig/erichmet with respect to profile shape f,m Thomas Prellberg, Göttige, Ja 4, 24 p.5/24
32 Simulatio Results Thomas Prellberg, Göttige, Ja 4, 24 p.6/24
33 ) Simulatio results: HP model Sequece I (4 Moomers, HPHPHHPHPHHPPH, d = 3):! ' ( %& # $ "! Pedagogical example, egieered for ative groud state Perfect agreemet with exact eumeratio Thomas Prellberg, Göttige, Ja 4, 24 p.7/24
34 # Simulatio results: HP model Sequece II (85 Moomers, d = 2):! " Ivestigated several other sequeces i d = 2 ad d = 3 Collapsed regime accessible Reproduced kow groud state eergies Obtaied C,m over large rage Thomas Prellberg, Göttige, Ja 4, 24 p.8/24
35 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Equilibratio with delay.: after t tours growth up to legth t Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
36 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size:,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
37 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size:,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
38 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 2,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
39 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 3,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
40 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 4,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
41 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 5,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
42 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 6,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
43 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 7,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
44 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 8,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
45 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 9,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
46 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size:,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
47 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size:,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
48 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 2,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
49 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 3,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
50 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 4,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
51 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 5,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
52 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 6,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
53 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 7,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
54 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 8,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
55 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 9,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
56 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 2,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
57 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 2,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
58 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 22,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
59 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 23,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
60 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 24,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
61 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 25,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
62 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 26,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
63 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 27,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
64 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 28,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
65 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 29,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
66 Simulatio: Equilibratio for 2d ISAW 2d ISAW simulatio up to = 24 Total sample size: 3,, log (C m ) S m m/ m/ Thomas Prellberg, Göttige, Ja 4, 24 p.9/24
67 " ( Simulatio results: 2d ISAW e+8 e+7 e d ISAW up to = 24 Oe simulatio suffices 3 orders of magitude! & '! #%$ Thomas Prellberg, Göttige, Ja 4, 24 p.2/24
68 % Simulatio results: 3d ISAW e+7 e d ISAW up to = 24 Oe simulatio suffices 4 orders of magitude # $ "! Thomas Prellberg, Göttige, Ja 4, 24 p.2/24
69 Summary Thomas Prellberg, Göttige, Ja 4, 24 p.22/24
70 Coclusio: A Promisig New Algorithm Reviewed Stochastic Growth Algorithms for Polymers Preseted flat histogram versio of PERM Oe simulatio for complete desity of states! (the rage ca also be selectively restricted) Applicatios: HP model, ISAW Thomas Prellberg, Göttige, Ja 4, 24 p.23/24
71 Coclusio: A Promisig New Algorithm Reviewed Stochastic Growth Algorithms for Polymers Preseted flat histogram versio of PERM Oe simulatio for complete desity of states! (the rage ca also be selectively restricted) Applicatios: HP model, ISAW Outlook: applicatios to further models, e.g. Square lattice trees / brached polymers: with A. Rechitzer Coective costat estimate 5.435(5) (compare to 5.434(7) with Pivot algorithm) Two-dimesioal desity of states with oe simulatio Absorbig collapsig polymers: with A. Owczarek Exteded Domb-Joyce model: with J. Krawczyk Thomas Prellberg, Göttige, Ja 4, 24 p.23/24
72 The Ed Thomas Prellberg, Göttige, Ja 4, 24 p.24/24
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